From jawbrey at oakland.edu Mon Jun 2 10:30:05 2003 From: jawbrey at oakland.edu (Jon Awbrey) Date: Wed Jan 21 20:34:24 2004 Subject: [Inquiry] Re: Differential Logic References: <3E982563.F67459A5@oakland.edu> <3EB73352.259E602@oakland.edu> <3ED7BACC.9E14FC98@oakland.edu> <3ED7C655.E94D97BB@oakland.edu> <3ED96C31.69BC3F4B@oakland.edu> Message-ID: <3EDB6D7D.DB32C5D6@oakland.edu> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Note D70 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Operator Diagrams for the Conjunction J = uv o o //\ /X\ ////\ /XXX\ //////\ oXXXXXo ////////\ /X\XXX/X\ //////////\ /XXX\X/XXX\ o///////////o oXXXXXoXXXXXo / \////////// \ / \XXX/X\XXX/ \ / \//////// \ / \X/XXX\X/ \ / \////// \ o oXXXXXo o / \//// \ / \ / \XXX/ \ / \ / \// \ / \ / \X/ \ / \ o o o o o o o o |\ / \ /| |\ / \ / \ / \ /| | \ / \ / | | \ / \ / \ / \ / | | \ / \ / | | o o o o | | \ / \ / | | |\ / \ / \ /| | | u \ / \ / v | |u | \ / \ / \ / | v| o-----o o-----o o--+--o o o--+--o \ / | \ / \ / | \ / | du \ / \ / dv | \ / o-----o o-----o \ / \ / \ / \ / o o U% $e$ $E$U% o------------------>o | | | | | | | | J | | $e$J | | | | | | v v o------------------>o X% $e$ $E$X% o o //\ /X\ ////\ /XXX\ //////\ /XXXXX\ ////////\ /XXXXXXX\ //////////\ /XXXXXXXXX\ ////////////o oXXXXXXXXXXXo ///////////// \ //\XXXXXXXXX/\\ ///////////// \ ////\XXXXXXX/\\\\ ///////////// \ //////\XXXXX/\\\\\\ ///////////// \ ////////\XXX/\\\\\\\\ ///////////// \ //////////\X/\\\\\\\\\\ o//////////// o o///////////o\\\\\\\\\\\o |\////////// / |\////////// \\\\\\\\\\/| | \//////// / | \//////// \\\\\\\\/ | | \////// / | \////// \\\\\\/ | | \//// / | \//// \\\\/ | | x \// / | x \// \\/ dx | o-----o / o-----o o-----o \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / o o Figure 57-1. Radius Operator Diagram for the Conjunction J = uv o o //\ /X\ ////\ /XXX\ //////\ oXXXXXo ////////\ //\XXX//\ //////////\ ////\X////\ o///////////o o/////o/////o / \////////// \ /\\/////\////\\ / \//////// \ /\\\\/////\//\\\\ / \////// \ o\\\\\o/////o\\\\\o / \//// \ / \\\\/ \//// \\\\/ \ / \// \ / \\/ \// \\/ \ o o o o o o o o |\ / \ /| |\ / \ /\\ / \ /| | \ / \ / | | \ / \ /\\\\ / \ / | | \ / \ / | | o o\\\\\o o | | \ / \ / | | |\ / \\\\/ \ /| | | u \ / \ / v | |u | \ / \\/ \ / | v| o-----o o-----o o--+--o o o--+--o \ / | \ / \ / | \ / | du \ / \ / dv | \ / o-----o o-----o \ / \ / \ / \ / o o U% $E$ $E$U% o------------------>o | | | | | | | | J | | $E$J | | | | | | v v o------------------>o X% $E$ $E$X% o o //\ /X\ ////\ /XXX\ //////\ /XXXXX\ ////////\ /XXXXXXX\ //////////\ /XXXXXXXXX\ ////////////o oXXXXXXXXXXXo ///////////// \ //\XXXXXXXXX/\\ ///////////// \ ////\XXXXXXX/\\\\ ///////////// \ //////\XXXXX/\\\\\\ ///////////// \ ////////\XXX/\\\\\\\\ ///////////// \ //////////\X/\\\\\\\\\\ o//////////// o o///////////o\\\\\\\\\\\o |\////////// / |\////////// \\\\\\\\\\/| | \//////// / | \//////// \\\\\\\\/ | | \////// / | \////// \\\\\\/ | | \//// / | \//// \\\\/ | | x \// / | x \// \\/ dx | o-----o / o-----o o-----o \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / o o Figure 57-2. Secant Operator Diagram for the Conjunction J = uv o o //\ //\ ////\ ////\ //////\ o/////o ////////\ /X\////X\ //////////\ /XXX\//XXX\ o///////////o oXXXXXoXXXXXo / \////////// \ /\\XXX/X\XXX/\\ / \//////// \ /\\\\X/XXX\X/\\\\ / \////// \ o\\\\\oXXXXXo\\\\\o / \//// \ / \\\\/ \XXX/ \\\\/ \ / \// \ / \\/ \X/ \\/ \ o o o o o o o o |\ / \ /| |\ / \ /\\ / \ /| | \ / \ / | | \ / \ /\\\\ / \ / | | \ / \ / | | o o\\\\\o o | | \ / \ / | | |\ / \\\\/ \ /| | | u \ / \ / v | |u | \ / \\/ \ / | v| o-----o o-----o o--+--o o o--+--o \ / | \ / \ / | \ / | du \ / \ / dv | \ / o-----o o-----o \ / \ / \ / \ / o o U% $D$ $E$U% o------------------>o | | | | | | | | J | | $D$J | | | | | | v v o------------------>o X% $D$ $E$X% o o //\ /X\ ////\ /XXX\ //////\ /XXXXX\ ////////\ /XXXXXXX\ //////////\ /XXXXXXXXX\ ////////////o oXXXXXXXXXXXo ///////////// \ //\XXXXXXXXX/\\ ///////////// \ ////\XXXXXXX/\\\\ ///////////// \ //////\XXXXX/\\\\\\ ///////////// \ ////////\XXX/\\\\\\\\ ///////////// \ //////////\X/\\\\\\\\\\ o//////////// o o///////////o\\\\\\\\\\\o |\////////// / |\////////// \\\\\\\\\\/| | \//////// / | \//////// \\\\\\\\/ | | \////// / | \////// \\\\\\/ | | \//// / | \//// \\\\/ | | x \// / | x \// \\/ dx | o-----o / o-----o o-----o \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / o o Figure 57-3. Chord Operator Diagram for the Conjunction J = uv o o //\ //\ ////\ ////\ //////\ o/////o ////////\ /X\////X\ //////////\ /XXX\//XXX\ o///////////o oXXXXXoXXXXXo / \////////// \ /\\XXX//\XXX/\\ / \//////// \ /\\\\X////\X/\\\\ / \////// \ o\\\\\o/////o\\\\\o / \//// \ / \\\\/\\////\\\\\/ \ / \// \ / \\/\\\\//\\\\\/ \ o o o o o\\\\\o\\\\\o o |\ / \ /| |\ / \\\\/ \\\\/ \ /| | \ / \ / | | \ / \\/ \\/ \ / | | \ / \ / | | o o o o | | \ / \ / | | |\ / \ / \ /| | | u \ / \ / v | |u | \ / \ / \ / | v| o-----o o-----o o--+--o o o--+--o \ / | \ / \ / | \ / | du \ / \ / dv | \ / o-----o o-----o \ / \ / \ / \ / o o U% $T$ $E$U% o------------------>o | | | | | | | | J | | $T$J | | | | | | v v o------------------>o X% $T$ $E$X% o o //\ /X\ ////\ /XXX\ //////\ /XXXXX\ ////////\ /XXXXXXX\ //////////\ /XXXXXXXXX\ ////////////o oXXXXXXXXXXXo ///////////// \ //\XXXXXXXXX/\\ ///////////// \ ////\XXXXXXX/\\\\ ///////////// \ //////\XXXXX/\\\\\\ ///////////// \ ////////\XXX/\\\\\\\\ ///////////// \ //////////\X/\\\\\\\\\\ o//////////// o o///////////o\\\\\\\\\\\o |\////////// / |\////////// \\\\\\\\\\/| | \//////// / | \//////// \\\\\\\\/ | | \////// / | \////// \\\\\\/ | | \//// / | \//// \\\\/ | | x \// / | x \// \\/ dx | o-----o / o-----o o-----o \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / o o Figure 57-4. Tangent Functor Diagram for the Conjunction J = uv Jon Awbrey o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at oakland.edu Mon Jun 2 12:40:39 2003 From: jawbrey at oakland.edu (Jon Awbrey) Date: Wed Jan 21 20:34:25 2004 Subject: [Inquiry] Re: Differential Logic References: <3E982563.F67459A5@oakland.edu> <3EB73352.259E602@oakland.edu> Message-ID: <3EDB8C17.8B094CD1@oakland.edu> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Note D71 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | The past and present wilt . . . . I have filled them and | emptied them, | And proceed to fill my next fold of the future. | | Walt Whitman, 'Leaves of Grass', [Whi, 87] Taking Aim at Higher Dimensional Targets In the next Subdivision I consider a logical transformation F that has the concrete type F : [u, v] -> [x, y] and the abstract type F : [B^2] -> [B^2]. From jawbrey at oakland.edu Mon Jun 2 16:40:02 2003 From: jawbrey at oakland.edu (Jon Awbrey) Date: Wed Jan 21 20:34:25 2004 Subject: [Inquiry] Re: Differential Logic References: <3E982563.F67459A5@oakland.edu> <3EB73352.259E602@oakland.edu> <3EDB8C17.8B094CD1@oakland.edu> Message-ID: <3EDBC432.F0D342DC@oakland.edu> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Note D72 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Taking Aim at Higher Dimensional Targets (concl.) Let us now refer to the dimension of the target space or codomain as the "toll" (or "tole") of a transformation, as distinguished from the dimension of the range or image that is customarily called the "rank". When we keep to transformations with a toll of 1, as J : [u, v] -> [x], we tend to get lazy about distinguishing a logical transformation from its component propositions. However, if we deal with transformations of a higher toll, this form of indolence can no longer be tolerated. Well, perhaps we can carry it a little further. After all, the operator result WJ : EU% -> EX% is a map of toll 2, and cannot be unfolded in one piece as a proposition. But when a map has rank 1, like !e!J : EU -> X c EX or dJ : EU -> dX c EX, we naturally choose to concentrate on the 1-dimensional range of the operator result WJ, ignoring the final difference in quality between the spaces X and dX, and view WJ as a proposition about EU. In this way, an initial ambivalence about the role of the operand J conveys a double duty to the result WJ. The pivot that is formed by our focus of attention is essential to the linkage that transfers this double moment, as the whole process takes its bearing and wheels around the precise measure of a narrow bead that we can draw on the range of WJ. This is the escapement that it takes to get away with what may otherwise seem to be a simple duplicity, and this is the tolerance that is needed to counterbalance a certain arrogance of equivocation, by all of which machinations we make ourselves free to indicate the operator results WJ as propositions or as transformations, indifferently. But that's it, and no further. Neglect of these distinctions in range and target universes of higher dimensions is bound to cause a hopeless confusion. To guard against these adverse prospects, Tables 58 and 59 lay the groundwork for discussing a typical map F : [B^2] -> [B^2], and begin to pave the way, to some extent, for discussing any transformation of the form F : [B^n] -> [B^k]. Table 58. Cast of Characters: Expansive Subtypes of Objects and Operators o------o-------------------------o------------------o----------------------------o | Item | Notation | Description | Type | o------o-------------------------o------------------o----------------------------o | | | | | | U% | = [u, v] | Source Universe | [B^n] | | | | | | o------o-------------------------o------------------o----------------------------o | | | | | | X% | = [x, y] | Target Universe | [B^k] | | | = [f, g] | | | | | | | | o------o-------------------------o------------------o----------------------------o | | | | | | EU% | = [u, v, du, dv] | Extended | [B^n x D^n] | | | | Source Universe | | | | | | | o------o-------------------------o------------------o----------------------------o | | | | | | EX% | = [x, y, dx, dy] | Extended | [B^k x D^k] | | | = [f, g, df, dg] | Target Universe | | | | | | | o------o-------------------------o------------------o----------------------------o | | | | | | F | F = : U% -> X% | Transformation, | [B^n] -> [B^k] | | | | or Mapping | | | | | | | o------o-------------------------o------------------o----------------------------o | | | | | | | f, g : U -> B | Proposition, | B^n -> B | | | | special case | | | f | f : U -> [x] c X% | of a mapping, | c (B^n, B^n -> B) | | | | or component | | | g | g : U -> [y] c X% | of a mapping. | = (B^n +-> B) = [B^n] | | | | | | o------o-------------------------o------------------o----------------------------o | | | | | | W | W : | Operator | | | | U% -> EU%, | | [B^n] -> [B^n x D^n], | | | X% -> EX%, | | [B^k] -> [B^k x D^k], | | | (U%->X%)->(EU%->EX%), | | ([B^n] -> [B^k]) | | | for each W among: | | -> | | | !e!, !h!, E, D, d | | ([B^n x D^n]->[B^k x D^k]) | | | | | | o------o-------------------------o------------------o----------------------------o | | | | | !e! | | Tacit Extension Operator !e! | | !h! | | Trope Extension Operator !h! | | E | | Enlargement Operator E | | D | | Difference Operator D | | d | | Differential Operator d | | | | | o------o-------------------------o------------------o----------------------------o | | | | | | $W$ | $W$ : | Operator | | | | U% -> $T$U% = EU%, | | [B^n] -> [B^n x D^n], | | | X% -> $T$X% = EX%, | | [B^k] -> [B^k x D^k], | | | (U%->X%)->($T$U%->$T$X%)| | ([B^n] -> [B^k]) | | | for each $W$ among: | | -> | | | $e$, $E$, $D$, $T$ | | ([B^n x D^n]->[B^k x D^k]) | | | | | | o------o-------------------------o------------------o----------------------------o | | | | | $e$ | | Radius Operator $e$ = | | $E$ | | Secant Operator $E$ = | | $D$ | | Chord Operator $D$ = | | $T$ | | Tangent Functor $T$ = | | | | | o------o-------------------------o-----------------------------------------------o Table 59. Synopsis of Terminology: Restrictive and Alternative Subtypes o--------------o----------------------o--------------------o----------------------o | | Operator | Proposition | Transformation | | | or | or | or | | | Operand | Component | Mapping | o--------------o----------------------o--------------------o----------------------o | | | | | | Operand | F = | F_i : <|u,v|> -> B | F : [u, v] -> [x, y] | | | | | | | | F = : U -> X | F_i : B^n -> B | F : B^n -> B^k | | | | | | o--------------o----------------------o--------------------o----------------------o | | | | | | Tacit | !e! : | !e!F_i : | !e!F : | | Extension | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> B | [u,v,du,dv]->[x, y] | | | (U%->X%)->(EU%->X%) | B^n x D^n -> B | [B^n x D^n]->[B^k] | | | | | | o--------------o----------------------o--------------------o----------------------o | | | | | | Trope | !h! : | !h!F_i : | !h!F : | | Extension | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx,dy] | | | (U%->X%)->(EU%->dX%) | B^n x D^n -> D | [B^n x D^n]->[D^k] | | | | | | o--------------o----------------------o--------------------o----------------------o | | | | | | Enlargement | E : | EF_i : | EF : | | Operator | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx,dy] | | | (U%->X%)->(EU%->dX%) | B^n x D^n -> D | [B^n x D^n]->[D^k] | | | | | | o--------------o----------------------o--------------------o----------------------o | | | | | | Difference | D : | DF_i : | DF : | | Operator | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx,dy] | | | (U%->X%)->(EU%->dX%) | B^n x D^n -> D | [B^n x D^n]->[D^k] | | | | | | o--------------o----------------------o--------------------o----------------------o | | | | | | Differential | d : | dF_i : | dF : | | Operator | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx,dy] | | | (U%->X%)->(EU%->dX%) | B^n x D^n -> D | [B^n x D^n]->[D^k] | | | | | | o--------------o----------------------o--------------------o----------------------o | | | | | | Remainder | r : | rF_i : | rF : | | Operator | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx,dy] | | | (U%->X%)->(EU%->dX%) | B^n x D^n -> D | [B^n x D^n]->[D^k] | | | | | | o--------------o----------------------o--------------------o----------------------o | | | | | | Radius | $e$ = : | | $e$F : | | Operator | | | | | | U%->EU%, X%->EX%, | | [u, v, du, dv] -> | | | (U%->X%)->(EU%->EX%) | | [x, y, dx, dy], | | | | | | | | | | [B^n x D^n] -> | | | | | [B^k x D^k] | | | | | | o--------------o----------------------o--------------------o----------------------o | | | | | | Secant | $E$ = : | | $E$F : | | Operator | | | | | | U%->EU%, X%->EX%, | | [u, v, du, dv] -> | | | (U%->X%)->(EU%->EX%) | | [x, y, dx, dy], | | | | | | | | | | [B^n x D^n] -> | | | | | [B^k x D^k] | | | | | | o--------------o----------------------o--------------------o----------------------o | | | | | | Chord | $D$ = : | | $D$F : | | Operator | | | | | | U%->EU%, X%->EX%, | | [u, v, du, dv] -> | | | (U%->X%)->(EU%->EX%) | | [x, y, dx, dy], | | | | | | | | | | [B^n x D^n] -> | | | | | [B^k x D^k] | | | | | | o--------------o----------------------o--------------------o----------------------o | | | | | | Tangent | $T$ = : | dF_i : | $T$F : | | Functor | | | | | | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u, v, du, dv] -> | | | (U%->X%)->(EU%->EX%) | | [x, y, dx, dy], | | | | | | | | | B^n x D^n -> D | [B^n x D^n] -> | | | | | [B^k x D^k] | | | | | | o--------------o----------------------o--------------------o----------------------o Jon Awbrey o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at oakland.edu Mon Jun 2 21:52:04 2003 From: jawbrey at oakland.edu (Jon Awbrey) Date: Wed Jan 21 20:34:25 2004 Subject: [Inquiry] Re: Differential Logic References: <3E982563.F67459A5@oakland.edu> <3EB73352.259E602@oakland.edu> Message-ID: <3EDC0D54.D068D971@oakland.edu> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Note D73 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Transformations of Type B^2 -> B^2 To take up a slightly more complex example, but one that remains simple enough to pursue through a complete series of developments, consider the transformation from U% = [u, v] to X% = [x, y] that is defined by the following system of equations: o-----------------------------------------------------------o | | | x = f(u, v) = ((u)(v)) | | | | y = g(u, v) = ((u, v)) | | | o-----------------------------------------------------------o The component notation F = = : U% -> X% allows us to give a name and a type to this transformation, and permits us to define it by means of the compact description that follows: o-----------------------------------------------------------o | | | = F = <((u)(v)), ((u, v))> | | | o-----------------------------------------------------------o The information that defines the logical transformation F can be represented in the form of a truth table, as in Table 60. To cut down on subscripts in this example I continue to use plain letter equivalents for all components of spaces and maps. Table 60. A Propositional Transformation o-------------o-------------o-------------o-------------o | u | v | f | g | o-------------o-------------o-------------o-------------o | | | | | | 0 | 0 | 0 | 1 | | | | | | | 0 | 1 | 1 | 0 | | | | | | | 1 | 0 | 1 | 0 | | | | | | | 1 | 1 | 1 | 1 | | | | | | o-------------o-------------o-------------o-------------o | | | ((u)(v)) | ((u, v)) | o-------------o-------------o-------------o-------------o Jon Awbrey o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at oakland.edu Tue Jun 3 10:46:34 2003 From: jawbrey at oakland.edu (Jon Awbrey) Date: Wed Jan 21 20:34:25 2004 Subject: [Inquiry] Re: Differential Logic References: <3E982563.F67459A5@oakland.edu> <3EB73352.259E602@oakland.edu> <3EDC0D54.D068D971@oakland.edu> Message-ID: <3EDCC2DA.ED1AF1C9@oakland.edu> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Note D74 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Transformations of Type B^2 -> B^2 (cont.) Figure 61 shows how one might paint a picture of the logical transformation F on the canvass that was earlier primed for this purpose (way back in Figure 30). o-----------------------------------------------------o | U | | | | o-----------o o-----------o | | / \ / \ | | / o \ | | / / \ \ | | / / \ \ | | o o o o | | | | | | | | | u | | v | | | | | | | | | o o o o | | \ \ / / | | \ \ / / | | \ o / | | \ / \ / | | o-----------o o-----------o | | | | | o-----------------------------------------------------o / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ o-------------------------o o-------------------------o | U | |\U \\\\\\\\\\\\\\\\\\\\\\| | o---o o---o | |\\\\\\o---o\\\o---o\\\\\\| | //////\ //////\ | |\\\\\/ \\/ \\\\\\| | ////////o///////\ | |\\\\/ o \\\\\| | //////////\///////\ | |\\\/ /\\ \\\\| | o///////o///o///////o | |\\o o\\\o o\\| | |// u //|///|// v //| | |\\| u |\\\| v |\\| | o///////o///o///////o | |\\o o\\\o o\\| | \///////\////////// | |\\\\ \\/ /\\\| | \///////o//////// | |\\\\\ o /\\\\| | \////// \////// | |\\\\\\ /\\ /\\\\\| | o---o o---o | |\\\\\\o---o\\\o---o\\\\\\| | | |\\\\\\\\\\\\\\\\\\\\\\\\\| o-------------------------o o-------------------------o \ | | / \ | | / \ | | / \ f | | g / \ | | / \ | | / \ | | / \ | | / \ | | / \ | | / o-------\----|---------------------------|----/-------o | X \ | | / | | \| |/ | | o-----------o o-----------o | | //////////////\ /\\\\\\\\\\\\\\ | | ////////////////o\\\\\\\\\\\\\\\\ | | /////////////////X\\\\\\\\\\\\\\\\\ | | /////////////////XXX\\\\\\\\\\\\\\\\\ | | o///////////////oXXXXXo\\\\\\\\\\\\\\\o | | |///////////////|XXXXX|\\\\\\\\\\\\\\\| | | |////// x //////|XXXXX|\\\\\\ y \\\\\\| | | |///////////////|XXXXX|\\\\\\\\\\\\\\\| | | o///////////////oXXXXXo\\\\\\\\\\\\\\\o | | \///////////////\XXX/\\\\\\\\\\\\\\\/ | | \///////////////\X/\\\\\\\\\\\\\\\/ | | \///////////////o\\\\\\\\\\\\\\\/ | | \////////////// \\\\\\\\\\\\\\/ | | o-----------o o-----------o | | | | | o-----------------------------------------------------o Figure 61. A Propositional Transformation Jon Awbrey o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at oakland.edu Tue Jun 3 13:00:20 2003 From: jawbrey at oakland.edu (Jon Awbrey) Date: Wed Jan 21 20:34:25 2004 Subject: [Inquiry] Re: Differential Logic References: <3E982563.F67459A5@oakland.edu> <3EB73352.259E602@oakland.edu> <3EDC0D54.D068D971@oakland.edu> <3EDCC2DA.ED1AF1C9@oakland.edu> Message-ID: <3EDCE234.D1E311FF@oakland.edu> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Note D75 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Transformations of Type B^2 -> B^2 (cont.) Figure 62 extracts the gist of Figure 61, epitomizing a style of diagram that is adequate for most purposes. o-------------------------o o-------------------------o | U | |\U \\\\\\\\\\\\\\\\\\\\\\| | o---o o---o | |\\\\\\o---o\\\o---o\\\\\\| | //////\ //////\ | |\\\\\/ \\/ \\\\\\| | ////////o///////\ | |\\\\/ o \\\\\| | //////////\///////\ | |\\\/ /\\ \\\\| | o///////o///o///////o | |\\o o\\\o o\\| | |// u //|///|// v //| | |\\| u |\\\| v |\\| | o///////o///o///////o | |\\o o\\\o o\\| | \///////\////////// | |\\\\ \\/ /\\\| | \///////o//////// | |\\\\\ o /\\\\| | \////// \////// | |\\\\\\ /\\ /\\\\\| | o---o o---o | |\\\\\\o---o\\\o---o\\\\\\| | | |\\\\\\\\\\\\\\\\\\\\\\\\\| o-------------------------o o-------------------------o \ / \ / \ / \ / \ / \ / \ f / \ g / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / o---------\-----/---------------------\-----/---------o | X \ / \ / | | \ / \ / | | o-----------o o-----------o | | //////////////\ /\\\\\\\\\\\\\\ | | ////////////////o\\\\\\\\\\\\\\\\ | | /////////////////X\\\\\\\\\\\\\\\\\ | | /////////////////XXX\\\\\\\\\\\\\\\\\ | | o///////////////oXXXXXo\\\\\\\\\\\\\\\o | | |///////////////|XXXXX|\\\\\\\\\\\\\\\| | | |////// x //////|XXXXX|\\\\\\ y \\\\\\| | | |///////////////|XXXXX|\\\\\\\\\\\\\\\| | | o///////////////oXXXXXo\\\\\\\\\\\\\\\o | | \///////////////\XXX/\\\\\\\\\\\\\\\/ | | \///////////////\X/\\\\\\\\\\\\\\\/ | | \///////////////o\\\\\\\\\\\\\\\/ | | \////////////// \\\\\\\\\\\\\\/ | | o-----------o o-----------o | | | | | o-----------------------------------------------------o Figure 62. A Propositional Transformation (Short Form) Jon Awbrey o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at oakland.edu Tue Jun 3 22:00:17 2003 From: jawbrey at oakland.edu (Jon Awbrey) Date: Wed Jan 21 20:34:26 2004 Subject: [Inquiry] Re: Differential Logic References: <3E982563.F67459A5@oakland.edu> <3EB73352.259E602@oakland.edu> <3EDC0D54.D068D971@oakland.edu> <3EDCC2DA.ED1AF1C9@oakland.edu> <3EDCE234.D1E311FF@oakland.edu> Message-ID: <3EDD60C0.F2108C0@oakland.edu> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Note D76 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Transformations of Type B^2 -> B^2 (cont.) Figure 63 give a more complete picture of the transformation F, showing how the points of U% are transformed into points of X%. The lines that cross from one universe to the other trace the action that F induces on points, in other words, they depict the aspect of the transformation that acts as a mapping from points to points, and chart its effects on the elements that are variously called cells, points, positions, or singular propositions. o-----------------------------------------------------o |`U` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `| |` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `| |` ` ` ` ` ` o-----------o ` o-----------o ` ` ` ` ` `| |` ` ` ` ` `/' ' ' ' ' ' '\`/' ' ' ' ' ' '\` ` ` ` ` `| |` ` ` ` ` / ' ' ' ' ' ' ' o ' ' ' ' ' ' ' \ ` ` ` ` `| |` ` ` ` `/' ' ' ' ' ' ' '/^\' ' ' ' ' ' ' '\` ` ` ` `| |` ` ` ` / ' ' ' ' ' ' ' /^^^\ ' ' ' ' ' ' ' \ ` ` ` `| |` ` ` `o' ' ' ' ' ' ' 'o^^^^^o' ' ' ' ' ' ' 'o` ` ` `| |` ` ` `|' ' ' ' ' ' ' '|^^^^^|' ' ' ' ' ' ' '|` ` ` `| |` ` ` `|' ' ' ' u ' ' '|^^^^^|' ' ' v ' ' ' '|` ` ` `| |` ` ` `|' ' ' ' ' ' ' '|^^^^^|' ' ' ' ' ' ' '|` ` ` `| |` `@` `o' ' ' ' @ ' ' 'o^^@^^o' ' ' @ ' ' ' 'o` ` ` `| |` ` \ ` \ ' ' ' | ' ' ' \^|^/ ' ' ' | ' ' ' / ` ` ` `| |` ` `\` `\' ' ' | ' ' ' '\|/' ' ' ' | ' ' '/` ` ` ` `| |` ` ` \ ` \ ' ' | ' ' ' ' | ' ' ' ' | ' ' / ` ` ` ` `| |` ` ` `\` `\' ' | ' ' ' '/|\' ' ' ' | ' '/` ` ` ` ` `| |` ` ` ` \ ` o---|-------o | o-------|---o ` ` ` ` ` `| |` ` ` ` `\` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` `| |` ` ` ` ` \ ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` `| o-----------\----|---------|---------|----------------o " " \ | | | " " " " \ | | | " " " " \ | | | " " " " \| | | " " o-------------------------o \ | | o-------------------------o | U | |\ | | |`U```````````````````````| | o---o o---o | | \ | | |``````o---o```o---o``````| | /'''''\ /'''''\ | | \ | | |`````/ \`/ \`````| | /'''''''o'''''''\ | | \ | | |````/ o \````| | /'''''''/'\'''''''\ | | \ | | |```/ /`\ \```| | o'''''''o'''o'''''''o | | \ | | |``o o```o o``| | |'''u'''|'''|'''v'''| | | \ | | |``| u |```| v |``| | o'''''''o'''o'''''''o | | \ | | |``o o```o o``| | \'''''''\'/'''''''/ | | \| | |```\ \`/ /```| | \'''''''o'''''''/ | | \ | |````\ o /````| | \'''''/ \'''''/ | | |\ | |`````\ /`\ /`````| | o---o o---o | | | \ | |``````o---o```o---o``````| | | | | \ * |`````````````````````````| o-------------------------o | | \ / o-------------------------o \ | | | \ / | / \ ((u)(v)) | | | \/ | ((u, v)) / \ | | | /\ | / \ | | | / \ | / \ | | | / \ | / \ | | | / * | / \ | | | / | | / \ | | |/ | | / \ | | / | | / \ | | /| | | / o-------\----|---|-------/-|---------|---|----/-------o | X \ | | / | | | / | | \| | / | | |/ | | o---|----/--o | o-------|---o | | /' ' | ' / ' '\|/` ` ` ` | ` `\ | | / ' ' | '/' ' ' | ` ` ` ` | ` ` \ | | /' ' ' | / ' ' '/|\` ` ` ` | ` ` `\ | | / ' ' ' |/' ' ' /^|^\ ` ` ` | ` ` ` \ | | @ o' ' ' ' @ ' ' 'o^^@^^o` ` ` @ ` ` ` `o | | |' ' ' ' ' ' ' '|^^^^^|` ` ` ` ` ` ` `| | | |' ' ' ' f ' ' '|^^^^^|` ` ` g ` ` ` `| | | |' ' ' ' ' ' ' '|^^^^^|` ` ` ` ` ` ` `| | | o' ' ' ' ' ' ' 'o^^^^^o` ` ` ` ` ` ` `o | | \ ' ' ' ' ' ' ' \^^^/ ` ` ` ` ` ` ` / | | \' ' ' ' ' ' ' '\^/` ` ` ` ` ` ` `/ | | \ ' ' ' ' ' ' ' o ` ` ` ` ` ` ` / | | \' ' ' ' ' ' '/ \` ` ` ` ` ` `/ | | o-----------o o-----------o | | | | | o-----------------------------------------------------o Figure 63. A Transformation of Positions o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at oakland.edu Wed Jun 4 08:28:06 2003 From: jawbrey at oakland.edu (Jon Awbrey) Date: Wed Jan 21 20:34:26 2004 Subject: [Inquiry] Lattices As Charts Message-ID: <3EDDF3E6.7160F9C3@oakland.edu> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o LAC. Note 1 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Here is a first attempt to combine the ideas of lattice theory and manifold theory for ontology applications. (I was going to call this "Atlases Of Lattices", but that acronym has been taken.) o-------------------------------------------------------------o | X | | | | o-------------o o-------------o | | / \ / \ | | / o \ | | / / \ \ | | / / \ \ | | / / \ \ | | o o W o o | | | | = | | | | | |U |^| V| | | | | U | | V | | | | | x y | | | | | | o o | | | | o o o o | | \ \ / / | | \ \ / / | | \ \ / / | | \ o / | | \ | / \ | / | | o------|------o o------|------o | | | | | | | | | o---------------------|-----------------|---------------------o | | f | | g | | o---------------------|-----o o-----|---------------------o | L_1 v | | v L_2 | | | | | | o-----------o | | o-----------o | | / f(U) \ | | / g(V) \ | | / o | | o \ | | / / \ | | / \ \ | | / / \ | | / \ \ | | o ^ ^ ^ o fx o | | o gy o ^ ^ ^ o | | | \ | / |^ ^ ^| | | |^ ^ ^| \ | / | | | | \|/ | \|/ | | !t! | | \|/ | \|/ | | | | o | o ------------> o | o | | | | /|\ | /|\ | | | | /|\ | /|\ | | | | / | \ |v v v| | | |v v v| / | \ | | | o v v v o fy o | | o gx o v v v o | | \ \ / | | \ / / | | \ \ / | | \ / / | | \ o | | o / | | \ / | | \ / | | o-----------o | | o-----------o | | | | | | | | | o---------------------------o o---------------------------o X is the manifold of what really exists. It is called "X" to serve as a constant reminder of how little we actually know about anything of that description. The chart maps f and g are the maps that different parties of mappers make of X, covering the domains U, V c X, respectively. Accordingly, we have f : U -> L_1 and g : V -> L_2, where the domains of f and g overlap in the intersection W = U |^| V that constitutes their common domain, but f and g can form very different pictures of W. In our application, let us assume that the chart codomains L_1 and L_2 are organized as lattices (partial orders, preorders, or what have you). I have depicted a situation where f orders f(x) above f(y) in its lattice, but g orders g(x) below g(y) in its lattice. Notice that there is really no contradiction here, as these orderings take place in two different "images" of X, but if mappers unthinkingly "retro-project" their maps back onto X, as they do when they insist that x >= y or x =< y, then there will appear to be a conflict. Jon Awbrey o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at oakland.edu Wed Jun 4 13:32:02 2003 From: jawbrey at oakland.edu (Jon Awbrey) Date: Wed Jan 21 20:34:26 2004 Subject: [Inquiry] Re: Differential Logic References: <3E982563.F67459A5@oakland.edu> <3EB73352.259E602@oakland.edu> <3EDC0D54.D068D971@oakland.edu> <3EDCC2DA.ED1AF1C9@oakland.edu> <3EDCE234.D1E311FF@oakland.edu> <3EDD60C0.F2108C0@oakland.edu> Message-ID: <3EDE3B22.60A09F83@oakland.edu> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Note D77 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Transformations of Type B^2 -> B^2 (cont.) Table 64 shows how the action of the transformation F on cells or points is computed in terms of coordinates. Table 64. Transformation of Positions o-----o----------o----------o-------o-------o--------o--------o-------------o | u v | x | y | x y | x(y) | (x)y | (x)(y) | X% = [x, y] | o-----o----------o----------o-------o-------o--------o--------o-------------o | | | | | | | | ^ | | 0 0 | 0 | 1 | 0 | 0 | 1 | 0 | | | | | | | | | | | | | 0 1 | 1 | 0 | 0 | 1 | 0 | 0 | F | | | | | | | | | = | | 1 0 | 1 | 0 | 0 | 1 | 0 | 0 | | | | | | | | | | | | 1 1 | 1 | 1 | 1 | 0 | 0 | 0 | ^ | | | | | | | | | | | o-----o----------o----------o-------o-------o--------o--------o-------------o | | ((u)(v)) | ((u, v)) | u v | (u,v) | (u)(v) | 0 | U% = [u, v] | o-----o----------o----------o-------o-------o--------o--------o-------------o Table 65 extends this scheme from single cells to arbitrary regions of the source and target universes, and illustrates a form of computation that can be used to determine how a logical transformation acts on all of the propositions in a universe of discourse, what is usually called the "induced action" of the transformation from universe to universe. Table 65. Induced Transformation of Propositions o------------o---------------------------------o------------o | X% | <--- F = <--- | U% | o------------o----------o-----------o----------o------------o | | u = | 1 1 0 0 | = u | | | | v = | 1 0 1 0 | = v | | | f_i o----------o-----------o----------o f_j | | | x = | 1 1 1 0 | = f | | | | y = | 1 0 0 1 | = g | | o------------o----------o-----------o----------o------------o | | | | | | | f_0 | () | 0 0 0 0 | () | f_0 | | | | | | | | f_1 | (x)(y) | 0 0 0 1 | () | f_0 | | | | | | | | f_2 | (x) y | 0 0 1 0 | (u)(v) | f_1 | | | | | | | | f_3 | (x) | 0 0 1 1 | (u)(v) | f_1 | | | | | | | | f_4 | x (y) | 0 1 0 0 | (u, v) | f_6 | | | | | | | | f_5 | (y) | 0 1 0 1 | (u, v) | f_6 | | | | | | | | f_6 | (x, y) | 0 1 1 0 | (u v) | f_7 | | | | | | | | f_7 | (x y) | 0 1 1 1 | (u v) | f_7 | | | | | | | o------------o----------o-----------o----------o------------o | | | | | | | f_8 | x y | 1 0 0 0 | u v | f_8 | | | | | | | | f_9 | ((x, y)) | 1 0 0 1 | u v | f_8 | | | | | | | | f_10 | y | 1 0 1 0 | ((u, v)) | f_9 | | | | | | | | f_11 | (x (y)) | 1 0 1 1 | ((u, v)) | f_9 | | | | | | | | f_12 | x | 1 1 0 0 | ((u)(v)) | f_14 | | | | | | | | f_13 | ((x) y) | 1 1 0 1 | ((u)(v)) | f_14 | | | | | | | | f_14 | ((x)(y)) | 1 1 1 0 | (()) | f_15 | | | | | | | | f_15 | (()) | 1 1 1 1 | (()) | f_15 | | | | | | | o------------o----------o-----------o----------o------------o Jon Awbrey o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at oakland.edu Wed Jun 4 21:44:01 2003 From: jawbrey at oakland.edu (Jon Awbrey) Date: Wed Jan 21 20:34:26 2004 Subject: [Inquiry] Re: Differential Logic References: <3E982563.F67459A5@oakland.edu> <3EB73352.259E602@oakland.edu> <3EDC0D54.D068D971@oakland.edu> <3EDCC2DA.ED1AF1C9@oakland.edu> <3EDCE234.D1E311FF@oakland.edu> <3EDD60C0.F2108C0@oakland.edu> <3EDE3B22.60A09F83@oakland.edu> Message-ID: <3EDEAE71.46BB1B1@oakland.edu> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Note D78 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Transformations of Type B^2 -> B^2 (cont.) Given the alphabets !U! = {u, v} and !X! = {x, y}, along with the corresponding universes of discourse U% and X% ~=~ [B^2], how many logical transformations of the general form G = : U% -> X% are there? Since G_1 and G_2 can be any propositions of the type B^2 -> B, there are 2^4 = 16 choices for each of the maps G_1 and G_2, and thus there are 2^4 * 2^4 = 2^8 = 256 different mappings altogether of the form G : U% -> X%. The set of all functions of a given type is customarily denoted by placing its type indicator in parentheses, in the present instance writing (U% -> X%) = {G : U% -> X%}, and so the cardinality of this "function space" can be most conveniently summed up by writing |(U% -> X%)| = |(B^2 -> B^2)| = 4^4 = 256. Given any transformation G = : U% -> X% of this type, one can define a couple of further transformations, related to G, that operate between the extended universes, EU% and EX%, of its source and target domains. First, the enlargement map (or the secant transformation) EG = : EU% -> EX% is defined by the following set of component equations: o-------------------------------------------------o | | | EG_i = G_i | | | o-------------------------------------------------o Second, the difference map (or the chordal transformation) DG = : EU% -> EX% is defined in component-wise fashion as the boolean sum (or mod 2 sum) of the initial and the enlarged propositions: o-------------------------------------------------o | | | DG_i = G_i + EG_i | | | | = G_i + G_i | | | o-------------------------------------------------o Maintaining a strict analogy with ordinary difference calculus would perhaps have us write DG_i = EG_i - G_i, but the sum and difference operations are the same thing in boolean arithmetic. It is more often natural in the logical context to consider an initial proposition q, then to compute the enlargement Eq, and finally to determine the difference Dq = q + Eq, so we let the variant order of terms reflect this sequence of considerations. Viewed in this light the difference operator D is imagined to be a function of very wide scope and polymorphic application, one that is able to realize the association between each transformation G and its difference map DG, for instance, taking the function space (U% -> X%) into (EU% -> EX%). Given the interpretive flexibility of contexts in which we are allowing a proposition to appear, it should be clear that an operator of this scope is not at all a trivial matter to define properly, and may take some trouble to work out. For the moment, let's content ourselves with returning to particular cases. Jon Awbrey o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at oakland.edu Thu Jun 5 09:36:26 2003 From: jawbrey at oakland.edu (Jon Awbrey) Date: Wed Jan 21 20:34:26 2004 Subject: [Inquiry] Re: Differential Logic References: <3E982563.F67459A5@oakland.edu> <3EB73352.259E602@oakland.edu> <3EDC0D54.D068D971@oakland.edu> <3EDCC2DA.ED1AF1C9@oakland.edu> <3EDCE234.D1E311FF@oakland.edu> <3EDD60C0.F2108C0@oakland.edu> <3EDE3B22.60A09F83@oakland.edu> <3EDEAE71.46BB1B1@oakland.edu> Message-ID: <3EDF556A.E2D8D8C5@oakland.edu> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Note D79 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Transformations of Type B^2 -> B^2 (cont.) In their application to the present example, namely, the logical transformation F = = <((u)(v)), ((u, v))>, the operators E and D respectively produce the enlarged map EF = and the difference map DF = , whose components can be given as follows, if the reader, in lieu of a special font for the logical parentheses, can forgive a syntactically bilingual formulation: o-------------------------------------------------o | | | Ef = ((u + du)(v + dv)) | | | | Eg = ((u + du, v + dv)) | | | o-------------------------------------------------o o-------------------------------------------------o | | | Df = ((u)(v)) + ((u + du)(v + dv)) | | | | Dg = ((u, v)) + ((u + du, v + dv)) | | | o-------------------------------------------------o But these initial formulas are purely definitional, and help us little in understanding either the purpose of the operators or the meaning of their results. Working symbolically, let us apply the same method to the separate components f and g that we earlier used on J. This work is recorded in Appendix 1 and a summary of the results is presented in Tables 66-i and 66-ii. Table 66-i. Computation Summary for f = ((u)(v)) o--------------------------------------------------------------------------------o | | | !e!f = uv. 1 + u(v). 1 + (u)v. 1 + (u)(v). 0 | | | | Ef = uv. (du dv) + u(v). (du (dv)) + (u)v.((du) dv) + (u)(v).((du)(dv)) | | | | Df = uv. du dv + u(v). du (dv) + (u)v. (du) dv + (u)(v).((du)(dv)) | | | | df = uv. 0 + u(v). du + (u)v. dv + (u)(v). (du, dv) | | | | rf = uv. du dv + u(v). du dv + (u)v. du dv + (u)(v). du dv | | | o--------------------------------------------------------------------------------o Table 66-ii. Computation Summary for g = ((u, v)) o--------------------------------------------------------------------------------o | | | !e!g = uv. 1 + u(v). 0 + (u)v. 0 + (u)(v). 1 | | | | Eg = uv.((du, dv)) + u(v). (du, dv) + (u)v. (du, dv) + (u)(v).((du, dv)) | | | | Dg = uv. (du, dv) + u(v). (du, dv) + (u)v. (du, dv) + (u)(v). (du, dv) | | | | dg = uv. (du, dv) + u(v). (du, dv) + (u)v. (du, dv) + (u)(v). (du, dv) | | | | rg = uv. 0 + u(v). 0 + (u)v. 0 + (u)(v). 0 | | | o--------------------------------------------------------------------------------o Jon Awbrey o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at oakland.edu Thu Jun 5 14:48:13 2003 From: jawbrey at oakland.edu (Jon Awbrey) Date: Wed Jan 21 20:34:27 2004 Subject: [Inquiry] Re: Differential Logic References: <3E982563.F67459A5@oakland.edu> <3EB73352.259E602@oakland.edu> <3EDC0D54.D068D971@oakland.edu> <3EDCC2DA.ED1AF1C9@oakland.edu> <3EDCE234.D1E311FF@oakland.edu> <3EDD60C0.F2108C0@oakland.edu> <3EDE3B22.60A09F83@oakland.edu> <3EDEAE71.46BB1B1@oakland.edu> <3EDF556A.E2D8D8C5@oakland.edu> Message-ID: <3EDF9E7D.1BEB2C08@oakland.edu> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Note D80 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Transformations of Type B^2 -> B^2 (cont.) Table 67 shows how to compute the analytic series for F = = <((u)(v)), ((u, v))> in terms of coordinates, and Table 68 recaps these results in symbolic terms, agreeing with earlier derivations. Table 67. Computation of an Analytic Series in Terms of Coordinates o--------o-------o-------o--------o-------o-------o-------o-------o | u v | du dv | u' v' | f g | Ef Eg | Df Dg | df dg | rf rg | o--------o-------o-------o--------o-------o-------o-------o-------o | | | | | | | | | | 0 0 | 0 0 | 0 0 | 0 1 | 0 1 | 0 0 | 0 0 | 0 0 | | | | | | | | | | | | 0 1 | 0 1 | | 1 0 | 1 1 | 1 1 | 0 0 | | | | | | | | | | | | 1 0 | 1 0 | | 1 0 | 1 1 | 1 1 | 0 0 | | | | | | | | | | | | 1 1 | 1 1 | | 1 1 | 1 0 | 0 0 | 1 0 | | | | | | | | | | o--------o-------o-------o--------o-------o-------o-------o-------o | | | | | | | | | | 0 1 | 0 0 | 0 1 | 1 0 | 1 0 | 0 0 | 0 0 | 0 0 | | | | | | | | | | | | 0 1 | 0 0 | | 0 1 | 1 1 | 1 1 | 0 0 | | | | | | | | | | | | 1 0 | 1 1 | | 1 1 | 0 1 | 0 1 | 0 0 | | | | | | | | | | | | 1 1 | 1 0 | | 1 0 | 0 0 | 1 0 | 1 0 | | | | | | | | | | o--------o-------o-------o--------o-------o-------o-------o-------o | | | | | | | | | | 1 0 | 0 0 | 1 0 | 1 0 | 1 0 | 0 0 | 0 0 | 0 0 | | | | | | | | | | | | 0 1 | 1 1 | | 1 1 | 0 1 | 0 1 | 0 0 | | | | | | | | | | | | 1 0 | 0 0 | | 0 1 | 1 1 | 1 1 | 0 0 | | | | | | | | | | | | 1 1 | 0 1 | | 1 0 | 0 0 | 1 0 | 1 0 | | | | | | | | | | o--------o-------o-------o--------o-------o-------o-------o-------o | | | | | | | | | | 1 1 | 0 0 | 1 1 | 1 1 | 1 1 | 0 0 | 0 0 | 0 0 | | | | | | | | | | | | 0 1 | 1 0 | | 1 0 | 0 1 | 0 1 | 0 0 | | | | | | | | | | | | 1 0 | 0 1 | | 1 0 | 0 1 | 0 1 | 0 0 | | | | | | | | | | | | 1 1 | 0 0 | | 0 1 | 1 0 | 0 0 | 1 0 | | | | | | | | | | o--------o-------o-------o--------o-------o-------o-------o-------o Table 68. Computation of an Analytic Series in Symbolic Terms o-----o-----o------------o----------o----------o----------o----------o----------o | u v | f g | Df | Dg | df | dg | rf | rg | o-----o-----o------------o----------o----------o----------o----------o----------o | | | | | | | | | | 0 0 | 0 1 | ((du)(dv)) | (du, dv) | (du, dv) | (du, dv) | du dv | () | | | | | | | | | | | 0 1 | 1 0 | (du) dv | (du, dv) | dv | (du, dv) | du dv | () | | | | | | | | | | | 1 0 | 1 0 | du (dv) | (du, dv) | du | (du, dv) | du dv | () | | | | | | | | | | | 1 1 | 1 1 | du dv | (du, dv) | () | (du, dv) | du dv | () | | | | | | | | | | o-----o-----o------------o----------o----------o----------o----------o----------o Jon Awbrey o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at oakland.edu Thu Jun 5 22:00:09 2003 From: jawbrey at oakland.edu (Jon Awbrey) Date: Wed Jan 21 20:34:27 2004 Subject: [Inquiry] Re: Differential Logic References: <3E982563.F67459A5@oakland.edu> <3EB73352.259E602@oakland.edu> <3EDC0D54.D068D971@oakland.edu> <3EDCC2DA.ED1AF1C9@oakland.edu> <3EDCE234.D1E311FF@oakland.edu> <3EDD60C0.F2108C0@oakland.edu> <3EDE3B22.60A09F83@oakland.edu> <3EDEAE71.46BB1B1@oakland.edu> <3EDF556A.E2D8D8C5@oakland.edu> <3EDF9E7D.1BEB2C08@oakland.edu> Message-ID: <3EE003B8.7476DAFC@oakland.edu> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Note D81 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Transformations of Type B^2 -> B^2 (cont.) Figure 69 gives a graphical picture of the difference map DF = for the transformation F = = <((u)(v)), ((u, v))>. This depicts the same information about Df and Dg that was given in the corresponding rows of the computation summary in Tables 66-i and 66-ii, excerpted here: o-------------------------------------------------------------------------o | | | Df = uv. du dv + u(v). du (dv) + (u)v.(du) dv + (u)(v).((du)(dv)) | | | | Dg = uv.(du, dv) + u(v).(du, dv) + (u)v.(du, dv) + (u)(v). (du, dv) | | | o-------------------------------------------------------------------------o o-----------------------------------o o-----------------------------------o | U | |`U`````````````````````````````````| | | |```````````````````````````````````| | ^ | |```````````````````````````````````| | | | |```````````````````````````````````| | o-------o | o-------o | |```````o-------o```o-------o```````| | ^ /`````````\|/`````````\ ^ | | ^ ```/ ^ \`/ ^ \``` ^ | | \ /```````````|```````````\ / | |``\``/ \ o / \``/``| | \/`````u`````/|\`````v`````\/ | |```\/ u \/`\/ v \/```| | /\``````````/`|`\``````````/\ | |```/\ /\`/\ /\```| | o``\````````o``@``o````````/``o | |``o \ o``@``o / o``| | |```\```````|`````|```````/```| | |``| \ |`````| / |``| | |````@``````|`````|``````@````| | |``| @-------->`<--------@ |``| | |```````````|`````|```````````| | |``| |`````| |``| | o```````````o` ^ `o```````````o | |``o o`````o o``| | \```````````\`|`/```````````/ | |```\ \```/ /```| | \```` ^ ````\|/```` ^ ````/ | |````\ ^ \`/ ^ /````| | \`````\`````|`````/`````/ | |`````\ \ o / /`````| | \`````\```/|\```/`````/ | |``````\ \ /`\ / /``````| | o-----\-o | o-/-----o | |```````o-----\-o```o-/-----o```````| | \ | / | |``````````````\`````/``````````````| | \ | / | |```````````````\```/```````````````| | \|/ | |````````````````\`/````````````````| | @ | |`````````````````@`````````````````| o-----------------------------------o o-----------------------------------o \ / \ / \ / \ / \ ((u)(v)) / \ ((u, v)) / \ / \ / \ / \ / o----------\-------------/-----------------------\-------------/----------o | X \ / \ / | | \ / \ / | | \ / \ / | | o----------------o o----------------o | | / \ / \ | | / o \ | | / / \ \ | | / / \ \ | | / / \ \ | | / / \ \ | | / / \ \ | | o o o o | | | | | | | | | | | | | | | f | | g | | | | | | | | | | | | | | | o o o o | | \ \ / / | | \ \ / / | | \ \ / / | | \ \ / / | | \ \ / / | | \ o / | | \ / \ / | | o----------------o o----------------o | | | | | | | o-------------------------------------------------------------------------o Figure 69. Difference Map of F = = <((u)(v)), ((u, v))> Jon Awbrey o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at oakland.edu Fri Jun 6 08:24:06 2003 From: jawbrey at oakland.edu (Jon Awbrey) Date: Wed Jan 21 20:34:27 2004 Subject: [Inquiry] Differential Logic Message-ID: <3EE095F6.9DC60D5E@oakland.edu> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o SUO Working Group, One way to think about mappings between different ontologies, and also about ontologies that develop through time -- the two problems are intimately related -- is in terms of transformations from universe of discourse to universe of discourse, the sort of thing that one is naturally tempted to call a "transformation of discourse". Re-starting from the ground up, as experience constantly teaches that we must, we may contemplate simple propositions and unanalyzed predications of the sort that one finds pictured in euler-venn diagrams and that one computes in terms of bits and boolean functions. So, working at the basement level, a mapping between two different universes of discourse, which may of course only be two different ways of describing the same universe of discourse, could be written F : U -> V, and a transformation that describes the changes that occur in a single universe of discourse, which may of course come in the corresponding varieties of "alias" and "alibi" flavors, could be written F : U -> U. Transformations like these can be very complex things to think about -- for instance, one may be thinking of a neuroid system of formal neurons that carry one bit each, and so one's universe is a state space U that is isomorphic to B^n, where B = {0, 1} and n is roughly 10^10, give or take -- so we usually end up having to approach such creatures, the transformations F : U -> U and F : U1 -> U2, in series of increasing orders of approximation. That is what differential calculus is all about. A "derivative" or a "differential" of a transformation F is a "locally linear approximation" to F. In many ways, one can think of differentiation as an operation that takes the global description of a transformation and distributes the information into locally relevant forms. These days, differential calculus and differential geometry are carried out in terms of a thing called the "tangent functor", which is the category theoretic expression of what we do when we take derivatives. A "functor" W is a "mapping of maps" or a "transformation of transformations", so W takes a map F : X -> Y into another map WF : WX -> WY. Roughly speaking then, the particular sort of functor that we will soon know and love as the "tangent functor" T is one that takes the map F : X -> Y and gives back the locally relevant version TF : TX -> TY. Cranking the analogy for logic produces the subject of "differential logic", which has been my pursuit for a decade or two. I am almost done serializing one of my more coherent, but also more detailed, papers on the subject, and I have appended the outline of links so far. Jon Awbrey o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Differential Logic and Dynamic Systems 0. Purpose 1. Review and Transition D01. http://suo.ieee.org/ontology/msg04799.html 2. Functional Conception of Propositional Calculus D02. http://suo.ieee.org/ontology/msg04800.html D03. http://suo.ieee.org/ontology/msg04801.html D04. http://suo.ieee.org/ontology/msg04802.html D05. http://suo.ieee.org/ontology/msg04803.html D06. http://suo.ieee.org/ontology/msg04804.html D07. http://suo.ieee.org/ontology/msg04805.html D08. http://suo.ieee.org/ontology/msg04806.html D09. http://suo.ieee.org/ontology/msg04807.html D10. http://suo.ieee.org/ontology/msg04808.html D11. http://suo.ieee.org/ontology/msg04809.html 3. Differential Extension of Propositional Calculus D12. http://suo.ieee.org/ontology/msg04810.html D13. http://suo.ieee.org/ontology/msg04811.html D14. http://suo.ieee.org/ontology/msg04812.html D15. http://suo.ieee.org/ontology/msg04813.html D16. http://suo.ieee.org/ontology/msg04814.html D17. http://suo.ieee.org/ontology/msg04815.html 4. Back to the Beginning: Some Exemplary Universes D18. http://suo.ieee.org/ontology/msg04816.html D19. http://suo.ieee.org/ontology/msg04817.html D20. http://suo.ieee.org/ontology/msg04818.html D21. http://suo.ieee.org/ontology/msg04819.html D22. http://suo.ieee.org/ontology/msg04820.html D23. http://suo.ieee.org/ontology/msg04821.html D24. http://suo.ieee.org/ontology/msg04822.html 5. Transformations of Discourse D25. http://suo.ieee.org/ontology/msg04823.html 5.1. Foreshadowing Transformations: Extensions and Projections of Discourse D26. http://suo.ieee.org/ontology/msg04824.html D27. http://suo.ieee.org/ontology/msg04825.html 5.2. Thematization of Functions: And a Declaration of Independence for Variables D28. http://suo.ieee.org/ontology/msg04826.html D29. http://suo.ieee.org/ontology/msg04827.html D30. http://suo.ieee.org/ontology/msg04828.html D31. http://suo.ieee.org/ontology/msg04829.html D32. http://suo.ieee.org/ontology/msg04830.html D33. http://suo.ieee.org/ontology/msg04832.html 5.3. Propositional Transformations D34. http://suo.ieee.org/ontology/msg04833.html D35. http://suo.ieee.org/ontology/msg04834.html D36. http://suo.ieee.org/ontology/msg04835.html 5.4. Analytic Expansions: Operators and Functors D37. http://suo.ieee.org/ontology/msg04836.html D38. http://suo.ieee.org/ontology/msg04837.html D39. http://suo.ieee.org/ontology/msg04838.html D40. http://suo.ieee.org/ontology/msg04839.html D41. http://suo.ieee.org/ontology/msg04840.html D42. http://suo.ieee.org/ontology/msg04841.html D43. http://suo.ieee.org/ontology/msg04842.html D44. http://suo.ieee.org/ontology/msg04843.html D45. http://suo.ieee.org/ontology/msg04844.html 5.5. Transformations of Type B^2 -> B^1 D46. http://suo.ieee.org/ontology/msg04845.html D47. http://suo.ieee.org/ontology/msg04846.html D48. http://suo.ieee.org/ontology/msg04847.html D49. http://suo.ieee.org/ontology/msg04848.html D50. http://suo.ieee.org/ontology/msg04849.html D51. http://suo.ieee.org/ontology/msg04850.html D52. http://suo.ieee.org/ontology/msg04851.html D53. http://suo.ieee.org/ontology/msg04852.html D54. http://suo.ieee.org/ontology/msg04853.html D55. http://suo.ieee.org/ontology/msg04854.html D56. http://suo.ieee.org/ontology/msg04855.html D57. http://suo.ieee.org/ontology/msg04856.html D58. http://suo.ieee.org/ontology/msg04857.html D59. http://suo.ieee.org/ontology/msg04858.html D60. http://suo.ieee.org/ontology/msg04859.html D61. http://suo.ieee.org/ontology/msg04860.html D62. http://suo.ieee.org/ontology/msg04861.html D63. http://suo.ieee.org/ontology/msg04862.html D64. http://suo.ieee.org/ontology/msg04863.html D65. http://suo.ieee.org/ontology/msg04864.html D66. http://suo.ieee.org/ontology/msg04865.html D67. http://suo.ieee.org/ontology/msg04866.html D68. http://suo.ieee.org/ontology/msg04867.html D69. http://suo.ieee.org/ontology/msg04868.html D70. http://suo.ieee.org/ontology/msg04869.html 5.6. Taking Aim at Higher Dimensional Targets D71. http://suo.ieee.org/ontology/msg04870.html D72. http://suo.ieee.org/ontology/msg04871.html 5.7. Transformations of Type B^2 -> B^2 D73. http://suo.ieee.org/ontology/msg04872.html D74. http://suo.ieee.org/ontology/msg04873.html D75. http://suo.ieee.org/ontology/msg04874.html D76. http://suo.ieee.org/ontology/msg04875.html D77. http://suo.ieee.org/ontology/msg04876.html D78. http://suo.ieee.org/ontology/msg04877.html D79. http://suo.ieee.org/ontology/msg04878.html D80. http://suo.ieee.org/ontology/msg04879.html D81. http://suo.ieee.org/ontology/msg04880.html o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at oakland.edu Fri Jun 6 11:28:05 2003 From: jawbrey at oakland.edu (Jon Awbrey) Date: Wed Jan 21 20:34:27 2004 Subject: [Inquiry] Re: Differential Logic -- Correction References: <3E982563.F67459A5@oakland.edu> <3EB73352.259E602@oakland.edu> <3EB7CC43.38A81EA5@oakland.edu> <3EB7EBAC.500A89A6@oakland.edu> <3EB80182.7DD2F152@oakland.edu> <3EB887A5.14C78909@oakland.edu> <3EB89D4F.B72C97D0@oakland.edu> <3EB91162.914A3F71@oakland.edu> Message-ID: <3EE0C115.112E7A70@oakland.edu> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Correction. The second paragraph should read as follows. I had this correct, though ambiguous, in my old version, but in trying to insert a clarifying remark, I inserted a confounding one instead. ~~ J.A. | Amazingly enough, these isomorphisms are themselves schematized by the | axioms and theorems of propositional logic. This fact is known as the | "propositions as types analogy" (PATA) [How]. In another formulation | it says that terms are to types as proofs are to propositions. This | principle seems to have more implications for our subject than I can | fully comprehend at present, though I sense that they must be crucial. | (Cf. [LaS, 42-46] and [SeH] for discussion and further references.) | To anticipate the bearing of these issues on our immediate topic, | Table 3 gives a partial overview of the Real to Boolean analogy | that may serve to illustrate the paradigm that I have in mind. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at oakland.edu Sat Jun 7 12:24:30 2003 From: jawbrey at oakland.edu (Jon Awbrey) Date: Wed Jan 21 20:34:27 2004 Subject: [Inquiry] Re: Differential Logic References: <3E982563.F67459A5@oakland.edu> <3EB73352.259E602@oakland.edu> <3EDC0D54.D068D971@oakland.edu> <3EDCC2DA.ED1AF1C9@oakland.edu> <3EDCE234.D1E311FF@oakland.edu> <3EDD60C0.F2108C0@oakland.edu> <3EDE3B22.60A09F83@oakland.edu> <3EDEAE71.46BB1B1@oakland.edu> <3EDF556A.E2D8D8C5@oakland.edu> <3EDF9E7D.1BEB2C08@oakland.edu> <3EE003B8.7476DAFC@oakland.edu> Message-ID: <3EE21FCE.DE4A05AD@oakland.edu> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Note D82 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Transformations of Type B^2 -> B^2 (cont.) Figure 70-a shows a graphical way of picturing the tangent functor map dF = for the transformation F = = <((u)(v)), ((u, v))>. This amounts to the same information about df and dg that was given in the computation summary of Tables 66-i and 66-ii, the relevant rows of which are repeated here: o-------------------------------------------------------------------------------o | | | df = uv. 0 + u(v). du + (u)v. dv + (u)(v).(du, dv) | | | | dg = uv.(du, dv) + u(v).(du, dv) + (u)v.(du, dv) + (u)(v).(du, dv) | | | o-------------------------------------------------------------------------------o o o / \ / \ / \ / \ / \ / O \ / \ o /@\ o / \ / \ / \ / \ / \ / \ / O \ / O \ / O \ o /@\ o o /@\ o /@\ o / \ / \ / \ \ / \ \ / \ / \ / \ / \ / \ / \ / \ / \ / O \ / O \ / O \ / \ / \ o /@ o /@\ o /@ o / \ / \ / \ \ / \ / \ \ / \ / \ / \ / \ / \ / \ / \ / O \ / O \ / O \ / O \ / O \ / O \ o /@ o /@ o o /@ o /@ o /@ o /@ o |\ / \ /| |\ / \ / / \ / / \ /| | \ / \ / | | \ / \ / \ / \ / | | \ / \ / | | \ / O \ / O \ / O \ / | | \ / \ / | | o /@ o @\ o /@ o | | \ / \ / | | |\ / \ / \ / \ / \ /| | | \ / \ / | | | \ / \ / \ / | | | u \ / O \ / v | | u | \ / O \ / O \ / | v | o-------o @\ o-------o o---+---o @\ o @\ o---+---o \ / | \ / \ / \ / \ / | \ / | \ / \ / | \ / | du \ / O \ / dv | \ / o-------o @\ o-------o \ / \ / \ / \ / \ / \ / o o U% $T$ $E$U% o------------------>o | | | | | | | | F | | $T$F | | | | | | v v o------------------>o X% $T$ $E$X% o o / \ / \ / \ / \ / \ / O \ / \ o /@\ o / \ / \ / \ / \ / \ / \ / O \ / O \ / O \ o /@\ o o /@\ o /@\ o / \ / \ / \ \ / \ / / \ / \ / \ / \ / \ / \ / \ / \ / O \ / O \ / O \ / \ / \ o /@ o /@\ o @\ o / \ / \ / \ \ / \ / \ / \ / / \ / \ / \ / \ / \ / \ / \ / O \ / O \ / O \ / O \ / O \ / O \ o /@ o @\ o o /@ o /@ o @\ o @\ o |\ / \ /| |\ / \ / \ / \ / \ / \ /| | \ / \ / | | \ / \ / \ / \ / | | \ / \ / | | \ / O \ / O \ / O \ / | | \ / \ / | | o /@ o @ o @\ o | | \ / \ / | | |\ / / \ / \ / \ \ /| | | \ / \ / | | | \ / \ / \ / | | | x \ / O \ / y | | x | \ / O \ / O \ / | y | o-------o @ o-------o o---+---o @ o @ o---+---o \ / | \ / / \ \ / | \ / | \ / \ / | \ / | dx \ / O \ / dy | \ / o-------o @ o-------o \ / \ / \ / \ / \ / \ / o o Figure 70-a. Tangent Functor Diagram for F = <((u)(v)), ((u, v))> Jon Awbrey o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at oakland.edu Sat Jun 7 22:18:03 2003 From: jawbrey at oakland.edu (Jon Awbrey) Date: Wed Jan 21 20:34:27 2004 Subject: [Inquiry] Re: Differential Logic References: <3E982563.F67459A5@oakland.edu> <3EB73352.259E602@oakland.edu> <3EDC0D54.D068D971@oakland.edu> <3EDCC2DA.ED1AF1C9@oakland.edu> <3EDCE234.D1E311FF@oakland.edu> <3EDD60C0.F2108C0@oakland.edu> <3EDE3B22.60A09F83@oakland.edu> <3EDEAE71.46BB1B1@oakland.edu> <3EDF556A.E2D8D8C5@oakland.edu> <3EDF9E7D.1BEB2C08@oakland.edu> <3EE003B8.7476DAFC@oakland.edu> <3EE21FCE.DE4A05AD@oakland.edu> Message-ID: <3EE2AAEB.A0416661@oakland.edu> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Note D83 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Transformations of Type B^2 -> B^2 (concl.) Figure 70-b shows another way to picture the action of the tangent functor on the logical transformation F = <((u)(v)), ((u, v))>, roughly in the style of the "bundle of universes" type of diagram. [NB. I can't really do justice to the original Figure in ascii graphics, but this collection of pictures may serve as a construction kit, with some assembly required, to convey the general idea.] o-----------------------o o-----------------------o o-----------------------o | dU | | dU | | dU | | o--o o--o | | o--o o--o | | o--o o--o | | /////\ /////\ | | /XXXX\ /XXXX\ | | /\\\\\ /\\\\\ | | ///////o//////\ | | /XXXXXXoXXXXXX\ | | /\\\\\\o\\\\\\\ | | //////// \//////\ | | /XXXXXX/ \XXXXXX\ | | /\\\\\\/ \\\\\\\\ | | o/////// \//////o | | oXXXXXX/ \XXXXXXo | | o\\\\\\/ \\\\\\\o | | |/////o o/////| | | |XXXXXo oXXXXX| | | |\\\\\o o\\\\\| | | |/du//| |//dv/| | | |XXXXX| |XXXXX| | | |\du\\| |\\dv\| | | |/////o o/////| | | |XXXXXo oXXXXX| | | |\\\\\o o\\\\\| | | o//////\ ///////o | | oXXXXXX\ /XXXXXXo | | o\\\\\\\ /\\\\\\o | | \//////\ //////// | | \XXXXXX\ /XXXXXX/ | | \\\\\\\\ /\\\\\\/ | | \//////o/////// | | \XXXXXXoXXXXXX/ | | \\\\\\\o\\\\\\/ | | \///// \///// | | \XXXX/ \XXXX/ | | \\\\\/ \\\\\/ | | o--o o--o | | o--o o--o | | o--o o--o | | | | | | | o-----------------------o o-----------------------o o-----------------------o = du' @ (u)(v) o-----------------------o dv' @ (u)(v) = = | dU' | = = | o--o o--o | = = | /////\ /\\\\\ | = = | ///////o\\\\\\\ | = = | ////////X\\\\\\\\ | = = | o///////XXX\\\\\\\o | = = | |/////oXXXXXo\\\\\| | = = = = = = = = = = = =|/du'/|XXXXX|\dv'\|= = = = = = = = = = = | |/////oXXXXXo\\\\\| | | o//////\XXX/\\\\\\o | | \//////\X/\\\\\\/ | | \//////o\\\\\\/ | | \///// \\\\\/ | | o--o o--o | | | o-----------------------o o-----------------------o o-----------------------o o-----------------------o | dU | | dU | | dU | | o--o o--o | | o--o o--o | | o--o o--o | | / \ /////\ | | /\\\\\ /XXXX\ | | /\\\\\ /\\\\\ | | / o//////\ | | /\\\\\\oXXXXXX\ | | /\\\\\\o\\\\\\\ | | / //\//////\ | | /\\\\\\//\XXXXXX\ | | /\\\\\\/ \\\\\\\\ | | o ////\//////o | | o\\\\\\////\XXXXXXo | | o\\\\\\/ \\\\\\\o | | | o/////o/////| | | |\\\\\o/////oXXXXX| | | |\\\\\o o\\\\\| | | | du |/////|//dv/| | | |\\\\\|/////|XXXXX| | | |\du\\| |\\dv\| | | | o/////o/////| | | |\\\\\o/////oXXXXX| | | |\\\\\o o\\\\\| | | o \//////////o | | o\\\\\\\////XXXXXXo | | o\\\\\\\ /\\\\\\o | | \ \///////// | | \\\\\\\\//XXXXXX/ | | \\\\\\\\ /\\\\\\/ | | \ o/////// | | \\\\\\\oXXXXXX/ | | \\\\\\\o\\\\\\/ | | \ / \///// | | \\\\\/ \XXXX/ | | \\\\\/ \\\\\/ | | o--o o--o | | o--o o--o | | o--o o--o | | | | | | | o-----------------------o o-----------------------o o-----------------------o = du' @ (u) v o-----------------------o dv' @ (u) v = = | dU' | = = | o--o o--o | = = | /////\ /\\\\\ | = = | ///////o\\\\\\\ | = = | ////////X\\\\\\\\ | = = | o///////XXX\\\\\\\o | = = | |/////oXXXXXo\\\\\| | = = = = = = = = = = = =|/du'/|XXXXX|\dv'\|= = = = = = = = = = = | |/////oXXXXXo\\\\\| | | o//////\XXX/\\\\\\o | | \//////\X/\\\\\\/ | | \//////o\\\\\\/ | | \///// \\\\\/ | | o--o o--o | | | o-----------------------o o-----------------------o o-----------------------o o-----------------------o | dU | | dU | | dU | | o--o o--o | | o--o o--o | | o--o o--o | | /////\ / \ | | /XXXX\ /\\\\\ | | /\\\\\ /\\\\\ | | ///////o \ | | /XXXXXXo\\\\\\\ | | /\\\\\\o\\\\\\\ | | /////////\ \ | | /XXXXXX//\\\\\\\\ | | /\\\\\\/ \\\\\\\\ | | o//////////\ o | | oXXXXXX////\\\\\\\o | | o\\\\\\/ \\\\\\\o | | |/////o/////o | | | |XXXXXo/////o\\\\\| | | |\\\\\o o\\\\\| | | |/du//|/////| dv | | | |XXXXX|/////|\\\\\| | | |\du\\| |\\dv\| | | |/////o/////o | | | |XXXXXo/////o\\\\\| | | |\\\\\o o\\\\\| | | o//////\//// o | | oXXXXXX\////\\\\\\o | | o\\\\\\\ /\\\\\\o | | \//////\// / | | \XXXXXX\//\\\\\\/ | | \\\\\\\\ /\\\\\\/ | | \//////o / | | \XXXXXXo\\\\\\/ | | \\\\\\\o\\\\\\/ | | \///// \ / | | \XXXX/ \\\\\/ | | \\\\\/ \\\\\/ | | o--o o--o | | o--o o--o | | o--o o--o | | | | | | | o-----------------------o o-----------------------o o-----------------------o = du' @ u (v) o-----------------------o dv' @ u (v) = = | dU' | = = | o--o o--o | = = | /////\ /\\\\\ | = = | ///////o\\\\\\\ | = = | ////////X\\\\\\\\ | = = | o///////XXX\\\\\\\o | = = | |/////oXXXXXo\\\\\| | = = = = = = = = = = = =|/du'/|XXXXX|\dv'\|= = = = = = = = = = = | |/////oXXXXXo\\\\\| | | o//////\XXX/\\\\\\o | | \//////\X/\\\\\\/ | | \//////o\\\\\\/ | | \///// \\\\\/ | | o--o o--o | | | o-----------------------o o-----------------------o o-----------------------o o-----------------------o | dU | | dU | | dU | | o--o o--o | | o--o o--o | | o--o o--o | | / \ / \ | | /\\\\\ /\\\\\ | | /\\\\\ /\\\\\ | | / o \ | | /\\\\\\o\\\\\\\ | | /\\\\\\o\\\\\\\ | | / / \ \ | | /\\\\\\/ \\\\\\\\ | | /\\\\\\/ \\\\\\\\ | | o / \ o | | o\\\\\\/ \\\\\\\o | | o\\\\\\/ \\\\\\\o | | | o o | | | |\\\\\o o\\\\\| | | |\\\\\o o\\\\\| | | | du | | dv | | | |\\\\\| |\\\\\| | | |\du\\| |\\dv\| | | | o o | | | |\\\\\o o\\\\\| | | |\\\\\o o\\\\\| | | o \ / o | | o\\\\\\\ /\\\\\\o | | o\\\\\\\ /\\\\\\o | | \ \ / / | | \\\\\\\\ /\\\\\\/ | | \\\\\\\\ /\\\\\\/ | | \ o / | | \\\\\\\o\\\\\\/ | | \\\\\\\o\\\\\\/ | | \ / \ / | | \\\\\/ \\\\\/ | | \\\\\/ \\\\\/ | | o--o o--o | | o--o o--o | | o--o o--o | | | | | | | o-----------------------o o-----------------------o o-----------------------o = du' @ u v o-----------------------o dv' @ u v = = | dU' | = = | o--o o--o | = = | /////\ /\\\\\ | = = | ///////o\\\\\\\ | = = | ////////X\\\\\\\\ | = = | o///////XXX\\\\\\\o | = = | |/////oXXXXXo\\\\\| | = = = = = = = = = = = =|/du'/|XXXXX|\dv'\|= = = = = = = = = = = | |/////oXXXXXo\\\\\| | | o//////\XXX/\\\\\\o | | \//////\X/\\\\\\/ | | \//////o\\\\\\/ | | \///// \\\\\/ | | o--o o--o | | | o-----------------------o o-----------------------o o-----------------------o o-----------------------o | U | |\U\\\\\\\\\\\\\\\\\\\\\| |\U\\\\\\\\\\\\\\\\\\\\\| | o--o o--o | |\\\\\\o--o\\\o--o\\\\\\| |\\\\\\o--o\\\o--o\\\\\\| | /////\ /////\ | |\\\\\/////\\/////\\\\\\| |\\\\\/ \\/ \\\\\\| | ///////o//////\ | |\\\\///////o//////\\\\\| |\\\\/ o \\\\\| | /////////\//////\ | |\\\////////X\//////\\\\| |\\\/ /\\ \\\\| | o//////////\//////o | |\\o///////XXX\//////o\\| |\\o /\\\\ o\\| | |/////o/////o/////| | |\\|/////oXXXXXo/////|\\| |\\| o\\\\\o |\\| | |//u//|/////|//v//| | |\\|//u//|XXXXX|//v//|\\| |\\| u |\\\\\| v |\\| | |/////o/////o/////| | |\\|/////oXXXXXo/////|\\| |\\| o\\\\\o |\\| | o//////\//////////o | |\\o//////\XXX///////o\\| |\\o \\\\/ o\\| | \//////\///////// | |\\\\//////\X////////\\\| |\\\\ \\/ /\\\| | \//////o/////// | |\\\\\//////o///////\\\\| |\\\\\ o /\\\\| | \///// \///// | |\\\\\\/////\\/////\\\\\| |\\\\\\ /\\ /\\\\\| | o--o o--o | |\\\\\\o--o\\\o--o\\\\\\| |\\\\\\o--o\\\o--o\\\\\\| | | |\\\\\\\\\\\\\\\\\\\\\\\| |\\\\\\\\\\\\\\\\\\\\\\\| o-----------------------o o-----------------------o o-----------------------o = u' o-----------------------o v' = = | U' | = = | o--o o--o | = = | /////\ /\\\\\ | = = | ///////o\\\\\\\ | = = | ////////X\\\\\\\\ | = = | o///////XXX\\\\\\\o | = = | |/////oXXXXXo\\\\\| | = = = = = = = = = = = =|/u'//|XXXXX|\\v'\|= = = = = = = = = = = | |/////oXXXXXo\\\\\| | | o//////\XXX/\\\\\\o | | \//////\X/\\\\\\/ | | \//////o\\\\\\/ | | \///// \\\\\/ | | o--o o--o | | | o-----------------------o Figure 70-b. Tangent Functor Ferris Wheel for F = <((u)(v)), ((u, v))> Jon Awbrey o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at oakland.edu Sat Jun 7 22:32:54 2003 From: jawbrey at oakland.edu (Jon Awbrey) Date: Wed Jan 21 20:34:27 2004 Subject: [Inquiry] Re: Differential Logic References: <3E982563.F67459A5@oakland.edu> <3EB73352.259E602@oakland.edu> Message-ID: <3EE2AE66.D7523827@oakland.edu> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Note D84 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Epilogue, Enchoiry, Exodus | It is time to explain myself . . . . let us stand up. | | Walt Whitman, 'Leaves of Grass', [Whi, 79] o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at oakland.edu Sun Jun 8 08:58:10 2003 From: jawbrey at oakland.edu (Jon Awbrey) Date: Wed Jan 21 20:34:27 2004 Subject: [Inquiry] Examples! Examples! Examples! Message-ID: <3EE340F2.726DAFF7@oakland.edu> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o EEE. Note 1 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o I am going to use this thread to collect a few examples of ontology related materials from various literatures and other sources, focusing on those that are concrete and simple enough to subject to a complete analysis from several different practical and theoretical points of view. I also want to use this material to explore the issues of mappings and transformations between different theories and vocabularies, whether in the formally extracted object languages or the semi-formally employed embedding languages. The spirit of the exercise that I have in mind is ably captured by Jim Schoening's suggestion about a problem that has been with us from the very beginning: Jim Schoening wrote: | SUO Group, | | We appear to have different opinions as to what | type of language to use to express the SUO content. | | I suggest the best way to resolve this is through | proposed content, not debate or a vote at this time. | As content is proposed, it should be expressed in a | proposed language. Content proposed in one language | could be translated and resubmitted in another. | This will demonstrate the maturity and features | of each language. We might find middle ground | or a composite solution, but at the least we | will see which language builds consensus. | | Subj: SUO: What Language to Use | From: Schoening, James <...> | Date: Sat, 4 Nov 2000 07:51:16 -0500 | http://suo.ieee.org/email/msg01949.html o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at oakland.edu Sun Jun 8 11:16:44 2003 From: jawbrey at oakland.edu (Jon Awbrey) Date: Wed Jan 21 20:34:27 2004 Subject: [Inquiry] Re: Examples! Examples! Examples! References: <3EE340F2.726DAFF7@oakland.edu> Message-ID: <3EE3616C.D082D47E@oakland.edu> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o EEE. Note 2 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o I am going to start with zeroth order theories, because those are the kind that I know how to analyze rather completely and also to express in computable if not always tractable form. It should be rather trivial for anybody working with first order theories to re-write all of these examples in their favorite language. If they do this, then it would provide us with some concrete material to explore the issues of logical transformation, including the subtopics of (1) mappings between ontology theories, (2) theory formation, (3) theory transformation, (4) representation invariant ontologies (RIO's). Example 1. John Sowa's "Top Level Categories" The first example that comes to mind is John Sowa's "Top Level Categories", in the version that is here: http://www.bestweb.net/~sowa/ontology/toplevel.htm There are several different way to express this content in my favorite zeroth order language -- called the "Cactus Language" because its strings are parsed into the species of graphs that graph-theorists call "cacti". I will outline the first version that I worked out and then translate it into ordinary syntax. The "alphabet" !A! (also called the "basis", "lexicon", "vocabulary", ...), is a finite set of strings, that can be regarded either as "sentences" or as "terms" depending on the level of application. In the present example, they appear to be the names of what some people call "distinctive features" of things that exist, or things that are covered by the ontology. Here, !A! is a set of 25 terms, !A! = {a_1, ..., a_25} = {"Abstract", ..., "Structure"}. After this point, one tends to drop the quotation marks around strings, using them only when necessary to avoid a clear and present danger of confusion. | !A! | | a_1 = "Abstract" | a_2 = "Actuality" | a_3 = "Continuant" | a_4 = "Description" | a_5 = "Form" | a_6 = "History" | a_7 = "Independent" | a_8 = "Intention" | a_9 = "Juncture" | a_10 = "Mediating" | a_11 = "Nexus" | a_12 = "Object" | a_13 = "Occurrent" | a_14 = "Participation" | a_15 = "Physical" | a_16 = "Prehension" | a_17 = "Process" | a_18 = "Proposition" | a_19 = "Purpose" | a_20 = "Reason" | a_21 = "Relative" | a_22 = "Schema" | a_23 = "Script" | a_24 = "Situation" | a_25 = "Structure" The "(finite) axiomset" $A$ for this example is just a (finite) set of zeroth order axioms or propositional constraints. Inasmuch as we can consider all of these propositions to be conjoined (conjuncted?) into a single proposition, we are at liberty to view the whole thing as a "one-axiom theory", if it makes us feel any better to do so. What the axiom does is this. It removes from immediate consideration particular regions of the universe of discourse that might otherwise be thought to have content. I use a percent sign suffix and square brackets around the alphabet to denote the universe of discourse that is generated by a given alphabet, writing A% = [!A!] = [a_1, ..., a_n]. The universe of discourse is a two-layer object, consisting of the set of "positions" or "cells" in the universe, written A = <|!A!|> = <|a_1, ..., a_n|>, and also the set of "propositions", that is, the mappings from points in A to the boolean domain B = {0, 1}, often pictured as various ways of shading a venn diagram, written A^ = (A -> B) = {f : A -> B}. Looked at this way, the universe of discourse for the TLC example potentially has 2^25 positions and 2^(2^25) propositions, but the axiomset $A$, which we may think of as singling out a single one of these propositions, eliminates all but a certain number of cells from consideration. This informational constraint can be thought of as a type of quotient operation, whose result we may indicate as A%/$A$, commonly read as "A% mod $A$". To be continued ... Jon Awbrey o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at oakland.edu Sun Jun 8 15:56:04 2003 From: jawbrey at oakland.edu (Jon Awbrey) Date: Wed Jan 21 20:34:28 2004 Subject: [Inquiry] Re: Examples! Examples! Examples! References: <3EE340F2.726DAFF7@oakland.edu> <3EE3616C.D082D47E@oakland.edu> Message-ID: <3EE3A2E4.5E4F29D4@oakland.edu> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o EEE. Note 3 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Example 1. John Sowa's "Top Level Categories" (cont.) Recall the alphabet !A! = {a_1, ..., a_25} = {"Abstract", ..., "Structure"} that we enumerated last time. These terms form what some people call the "non-logical vocabulary" of the subject matter in question, and I suppose that's a useful enough term for the moment. When we view things purely graph-theoretically, these brands of labels will be called "paints". In the Cactus Language, which can be interpreted in a couple of different ways for zeroth order logic, the "logical vocabulary", if you can even call it that, consists of just four symbols, specifically, the blank space character " ", the left parenthesis "(", the comma ",", and the right parenthesis ")". When I use the graph-theoretic idiom, I tend to call these "marks", as in punctuation marks, or "measures". The two different interpretations that I mentioned correspond to what C.S. Peirce called the "Entitative" and the "Existential" interpretations of his graphical syntax. I will use the existential reading for this example. In the existential interpretation, the blank symbol stands for two things: 1. Taken by itself it represents a boolean value of 1 or true. 2. Placed between terms it serves as a conjunctive connective. The other symbols operate in tandem as a bracket of the form "( , , , )", with any finite number k = 0, 1, 2, 3, ... of intervening argument places. In the existential interpretation, this so-called "boundary bracket" has the following basis for grounding its meaning: 1. For k = 0. The expression "()" means "false". 2. For k > 0. The expression "(e_1, ..., e_k)" means that just one of the k expressions e_j is false. This gives us enough to write out an axiom for TLC. The first version I got just by looking at the lattice diagram and trying to figure out what it meant in terms of logical constraints on the logical features that are represented by the terms of the alphabet !A!. Table 1. TLC in Cactus Language (Version 1) o------------------------------------------------------------------------o | | | (( Object ),( Process ),( Schema ),( Script ), | | ( Juncture ),( Participation ),( Description ),( History ), | | ( Structure ),( Situation ),( Reason ),( Purpose )) | | | | ( Independent ,( Actuality ),( Form )) | | ( Relative ,( Prehension ),( Proposition )) | | ( Mediating ,( Nexus ),( Intention )) | | | | ( Physical ,( Actuality ),( Prehension ),( Nexus )) | | ( Abstract ,( Form ),( Proposition ),( Intention )) | | | | ( Continuant ,( Object ),( Schema ),( Juncture ), | | ( Description ),( Structure ),( Reason )) | | | | ( Occurrent ,( Process ),( Script ),( Participation ), | | ( History ),( Situation ),( Purpose )) | | | | ( Actuality ,( Object ),( Process )) | | ( Form ,( Schema ),( Script )) | | ( Prehension ,( Juncture ),( Participation )) | | ( Proposition ,( Description ),( History )) | | ( Nexus ,( Structure ),( Situation )) | | ( Intention ,( Reason ),( Purpose )) | | | o------------------------------------------------------------------------o To read this, one can think this way: 1. An expression of the form (x_1, x_2, ..., x_k), if you take it as asserting or imposing a logical constraint on the argument features x_1, ..., x_k, means that just one of the x_j is false. 2. In particular, (x) means that x is false. 3. Consequently, an expression of the form ((x_1),(x_2), ..., (x_k)) means that exactly one of the expressions (x_1),(x_2), ..., (x_k) is false, which means that just one of the x_j is true. This is precisely equivalent to saying that the universe is partitioned into mutually exclusive and exhaustive classes each of which falls under exactly one term of the list x_1, x_2, ..., x_k. 4. Now look at expressions of the form (x, (x_1),(x_2), ..., (x_k)), where the term "x" appears at depth one and all the rest of the terms x_1, ..., x_k appear at depth two. Consider the cases: a. Suppose x is true. Then the expression reduces to the form of a partition ((x_1),(x_2), ..., (x_k)). b. Suppose x is false. Then it's the only one that is, and the expression reduces to (x_1)(x_2) ... (x_k), the conjunction of the negations of the other terms. In effect, one has said that the things for which the "genus" term x is true are partitioned among the "species" terms x_1, x_2, ..., x_k, and the the things for which x is false can have none of the terms x_1, x_2, ..., x_k apply to them. To sum it up, we can read a clause like: | (( Object ),( Process ),( Schema ),( Script ), | ( Juncture ),( Participation ),( Description ),( History ), | ( Structure ),( Situation ),( Reason ),( Purpose )) as asserting or imposing an absolute partition on the universe, and we can read each one of the following clauses: | ( Independent ,( Actuality ),( Form )) | ( Relative ,( Prehension ),( Proposition )) | ( Mediating ,( Nexus ),( Intention )) | | ( Physical ,( Actuality ),( Prehension ),( Nexus )) | ( Abstract ,( Form ),( Proposition ),( Intention )) | | ( Continuant ,( Object ),( Schema ),( Juncture ), | ( Description ),( Structure ),( Reason )) | | ( Occurrent ,( Process ),( Script ),( Participation ), | ( History ),( Situation ),( Purpose )) | | ( Actuality ,( Object ),( Process )) | ( Form ,( Schema ),( Script )) | ( Prehension ,( Juncture ),( Participation )) | ( Proposition ,( Description ),( History )) | ( Nexus ,( Structure ),( Situation )) | ( Intention ,( Reason ),( Purpose )) as asserting or imposing a relative partition on the universe. Jon Awbrey o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at oakland.edu Sun Jun 8 23:00:04 2003 From: jawbrey at oakland.edu (Jon Awbrey) Date: Wed Jan 21 20:34:28 2004 Subject: [Inquiry] Re: Examples! Examples! Examples! References: <3EE340F2.726DAFF7@oakland.edu> <3EE3616C.D082D47E@oakland.edu> <3EE3A2E4.5E4F29D4@oakland.edu> Message-ID: <3EE40644.BA8EA54D@oakland.edu> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o EEE. Note 4 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Example 1. John Sowa's "Top Level Categories" (cont.) So far in our formalization of TLC, we have the following ingredients: 1. An alphabet !A! = {a_1, ..., a_25} of 25 descriptives terms. 2. An alphabet !M! = {m_1, m_2, m_3, m_4} = {" ", "(", ",", ")"} of punctuation marks. 3. The full alphabet !B! = !B!(!A!) = !A! |_| !M!. 4. The things that we think of as "sentences" or (propositional) "expressions" are strings in a formal language L(!A!) that are formed from the elements of !B!(!A!) according to the prescriptions of a certain formal grammar. 5. In particular, the axiom that I gave for (my present understanding of) TLC, which is the sole member of its axiomset $A$, is a particular sentence !a! in L(!A!) c !B!*. Namely, !a! is the string displayed in Table 1, here: http://suo.ieee.org/email/msg09670.html Aside. It's not really important right now, but more details about the syntax, semantics, and pragmatics of the Cactus Language can be found here: http://www.altheim.com/cs/cactus.html This is the sort of computationally finitary specificity that we should expect to get when we ask questions about what's in the box of this or that module of this or that lattice of theories. References to theories in general, which are, in general, infinite sets of sentences, just won't cut it when it comes down to the e-bearing wire. Well, that was a little more formal detail than I had in mind for this thread, which I will try to keep more to the intuition building types of concrete observations. Jon Awbrey o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at oakland.edu Mon Jun 9 10:08:54 2003 From: jawbrey at oakland.edu (Jon Awbrey) Date: Wed Jan 21 20:34:28 2004 Subject: [Inquiry] Re: Examples! Examples! Examples! References: <3EE340F2.726DAFF7@oakland.edu> <3EE3616C.D082D47E@oakland.edu> <3EE3A2E4.5E4F29D4@oakland.edu> <3EE40644.BA8EA54D@oakland.edu> Message-ID: <3EE4A306.83620651@oakland.edu> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o EEE. Note 5 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Generalities: Informing the Media There are a few dimensions or distinctions of a practical nature that we need to keep an eye out for as we work through this set of examples. I will discuss these issues here in no particular order. 1. Casual/Formal There is the distinction between the "casual" and the "formal", in other words, the line that we draw somewhere on a spectrum that stretches from the haunts of the informal, the partially formalized, and the possibly even never wholly formalizable, into the regions of the completely formalized formal object of all-around reflection and enough-in-depth contemplation. Of course, we should not expect to formalize this boundary -- it's a slippery slope in both directions -- but it's still a good idea to maintain a rough sense of orientation with respect to it. A useful guideline is provided by keeping track of the number of elements in a given space. Spaces that have infinite or very large cardinalities are things that we may hope to denote or to mention in formal terms, even if many times we only wave our hands at them, but we do not come close to touching them the way that we can manage and manipulate finitary signs. For example, the numbers of positions and propositions in the TLC example are 2^25 = 33,554,432 and 2^(2^25) = 2^(33,554,432), respectively. A person might just consider making a truth table with 33 million rows of binary vectors, but is not likely to finish filling it in with all of the possible columns of boolean functions. 3. Mention/Use Let me adopt the familiar distinction between mention and use and use it in a way that has pragmatic computational imports. In the sense that I have in mind, we use the finitary signs that are within our grasp to mention the more intangible objects that are within our reach only in a mediate or a virtual sense. 4. Object/Sign That brings up the distinction between objects and signs, which is another one of those overlapping waves on the shoreline types of distinctions, since some things can be signs one moment and then objects the next, for instance, as we reflect on them and even formalize them to some degree of adequacy, while other things, though mentionable, are hardly a bit more than that, leaving us in the waving hands sort of usage with regard to them. 5. Generator/Relation In algebra, generally speaking, things that increase the number of elements in a space are called "generators" while things that decrease the number of elements in a space are called "relations", with a specific technical sense that is usually understood in situ, but if it isn't, then the word "relator" may be used in its place. A specification of an algebraic structure in terms of generators and relations is called a "presentation", and is customarily presented as a divided set {g_1, ..., g_m | r_1, ..., r_n}, with two finite sets, of generators g_i and relators r_j. For example, the TLC alphabet !A! = {a_1, ..., a_25} is a set of generators that can be thought of as generating the space of positions A = <|!A!|> = <|a_1, ..., a_25|>, using brackets of the form "<|...|>" as "generator bars", and inducing the generation of the space of propositions A^ = (A -> B) = {f : A -> B}, while the TLC axiom !a! in Table 1 can be thought of as a relator that shrinks both of these spaces more effectively than a $120/hr analyst. However, in the kinds of examples that we are concerned with, there are actually two sorts of spaces to worry about, which brings to the forefront the next dimension or distinction. 6. Model/Theory This is really just a special case of the object/sign distinction, but it's a good idea to emphasize it once more in this particular context of use. In the TLC example, the "model space" is the universe of discourse A% = [!A!] = [a_1, ..., a_25] that combines the positions of A with the propositions of A^ = (A -> B). The "syntactic space" is the formal language L = L(!A!) c (!A! |_| !M!)* of well-formed strings in the Cactus Language, or pick your own favorite language for the task, that we may variously describe as "expressions", "formulas", "sentences", "terms", "wffs", or whatever fits the moment. A "zeroth order theory" (ZOT) is an arbitrary subset of L. So we have, just for starters, at least two lattices: 1. The model lattice is (Pow(A), c), ordered by set-theoretic inclusion or the "contained as a subset" relation that is here denoted by "c". This is isomorphic to the proposition lattice (A^, =>) that consists of the propositions f in (A -> B), ordered by the logical implication relation that is here denoted by "=>". 2. The theory lattice is (Pow(L), c), ordered by inclusion as usual. But wait, there's more ... Jon Awbrey o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at oakland.edu Mon Jun 9 14:24:09 2003 From: jawbrey at oakland.edu (Jon Awbrey) Date: Wed Jan 21 20:34:28 2004 Subject: [Inquiry] Re: Examples! Examples! Examples! References: <3EE340F2.726DAFF7@oakland.edu> <3EE3616C.D082D47E@oakland.edu> <3EE3A2E4.5E4F29D4@oakland.edu> <3EE40644.BA8EA54D@oakland.edu> <3EE4A306.83620651@oakland.edu> Message-ID: <3EE4DED9.D3969CB8@oakland.edu> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o EEE. Note 6 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Example 1. John Sowa's "Top Level Categories" (cont.) Let's do a survey of the different sorts of lattices and quotient operations that we can find in the TLC example. I will leave the language just a bit loose at this point -- we can settle later on what we want to call "the" most pertinent lattice or quotient in each case. One thing is clear, though, all of these objects and operations exist, either in the platonic or sublunar environment, no matter what we decide to call them at the end of the day, and so it will be necessary to devote some attention to each of them, according to the parts they severally play in understanding the example. Further, I will continue with the common sort of casually constructive "models on the cuff" or "model theory on the installment plan" that is customary among mathematical and computer science folks, beginning with a working knowledge of sets, functions, relations, categories, and so on, and giving what passes for constructions of each and every object or operator that comes into play. For the sake of supporting an intuition that is soon to worked to the max, let's draw some pictures for the case of an alphabet with just one term !X!(1) = {x_1}, for convenience using the nickname x = x_1. The space X = <|!X!|> = <|x_1|> = <|x|> has 2^1 = 2 cells or positions, pictured in the venn diagram of Figure 2. o-------------------------------------------------o | X | | | | o-------------o | | / \ | | / \ | | / \ | | / \ | | o o | | | | | | (x) | x | | | | | | | o o | | \ / | | \ / | | \ / | | \ / | | o-------------o | | | | | o-------------------------------------------------o Figure 2. One-Dimensional Universe Of Discourse We can enumerate the 2 positions of X either in the coordinated form, as X = {<0>, <1>}, or in the conjunctive form, as X = {(x), x }. The most obvious lattice that arises from the space X is the "power set" lattice Pow(X) = 2^X = {W : W c X}, ordered by the set-theoretic inclusion relation "c". Since this is isomorphic to the proposition lattice (X^, =>), ordered by the logical implication "=>", the basic structure of both lattices is captured by the 2-ptych of 4-gons depicted in Figure 3. o-------------------------------------o-------------------------------------o | | | | X | 1 : X -> B | | ((u), u } | (( )) | | @ | @ | | / \ | / \ | | / \ | / \ | | / \ | / \ | | / \ | / \ | | / \ | / \ | | / \ | / \ | | / \ | / \ | | {(u)} @ @ { u } | (u) @ @ u | | \ / | \ / | | \ / | \ / | | \ / | \ / | | \ / | \ / | | \ / | \ / | | \ / | \ / | | \ / | \ / | | @ | @ | | | ( ) | | { } | 0 : X -> B | | | | o-------------------------------------o-------------------------------------o Figure 3. Lattice of Subsets Pow(X) and Lattice of Propositions X^ = (X -> B) Details, Details, ... Jon Awbrey o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at oakland.edu Mon Jun 9 16:20:51 2003 From: jawbrey at oakland.edu (Jon Awbrey) Date: Wed Jan 21 20:34:28 2004 Subject: [Inquiry] Re: Examples! Examples! Examples! References: <3EE340F2.726DAFF7@oakland.edu> <3EE3616C.D082D47E@oakland.edu> <3EE3A2E4.5E4F29D4@oakland.edu> <3EE40644.BA8EA54D@oakland.edu> <3EE4A306.83620651@oakland.edu> <3EE4DED9.D3969CB8@oakland.edu> Message-ID: <3EE4FA33.EFD1998E@oakland.edu> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o EEE. Note 7 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Example 1. John Sowa's "Top Level Categories" (cont.) Generally speaking, if expressing it very roughly, a quotient operation is the act of ignoring certain distinctions that might otherwise be made. For example, given the integers Z = {..., -2, -1, 0, 1, 2, ...}, taking the "integers modulo 2", written Z_2 ~=~ {0, 1}, amounts to neglecting the distinctions that are otherwise made between the members of the set 2Z = {..., -4, -2, 0, 2, 4, ...}, commonly known as "even integers", and consequently collapsing the distinctions that are normally made between the members of the set 2Z + 1 = {..., -3, -1, 1, 3, ...}, commonly known as "odd integers", and thereby resulting in but a single distinction, that between the "equivalence classes" or the "residue classes" of Evens and Odds, respectively. Drawing on the primer of the 1-dimensional universe X% = [x], we may derive some clue what the spaces Pow(A) and A^ look like as lattices: Pow(A) has the set A at the top and the empty set {} at the bottom, while A^ has the constantly true proposition (()) = 1 : A -> B at the top and the constantly false proposition () = 0 : A -> B at the bottom. Two down, 2^(2^25) - 2 to go! Which goes a long way to explain why we find the delicately balanced opposition between generators and relations to be an essential tension of the subject. Jon Awbrey o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at oakland.edu Mon Jun 9 20:04:23 2003 From: jawbrey at oakland.edu (Jon Awbrey) Date: Wed Jan 21 20:34:28 2004 Subject: [Inquiry] Re: Examples! Examples! Examples! References: <3EE340F2.726DAFF7@oakland.edu> <3EE3616C.D082D47E@oakland.edu> <3EE3A2E4.5E4F29D4@oakland.edu> <3EE40644.BA8EA54D@oakland.edu> <3EE4A306.83620651@oakland.edu> <3EE4DED9.D3969CB8@oakland.edu> <3EE4FA33.EFD1998E@oakland.edu> Message-ID: <3EE52E97.598A699D@oakland.edu> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o EEE. Note 8 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Example 1. John Sowa's "Top Level Categories" (cont.) Just for orientation, Figure 4 sketches the hi and lo points of the "unreduced extensional lattices" for the TLC example, namely, the lattice of subsets Pow(A) and the lattice of propositions A^. I have written the coordinate 25-tuples of A as commafree strings. o-------------------------------------o-------------------------------------o | | | | A | (( )) | | | | | <|a_1, ..., a_25|> | 1 : A -> B | | | | | {<0000000000000000000000000>, | {<0000000000000000000000000> ~> 1, | | <0000000000000000000000001>, | <0000000000000000000000001> ~> 1, | | ... | ... | | <1111111111111111111111110>, | <1111111111111111111111110> ~> 1, | | <1111111111111111111111111>} | <1111111111111111111111111> ~> 1} | | | | | @ | @ | | / \ | / \ | | / \ | / \ | | / \ | / \ | | / \ | / \ | | / 2^A \ | / A^ \ | | / \ | / \ | | / Pow (A) \ | / (A --> B) \ | | ... ... | ... ... | | \ 2^(2^25) / | \ 2^(2^25) / | | \ subsets / | \ functions / | | \ / | \ / | | \ / | \ / | | \ / | \ / | | \ / | \ / | | \ / | \ / | | @ | @ | | | | | { } | ( ) | | | | | | 0 : A -> B | | | | | | {<0000000000000000000000000> ~> 0, | | | <0000000000000000000000001> ~> 0, | | | ... | | | <1111111111111111111111110> ~> 0, | | | <1111111111111111111111111> ~> 0} | | | | o-------------------------------------o-------------------------------------o Figure 4. Lattice of Subsets Pow(A) and Lattice of Propositions A^ = (A -> B) That should be enough, lattice hope, about the spaces of "interpretations", truth value assignments, or propositional models for a while. Next we need to establish a base camp for the assault on the summit of TLC's theory space. Jon Awbrey o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at oakland.edu Tue Jun 10 10:48:02 2003 From: jawbrey at oakland.edu (Jon Awbrey) Date: Wed Jan 21 20:34:28 2004 Subject: [Inquiry] Re: Examples! Examples! Examples! References: <3EE340F2.726DAFF7@oakland.edu> <3EE3616C.D082D47E@oakland.edu> <3EE3A2E4.5E4F29D4@oakland.edu> <3EE40644.BA8EA54D@oakland.edu> <3EE4A306.83620651@oakland.edu> <3EE4DED9.D3969CB8@oakland.edu> <3EE4FA33.EFD1998E@oakland.edu> <3EE52E97.598A699D@oakland.edu> Message-ID: <3EE5FDB2.94ABBDEF@oakland.edu> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o EEE. Note 9 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o What's Your Semiotic Quotient? We are about to undertake the practice of two "forms of ignorance" (FOI's) -- now "Ignoring Stuff" is everybody's information management tool of choice, at least quasi modo infantes, but like every faith that entertains service to a duplicity of masters it is crucial to observe, with all due constancy and fidelity, exactly which is which. FOI 1. There's the stuff that we ignore because we just can't help it. We are faced with, entangled by a situation of data overload -- I don't call it information overload, because information, if we really had it, would be the cure. In the picture of TLC that I drew in Figure 4, this is represented by the ellipses "..." that are drawn in lieu of the big middle regions of the extensional lattices. FOI 2. There are the distinctions that we have deliberately decided to collapse, ignore, neglect, or quotient out of mind, out of sight, for the time being, or for the sake of the immediate end in view. The latter more discerning form of ignorance is what we need to take up next. I will approach the syntactic and theoretical domains of the TLC example in a series of stages, using the few pieces of scaffolding that I usually use for these types of problems, as indicated in the next couple of Figures. o-----------------------------o-----------------------------o | Object Domain | Syntactic Domain | o-----------------------------o-----------------------------o | | | o-----------o | | o~~~~~~~~~~~~~~~~~~~~~~~~/| s s s ... |\ | | / \ / o-----------o \ | | / \ / \ | | / \ o-----------o \ | | o~~~~~~~~~~~~~~~~~~~~~~~~~| s s s ... | \ | | \ \ o-----------o \ | | \ \ \ \ | | \ \ \ o-----------o | | \ o~~~~~~~~~~~~~~~~\~~~~~~~~| s s s ... | | | \ / \ o-----------o | | \ / \ / | | \ / \ o-----------o / | | o~~~~~~~~~~~~~~~~~~~~~~~~\| s s s ... |/ | | o-----------o | | | o-----------------------------------------------------------o Figure 5. Lattice of Objects Inducing a Partition of Signs Figure 5 depicts the generic situation, where we form a quotient in a space of signs, that is, any species of syntactic elements, in a way that corresponds to the elements in a space of objects, in this case organized in the structure of a lattice ordering. o--------------------------o---------------o--------------------------o | Syntactic Domain 1 | Object Domain | Syntactic Domain 2 | o--------------------------o---------------o--------------------------o | | | o-----------o o-----------o | | /| s s s ... |\~~~~~~~~~~~~~o~~~~~~~~~~~~~/| s s s ... |\ | | / o-----------o \ / \ / o-----------o \ | | / \ / \ / \ | | o-----------o \ / \ o-----------o \ | | | s s s ... |~~~~~~~~\~~~~~o~~~~~~~\~~~~~~| s s s ... | \ | | o-----------o \ \ \ o-----------o \ | | \ \ \ \ \ \ | | \ o-----------o \ \ \ o-----------o | | \ | s s s ... |~~~~~~\~~~~~~~o~~~~~\~~~~~~~~| s s s ... | | | \ o-----------o \ / \ o-----------o | | \ / \ / \ / | | \ o-----------o / \ / \ o-----------o / | | \| s s s ... |/~~~~~~~~~~~~~o~~~~~~~~~~~~~\| s s s ... |/ | | o-----------o o-----------o | | | o---------------------------------------------------------------------o Figure 6. Lattice of Objects Inducing a Diversity of Sign Partitions Figure 6 illustrates one potential direction of thickening plot, where we have two different languages or systems of signs, each of them falling into equivalence classes of signs according to their separate notions of equivalence. If these equivalence classes correspond to objects of the relevant object domain, the way that I have drawn them in the Figure, then they are called "referential equivalence classes" (REC's). However, it would also be possible to form up equivalence classes of signs according to any plan that strikes one's fancy, whether it was modeled on objective reality or not, and these more capricious sorts of equivalence classes are referred to as "semiotic equivalence classes" (SEC's). Jon Awbrey o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at oakland.edu Tue Jun 10 21:40:46 2003 From: jawbrey at oakland.edu (Jon Awbrey) Date: Wed Jan 21 20:34:28 2004 Subject: [Inquiry] Re: Examples! Examples! Examples! References: <3EE340F2.726DAFF7@oakland.edu> <3EE3616C.D082D47E@oakland.edu> <3EE3A2E4.5E4F29D4@oakland.edu> <3EE40644.BA8EA54D@oakland.edu> <3EE4A306.83620651@oakland.edu> <3EE4DED9.D3969CB8@oakland.edu> <3EE4FA33.EFD1998E@oakland.edu> <3EE52E97.598A699D@oakland.edu> <3EE5FDB2.94ABBDEF@oakland.edu> Message-ID: <3EE696AE.6A3BE29C@oakland.edu> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o EEE. Note 10 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Logical Equivalence Classes To flesh out the sketches that I drew last time, intended to suggest how quotient structures on spaces of signs can mirror the forms of spaces of objects, Figure 7 fills in a sample of the concrete details for two particular languages for zeroth order logic, Language 1 being one of the more familiar syntaxes and Language 2 being the Cactus Language earlier described. o-----------------------------o-------------------o-----------------------------o | Language 1 | Object Domain | Language 2 | o-----------------------------o-------------------o-----------------------------o | | | o----------o o----------o | | /| "T" |\ 1 /| " " |\ | | / | "x => x" |~\~~~~~~~~~~~~~~~~o~~~~~~~~~~~~~~~~/~| "(x(x))" | \ | | / | ... | \ / \ / | ... | \ | | / o----------o \ / \ / o----------o \ | | / \ / \ / \ | | / o----------o / \ / o----------o | | / | "x" | / \ x / | "x" | | | / | "T => x" |~~~~~~/~~~~~~~~~~~o~~~~/~~~~~~~~~~~~~| "((x))" | | | / | | / / / | ... | | | o----------o o----------o / / o----------o o----------o | | | "~x" | / / / | "(x)" | / | | | "x => F" |~~~~~~~~~~~~~/~~~~o~~~~~~~~~~~/~~~~~~| "(x(()))"| / | | | ... | / (x) \ / | | / | | o----------o / \ / o----------o / | | \ / \ / \ / | | \ o----------o / \ / \ o----------o / | | \ | "F" | / \ / \ | "()" | / | | \ | "x & ~x" |~/~~~~~~~~~~~~~~~~o~~~~~~~~~~~~~~~~\~| "x(x)" | / | | \| ... |/ 0 \| ... |/ | | o----------o o----------o | | | o-------------------------------------------------------------------------------o Figure 7. Lattice of Objects Inducing a Diversity of Sign Partitions In this illustration, the object structure is the lattice of four propositions (X^, =>) that inheres in the 1-dimensional universe of discourse X% = [x], and the quotient structures on the syntactic spaces are induced by the equivalence relation of logical equivalence (<=>). Whether these amount to REC's or SEC's is a little bit of a chicken or the egg question -- for the time being it will do to call them "logical equivalence classes" (LEC's). One of the things that we ought to observe right off is that the correspondence from signs to objects is infinity-to-one in its cardinality. Jon Awbrey o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at oakland.edu Wed Jun 11 10:40:13 2003 From: jawbrey at oakland.edu (Jon Awbrey) Date: Wed Jan 21 20:34:28 2004 Subject: [Inquiry] Re: Examples! Examples! Examples! References: <3EE340F2.726DAFF7@oakland.edu> <3EE3616C.D082D47E@oakland.edu> <3EE3A2E4.5E4F29D4@oakland.edu> <3EE40644.BA8EA54D@oakland.edu> <3EE4A306.83620651@oakland.edu> <3EE4DED9.D3969CB8@oakland.edu> <3EE4FA33.EFD1998E@oakland.edu> <3EE52E97.598A699D@oakland.edu> <3EE5FDB2.94ABBDEF@oakland.edu> <3EE696AE.6A3BE29C@oakland.edu> Message-ID: <3EE74D5D.A734821@oakland.edu> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o EEE. Note 11 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Intermezzo The time has come -- to speak of many things? no that's the other introit -- to assemble the players on a single stage, and perhaps to qualify a few of their names in the interest of more robust unions in future transformances. I do not apologize for the fact that I must appear to focus so much on the understudies, when there are such superstars to be seen, as I have learned that a recursive performance smashes or crashes on its foots and its steps. To preserve the characters "A" and "X" from the career blights of excessive type-casting, let's reissue the 1-dimensional universe formerly known as X% under a more stock title, say, X1% = [x_1], relegating it to the nuances of context whether it is permissible to call it under the nickname of X% = [x], and let's recall the alphabet of TLC as !TLC! = {a_1, ..., a_25}, assigning the now formal variables a_j to the same collection of distinctive features as they had before, but repackaging the universe of discourse A% under TLC%. Further, to conform to a previously established practice, I will designate the set of "painted cacti" on the set of "paints" !P! = {p_1, ..., p_n} as !C! = !C!(!P!) c !A!*, where "!A!" reverts to its generic use for denoting a contextually relevant alphabet, in this context giving !A! = !M! |_| !P|, where !M! is the set of marks mentioned before, !M! = {" ", "(", ",", ")"}, and where !P! = !X! or !P! = !TLC!, as the case may be. I know this is tedious, but computers are so darn picky about this stuff. Jon Awbrey o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at oakland.edu Wed Jun 11 15:52:08 2003 From: jawbrey at oakland.edu (Jon Awbrey) Date: Wed Jan 21 20:34:28 2004 Subject: [Inquiry] Re: Examples! Examples! Examples! References: <3EE340F2.726DAFF7@oakland.edu> <3EE3616C.D082D47E@oakland.edu> <3EE3A2E4.5E4F29D4@oakland.edu> <3EE40644.BA8EA54D@oakland.edu> <3EE4A306.83620651@oakland.edu> <3EE4DED9.D3969CB8@oakland.edu> <3EE4FA33.EFD1998E@oakland.edu> <3EE52E97.598A699D@oakland.edu> <3EE5FDB2.94ABBDEF@oakland.edu> <3EE696AE.6A3BE29C@oakland.edu> <3EE74D5D.A734821@oakland.edu> Message-ID: <3EE79678.FBF11EA2@oakland.edu> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o EEE. Note 12 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Now you knew that a Peircean could not be content with a 2-ptych. So here is the 3-ptych that puts the 1-dimensional space X1 into the line-up with the lattice X1^ = (X1 -> B) ~=~ Pow(X1) and the quotient lattice !C!(!P!)/((,)), where !P! = {p_1} = {"x"}, plus or minus a sprinkling of quotation marks syntactically salted or sugared according to the arguments of one's idiosyntactic taste. o-------------------o-----------------------------o-----------------------------o | | | | | | | o-----------o | | | | /| " " |\ | | | o | / | "(x(x))" | \ | | | / \ | / | ... | \ | | | / \ | / o-----------o \ | | | / \ | / \ | | | / \ | / \ | | | / \ | o-----------o o-----------o | | | / \ | | "(x)" | | "x" | | | (x) o---o x | (x) o o x | | "(x(()))" | | "((x))" | | | | \ / | | ... | | ... | | | | \ / | o-----------o o-----------o | | | \ / | \ / | | | \ / | \ / | | | \ / | \ o-----------o / | | | \ / | \ | "()" | / | | | o | \ | "x(x)" | / | | | | \| ... |/ | | | ( ) | o-----------o | | | | | o-------------------o-----------------------------o-----------------------------o Figure 8. Chart Target X1, Object Lattice X1^, Semiotic Quotient !C!("x")/((,)) Let's not worry too much about the names, as I am having to neologize them as I go. Jon Awbrey o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at oakland.edu Wed Jun 11 23:40:50 2003 From: jawbrey at oakland.edu (Jon Awbrey) Date: Wed Jan 21 20:34:29 2004 Subject: [Inquiry] Re: Examples! Examples! Examples! References: <3EE340F2.726DAFF7@oakland.edu> <3EE3616C.D082D47E@oakland.edu> <3EE3A2E4.5E4F29D4@oakland.edu> <3EE40644.BA8EA54D@oakland.edu> <3EE4A306.83620651@oakland.edu> <3EE4DED9.D3969CB8@oakland.edu> <3EE4FA33.EFD1998E@oakland.edu> <3EE52E97.598A699D@oakland.edu> <3EE5FDB2.94ABBDEF@oakland.edu> <3EE696AE.6A3BE29C@oakland.edu> <3EE74D5D.A734821@oakland.edu> <3EE79678.FBF11EA2@oakland.edu> Message-ID: <3EE80452.61D73F6F@oakland.edu> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o EEE. Note 13 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Example 1. John Sowa's "Top Level Categories" (cont.) Returning to the main attraction of the TLC example, the thing that I would like to really understand as thoroughly as possible is how exactly the adoption, the assertion, or the assumption of a particular sentence in the role of an axiom modulates the space of models, that is, the space that is via the medium of that act being conceived. For ease of reference the details of the formalization are repeated here in their revised formulation. | The Alphabet !TLC! | | a_1 = "Abstract" | a_2 = "Actuality" | a_3 = "Continuant" | a_4 = "Description" | a_5 = "Form" | a_6 = "History" | a_7 = "Independent" | a_8 = "Intention" | a_9 = "Juncture" | a_10 = "Mediating" | a_11 = "Nexus" | a_12 = "Object" | a_13 = "Occurrent" | a_14 = "Participation" | a_15 = "Physical" | a_16 = "Prehension" | a_17 = "Process" | a_18 = "Proposition" | a_19 = "Purpose" | a_20 = "Reason" | a_21 = "Relative" | a_22 = "Schema" | a_23 = "Script" | a_24 = "Situation" | a_25 = "Structure" !A!(!TLC!) = !TLC! |_| !M!, where !M! = {" ", "(", ",", ")"}. !C!(!TLC!) c !A!*, in accord with the applicable grammar. $A$(!TLC!) = {!a!_1} = {the axiom inscribed in Table 1}. Table 1. TLC in Cactus Language (Version 1) o-----------------------------------------------------------------------o | | | (( Object ),( Process ),( Schema ),( Script ), | | ( Juncture ),( Participation ),( Description ),( History ), | | ( Structure ),( Situation ),( Reason ),( Purpose )) | | | | ( Independent ,( Actuality ),( Form )) | | ( Relative ,( Prehension ),( Proposition )) | | ( Mediating ,( Nexus ),( Intention )) | | | | ( Physical ,( Actuality ),( Prehension ),( Nexus )) | | ( Abstract ,( Form ),( Proposition ),( Intention )) | | | | ( Continuant ,( Object ),( Schema ),( Juncture ), | | ( Description ),( Structure ),( Reason )) | | | | ( Occurrent ,( Process ),( Script ),( Participation ), | | ( History ),( Situation ),( Purpose )) | | | | ( Actuality ,( Object ),( Process )) | | ( Form ,( Schema ),( Script )) | | ( Prehension ,( Juncture ),( Participation )) | | ( Proposition ,( Description ),( History )) | | ( Nexus ,( Structure ),( Situation )) | | ( Intention ,( Reason ),( Purpose )) | | | o-----------------------------------------------------------------------o We are, in effect, back at the observation that I made at the beginning: | Looked at this way, the universe of discourse for the TLC example | potentially has 2^25 positions and 2^(2^25) propositions, but the | axiomset $A$, which we may think of as singling out a single one | of these propositions, eliminates all but a certain number of | cells from consideration. This informational constraint can | be thought of as a sort of quotient operation, whose result | we can indicate as TLC%/$A$, commonly read as "A% mod $A$". But what is most earnestly desired here is a way of intuitively visualizing, and with a bit of luck quickly computing, the indicated quotient operation. The mind continues to boggle, and when that happens I am usually reduced to looking at a reduced example or else to drawing on experience with related objects. If it were a mathematical "group" G, and I was trying to compute or at least to intuit the "quotient group" G/K, which makes sense "modulo" a "normal subgroup" K, then there are a couple of pictures that would leap to mind, one of them being category-theoretic and the other being by analogy geometric. The latter I would sketch this way: G o |\ | \ | \ K o \ |\ o G/K | \ | | \ | | \| 1 o----o 1 This makes salient the analogy or the proportion where G : K :: (G/K) : 1 and G : (G/K) :: K : 1. It memoizes the notion that K being mapped onto the trivial group {1}, that wholly consists of a lone identity element 1, is what induces the mapping of the big group G to its quotient G/K. A structure that forms the inverse image of the identity element under a transformation is called the "kernel" of that transformation, which is why the letter "K" came first to mind. According to this paradigmatic example, if I were to try and use the potential analogy with group theory, I would be led to ask: What is the arrow, the mapping, morphism, or transformation, and what is the kernel in this case? On that note, I think that I will call it a day, and see if addresses the problem to sleep on it. Jon Awbrey o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at oakland.edu Thu Jun 12 14:04:31 2003 From: jawbrey at oakland.edu (Jon Awbrey) Date: Wed Jan 21 20:34:29 2004 Subject: [Inquiry] Re: Examples! Examples! Examples! References: <3EE340F2.726DAFF7@oakland.edu> <3EE3616C.D082D47E@oakland.edu> <3EE3A2E4.5E4F29D4@oakland.edu> <3EE40644.BA8EA54D@oakland.edu> <3EE4A306.83620651@oakland.edu> <3EE4DED9.D3969CB8@oakland.edu> <3EE4FA33.EFD1998E@oakland.edu> <3EE52E97.598A699D@oakland.edu> <3EE5FDB2.94ABBDEF@oakland.edu> <3EE696AE.6A3BE29C@oakland.edu> <3EE74D5D.A734821@oakland.edu> <3EE79678.FBF11EA2@oakland.edu> <3EE80452.61D73F6F@oakland.edu> Message-ID: <3EE8CEBF.C5962085@oakland.edu> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o EEE. Note 14 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Example 1. John Sowa's "Top Level Categories" (cont.) Solvitur somnambulando (it is solved by sleep-walking), but for a spell so long solely as a sleep walk can run. It strikes me and all my innominate funny bones how the mind's eye can be so besooted with sclerotic motes that it can hardly fathom what's lying smack dab in front of the mind's face, right up to, and under the mind's nose. And yet I must leave that sleeping hemisphere to sleep on it, earnestly awaiting the outcoma of its long longed-for waking, and return to some of the wool that I previously gathered on the TLC example, reweaving through the text the mutations of notation that are enscribed to the palimpsest of adaptations. One "topic of pressingly obvious significance" (TOPOS) is the dimension or distinction that stretches between models and theories, especially insofar as it strikes me that I have yet to broach the lattice of theories that is generated by the alphabet !TLC!. And so to that task I now turn, with this review and revision: The "space of models", or more precisely, a coordinate chart over the space of models, is the universe of discourse TLC% = [!TLC!] = [a_1, ..., a_25]. This lays the maps of TLC^ = (TLC -> B) ~=~ Pow(TLC) ~=~ (B^25 -> B) over the coordinate positions of TLC = <|a_1, ..., a_25|> ~=~ B^25, and so it is assigned the "stereo type" TLC% : (TLC, TLC^) ~=~ (B^25, (B^25 -> B)) abbreviated as TLC% : (TLC +-> B) ~=~ (B^25 +-> B), or more briefly as TLC% : [TLC] ~=~ [B^25]. A synopsis of these notations and further discussion can be found here: D01. http://suo.ieee.org/ontology/msg04799.html D02. http://suo.ieee.org/ontology/msg04800.html D03. http://suo.ieee.org/ontology/msg04801.html The "syntactic space" is the formal language L = L(!TLC!) c (!TLC! |_| !M!)* of grammatical strings in the Cactus Language that we may variously describe as "expressions", "formulas", "sentences", "terms", "wffs", or whatever fits. A "zeroth order theory" about TLC is a set of sentences $T$ c L = L(!TLC!), and thus there is a "lattice at large" of theories in the medium of !TLC!, namely, (Pow(L), c). So we have for starters at least two lattices: 1. One model lattice is (Pow(TLC), c), ordered by set-theoretic inclusion or the "contained as a subset" relation that is here symbolized by "c". This is isomorphic to the proposition lattice (TLC^, =>) that consists of the propositions in (TLC -> B), ordered by the logical implication relation that is here symbolized by "=>". 2. One theory lattice is (Pow(L), c), ordered by inclusion as usual. Jon Awbrey o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at oakland.edu Thu Jun 12 22:08:27 2003 From: jawbrey at oakland.edu (Jon Awbrey) Date: Wed Jan 21 20:34:29 2004 Subject: [Inquiry] Re: Examples! Examples! Examples! References: <3EE340F2.726DAFF7@oakland.edu> <3EE3616C.D082D47E@oakland.edu> <3EE3A2E4.5E4F29D4@oakland.edu> <3EE40644.BA8EA54D@oakland.edu> <3EE4A306.83620651@oakland.edu> <3EE4DED9.D3969CB8@oakland.edu> <3EE4FA33.EFD1998E@oakland.edu> <3EE52E97.598A699D@oakland.edu> <3EE5FDB2.94ABBDEF@oakland.edu> <3EE696AE.6A3BE29C@oakland.edu> <3EE74D5D.A734821@oakland.edu> <3EE79678.FBF11EA2@oakland.edu> <3EE80452.61D73F6F@oakland.edu> <3EE8CEBF.C5962085@oakland.edu> Message-ID: <3EE9402B.5C056F54@oakland.edu> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o EEE. Note 15 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Here is a way to visualize two sorts of lattice quotients, operating in the syntactic space !C!(x) of sentences involving a single variable "x" and the objective space X^ = (X -> B) of propositions that are intended as the denotations of these expressions, respectively. For simplicity, I have shown how these operations look over the 1-dimensional universe. Table 9-1a shows a sample of expressions in the cactus language !C!(x), namely, with labels from the palette of one paint !P! = {p_1} = {"x"}. The formal language !C!(x) is just an amorphous set of sentences, but if we partition it into logical equivalence classes by applying the relevant set of axioms $A$(=) for logical equivalence (<=>), then the result !C!(x)/$A$(=) can be organized as a lattice structure. Table 9-1b shows the result of asserting or assuming the proposition x, that is, imposing the relation "x = 1" or "x is true". This amounts to coalescing the logical equivalence classes even further, tantamount to pretending that x is an alias for the blank space that signifies truth. Table 9-2 shows the images of these actions in the corresponding objective spaces, tracing the quotient mapping from X^ to X^/x that identifies x with 1 and (x) with 0. Table 9-1. Syntactic Quotient Lattices o-----------------------------o-----------------------------o | a. !C!(x)/$A$(=) | b. C!{x)/($A$(=), x} | o-----------------------------o-----------------------------o | | | | o-----------o | o-----------o | | /| " " |= | | " " | | | / | "(x(x))" | = | | "(x(x))" | | | / | ... | = | | ... | | | / o-----------o = | | "x" | | | / = | | "((x))" | | | / = | | ... | | | o-----------o o-----------o | o-----o-----o | | | "(x)" | | "x" | | | | | | "(x(()))" | | "((x))" | | | | | | ... | | ... | | | | | o-----------o o-----------o | o-----o-----o | | = / | | "()" | | | = / | | "x(x)" | | | = o-----------o / | | ... | | | = | "()" | / | | "(x)" | | | = | "x(x)" | / | | "(x(()))" | | | =| ... |/ | | ... | | | o-----------o | o-----------o | | | | o-----------------------------o-----------------------------o Table 9-2. Objective Quotient Lattices o-----------------------------o-----------------------------o | a. X^ ~=~ 2^X | b. X^/x | o-----------------------------o-----------------------------o | | | | | x | | | | | o | o | | / = | | | | / = | | | | / = | | | | / = | | | | / = | | | | / = | | | | (x) o o x | | | | = / | | | | = / | | | | = / | | | | = / | | | | = / | | | | = / | | | | o | o | | | | | ( ) | (x) | | | | o-----------------------------o-----------------------------o Jon Awbrey o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at oakland.edu Fri Jun 13 11:30:07 2003 From: jawbrey at oakland.edu (Jon Awbrey) Date: Wed Jan 21 20:34:29 2004 Subject: [Inquiry] Re: Examples! Examples! Examples! References: <3EE340F2.726DAFF7@oakland.edu> <3EE3616C.D082D47E@oakland.edu> <3EE3A2E4.5E4F29D4@oakland.edu> <3EE40644.BA8EA54D@oakland.edu> <3EE4A306.83620651@oakland.edu> <3EE4DED9.D3969CB8@oakland.edu> <3EE4FA33.EFD1998E@oakland.edu> <3EE52E97.598A699D@oakland.edu> <3EE5FDB2.94ABBDEF@oakland.edu> <3EE696AE.6A3BE29C@oakland.edu> <3EE74D5D.A734821@oakland.edu> <3EE79678.FBF11EA2@oakland.edu> <3EE80452.61D73F6F@oakland.edu> <3EE8CEBF.C5962085@oakland.edu> <3EE9402B.5C056F54@oakland.edu> Message-ID: <3EE9FC0F.4E7FC761@oakland.edu> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o EEE. Note 16 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Last time I made reference to a set of axioms $A$(=) that are here being interpreted as defining logical equivalence for the !P!-parameterized family of formal languages !C!(!P!) that are here being interpreted as denoting the propositions of zeroth order logic. These axioms derive from the formal systems of Charles Sanders Peirce and George Spencer Brown, and there is an account of their nature and use to be found here: PERS. Propositional Equation Reasoning Systems 01. http://suo.ieee.org/ontology/msg04522.html 02. http://suo.ieee.org/ontology/msg04523.html 03. http://suo.ieee.org/ontology/msg04524.html 04. http://suo.ieee.org/ontology/msg04525.html 05. http://suo.ieee.org/ontology/msg04526.html 06. http://suo.ieee.org/ontology/msg04527.html 07. http://suo.ieee.org/ontology/msg04528.html 08. http://suo.ieee.org/ontology/msg04529.html 09. http://suo.ieee.org/ontology/msg04530.html 10. http://suo.ieee.org/ontology/msg04531.html 11. http://suo.ieee.org/ontology/msg04532.html 12. http://suo.ieee.org/ontology/msg04533.html 13. http://suo.ieee.org/ontology/msg04534.html 14. http://suo.ieee.org/ontology/msg04536.html 15. http://suo.ieee.org/ontology/msg04537.html 16. http://suo.ieee.org/ontology/msg04538.html Jon Awbrey o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at oakland.edu Sun Jun 15 15:00:34 2003 From: jawbrey at oakland.edu (Jon Awbrey) Date: Wed Jan 21 20:34:29 2004 Subject: [Inquiry] Probability And Statistics Message-ID: <3EECD062.3DC143CD@oakland.edu> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o PAS. Probability And Statistics o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o PAS. Note 1 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Excerpts from 'Introduction to Probability Theory' by Paul G. Hoel, Sidney C. Port, Charles J. Stone. | 1.2. Probability Spaces | | Our purpose in this section is to develop the formal | mathematical structure, called a probability space, | that forms the foundation for the mathematical | treatment of random phenomena. | | Envision some real or imaginary experiment that we are trying to model. | The first thing we must do is decide on the possible outcomes of the | experiment. It is not too serious if we admit more things into our | consideration than can really occur, but we want to make sure that | we do not exclude things that might occur. Once we decide on the | possible outcomes, we choose a set !W! [Omega] whose points !w! | [omega] are associated with these outcomes. From the strictly | mathematical point of view, however, !W! is just an abstract | set of points. | | We next take a nonempty collection $A$ of subsets of !W! that is | to represent the collection of "events" to which we wish to assign | probabilities. By definition, now, an 'event' means a set A in $A$. | The statement 'the event A occurs' means that the outcome of our | experiment is represented by some point !w! in A. Again, from | the strictly mathematical point of view, $A$ is just a specified | collection of subsets of the set !W!. Only sets A in $A$, i.e., | events, will be assigned probabilities. In our model in Example 1, | $A$ consisted of all subsets of !W!. In the general situation when | !W! does not have a finite number of points, as in Example 2, it may | not be possible to choose $A$ in this manner. | | Hoel, Port, Stone, 'Probability Theory', p. 6. | | Hoel, P.G., Port, S.C., & Stone, C.J., |'Introduction to Probability Theory', | Houghton Mifflin, Boston, MA, 1971. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at oakland.edu Sun Jun 15 23:28:07 2003 From: jawbrey at oakland.edu (Jon Awbrey) Date: Wed Jan 21 20:34:29 2004 Subject: [Inquiry] Re: Probability And Statistics References: <3EECD062.3DC143CD@oakland.edu> Message-ID: <3EED4757.4F7AB578@oakland.edu> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o PAS. Note 2 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | 1.2. Probability Spaces (cont.) | | The next question is, what should the collection $A$ be? | It is quite reasonable to demand that $A$ be closed under | finite unions and finite intersections of sets in $A$ as | well as under complementation. | | For example, if A and B are two events, then A |_| B occurs if the | outcome of our experiment is either represented by a point in A or | a point in B. Clearly, then, if it is going to be meaningful to | talk about the probabilities that A and B occur, it should also | be meaningful to talk about the probability that either A or B | occurs, i.e., that the event A |_| B occurs. Since only sets | in $A$ will be assigned probabilities, we should require that | A |_| B is in $A$ whenever A and B are members of $A$. | | Now A |^| B occurs if the outcome of our experiment is represented | by some point that is in both A and B. A similar line of reasoning | to that used for A |_| B convinces us that we should have A |^| B | in $A$ whenever A, B are in $A$. | | Finally, to say that the event A does not occur is to say that | the outcome of our experiment is not represented by a point in A, | so that it must be represented by some point in A^c. It would be | the height of folly to say that we could talk about the probability | of A but not of A^c. Thus we shall demand that whenever A is in $A$ | so is A^c. | | We have thus arrived at the conclusion that $A$ | should be a nonempty collection of subsets of !W! | having the following properties: | | 1. If A is in $A$ so is A^c. | | 2. If A and B are in $A$ so are A |_| B and A |^| B. | | An easy induction argument shows that | if A_1, A_2, ..., A_n are sets in $A$ | then so are: | | |_| (i = 1 to n) A_i | | and | | |^| (i = 1 to n) A_i. | | Here we use the shorthand notation: | | |_| (i = 1 to n) A_i = A_1 |_| A_2 |_| ... |_| A_n | | and | | |^| (i = 1 to n) A_i = A_1 |^| A_2 |^| ... |^| A_n. | | Also, since A |^| A^c = {} and A |_| A^c = !W!, we see | that both the empty set {} and the set !W! must be in $A$. | | Hoel, Port, Stone, 'Probability Theory', pp. 6-7. | | Hoel, P.G., Port, S.C., & Stone, C.J., |'Introduction to Probability Theory', | Houghton Mifflin, Boston, MA, 1971. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at oakland.edu Mon Jun 16 00:14:07 2003 From: jawbrey at oakland.edu (Jon Awbrey) Date: Wed Jan 21 20:34:29 2004 Subject: [Inquiry] Re: Examples! Examples! Examples! References: <3EE340F2.726DAFF7@oakland.edu> <3EE3616C.D082D47E@oakland.edu> <3EE3A2E4.5E4F29D4@oakland.edu> <3EE40644.BA8EA54D@oakland.edu> <3EE4A306.83620651@oakland.edu> <3EE4DED9.D3969CB8@oakland.edu> <3EE4FA33.EFD1998E@oakland.edu> <3EE52E97.598A699D@oakland.edu> <3EE5FDB2.94ABBDEF@oakland.edu> <3EE696AE.6A3BE29C@oakland.edu> <3EE74D5D.A734821@oakland.edu> <3EE79678.FBF11EA2@oakland.edu> <3EE80452.61D73F6F@oakland.edu> <3EE8CEBF.C5962085@oakland.edu> <3EE9402B.5C056F54@oakland.edu> <3EE9FC0F.4E7FC761@oakland.edu> Message-ID: <3EED521F.1057451D@oakland.edu> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Incidentally, I have started documenting the work that I did all through the 80's to program a processor for the Cactus Language. It's really just a prototype "test of concept" type thing, but there are a few tricks to it that may be of interest. Here are the anchors of some pertinent threads: http://stderr.org/pipermail/inquiry/2003-March/000100.html -- exposition http://stderr.org/pipermail/inquiry/2003-March/000115.html -- source code http://stderr.org/pipermail/inquiry/2003-March/000120.html -- commentary http://stderr.org/pipermail/inquiry/2003-March/000141.html -- motivation Jon Awbrey o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at oakland.edu Mon Jun 16 07:50:21 2003 From: jawbrey at oakland.edu (Jon Awbrey) Date: Wed Jan 21 20:34:29 2004 Subject: [Inquiry] Re: Probability And Statistics References: <3EECD062.3DC143CD@oakland.edu> <3EED4757.4F7AB578@oakland.edu> Message-ID: <3EEDBD0D.3F145C05@oakland.edu> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o PAS. Note 3 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | 1.2. Probability Spaces (cont.) | | A nonempty collection of subsets of a given set !W! that is closed under | finite set theoretic operations is called a 'field of subsets' of !W!. | It therefore seems we should demand that $A$ be a field of subsets. | It turns out, however, that for certain mathematical reasons just | taking $A$ to be a field of subsets of !W! is insufficient. | What we will actually demand of the collection $A$ is | more stringent. We will demand that $A$ be closed | not only under finite set theoretic operations | but under countably infinite set theoretic | operations as well. In other words if | {A_n}, n >= 1, is a sequence of sets | in $A$, we will demand that: | | |_| (n = 1 to oo) A_n is in $A$ | | and | | |^| (n = 1 to oo) A_n is in $A$. | | Here we are using the shorthand notation: | | |_| (i = 1 to oo) A_i = A_1 |_| A_2 |_| ... | | to denote the union of all the sets of the sequence, and: | | |^| (i = 1 to oo) A_i = A_1 |^| A_2 |^| ... | | to denote the intersection of all the sets of the sequence. | | A collection of subsets of a given set !W! that is closed | under countable set theory operations is called a !s!-field | of subsets of !W!. (The !s! [sigma] is put in to distinguish | such a collection from a field of subsets.) More formally we | have the following: | | Definition 1. | | A nonempty collection of subsets $A$ of a set !W! | is called a !s!-field of subsets of !W! provided | that the following two properties hold: | | 1. If A is in $A$, then A^c is also in $A$. | | 2. If A_n is in $A$, n = 1, 2, ..., then: | | |_| (i = 1 to oo) A_i | | and | | |^| (i = 1 to oo) A_i | | are both in $A$. | | Hoel, Port, Stone, 'Probability Theory', p. 7. | | Hoel, P.G., Port, S.C., & Stone, C.J., |'Introduction to Probability Theory', | Houghton Mifflin, Boston, MA, 1971. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at oakland.edu Mon Jun 16 15:46:15 2003 From: jawbrey at oakland.edu (Jon Awbrey) Date: Wed Jan 21 20:34:29 2004 Subject: [Inquiry] Re: Probability And Statistics References: <3EECD062.3DC143CD@oakland.edu> <3EED4757.4F7AB578@oakland.edu> <3EEDBD0D.3F145C05@oakland.edu> Message-ID: <3EEE2C97.47628FAF@oakland.edu> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o PAS. Note 4 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | 1.2. Probability Spaces (cont.) | | We now come to the assignment of probabilities to events. | As was made clear in the examples of the preceding section, | the probability of an event is a nonnegative real number. For | an event A, let P(A) denote this number. Then 0 =< P(A) =< 1. | The set !W! representing every possible outcome should, of course, | be assigned the number 1, so P(!W!) = 1. | | In our discussion of Example 1 we showed that the probability of events | satisfies the property that if A and B are any two disjoint events then | P(A |_| B) = P(A) + P(B). Similarly, in Example 2 we showed that if | A and B are two disjoint intervals, then we should also require that: | | P(A |_| B) = P(A) + P(B). | | It now seems reasonable in general to demand that if A and B are disjoint | events then P(A |_| B) = P(A) + P(B). By induction, it would then follow | that if A_1, A_2, ..., A_n are any n mutually disjoint sets (that is, if | A_i |^| A_j = {} whenever i =/= j), then: | | P(|_| (i = 1 to n) A_i) = Sum (i = 1 to n) P(A_i). | | Actually, again for mathematical reasons, we will | in fact demand that this additivity property hold | for countable collections of disjoint events. | | Definition 2. | | A probability measure P on a !s!-field of | subsets $A$ of a set !W! is a real-valued | function having domain $A$ satisfying the | following properties: | | 1. P(!W!) = 1. | | 2. P(A) >= 0 for all A in $A$. | | 3. If A_n, n = 1, 2, 3, ..., are | mutually disjoint sets in $A$, | then: | | P(|_| (n = 1 to oo) A_n) = Sum (n = 1 to oo) P(A_n). | | A probability space, denoted by (!W!, $A$, P), | is a set !W!, a !s!-field of subsets $A$, and | a probability measure P defined on $A$. | | Hoel, Port, Stone, 'Probability Theory', p. 8. | | Hoel, P.G., Port, S.C., & Stone, C.J., |'Introduction to Probability Theory', | Houghton Mifflin, Boston, MA, 1971. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at oakland.edu Mon Jun 16 22:45:33 2003 From: jawbrey at oakland.edu (Jon Awbrey) Date: Wed Jan 21 20:34:30 2004 Subject: [Inquiry] Re: Examples! Examples! Examples! References: <3EE340F2.726DAFF7@oakland.edu> <3EE3616C.D082D47E@oakland.edu> <3EE3A2E4.5E4F29D4@oakland.edu> <3EE40644.BA8EA54D@oakland.edu> <3EE4A306.83620651@oakland.edu> <3EE4DED9.D3969CB8@oakland.edu> <3EE4FA33.EFD1998E@oakland.edu> <3EE52E97.598A699D@oakland.edu> <3EE5FDB2.94ABBDEF@oakland.edu> <3EE696AE.6A3BE29C@oakland.edu> <3EE74D5D.A734821@oakland.edu> <3EE79678.FBF11EA2@oakland.edu> <3EE80452.61D73F6F@oakland.edu> <3EE8CEBF.C5962085@oakland.edu> <3EE9402B.5C056F54@oakland.edu> <3EE9FC0F.4E7FC761@oakland.edu> <3EED521F.1057451D@oakland.edu> Message-ID: <3EEE8EDD.7FC1EB82@oakland.edu> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o EEE. Note 18 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o I'm still waiting for the flash or insight that would show me the structure of (2^L(!TLC!), c), the lattice of theories over the TLC lexicon !TLC! = {a_1, ..., a_25}, and help me to see at one glance just how the TLC axiom !a! acts to modulate its form. In the meantime, let's go back to the more concrete images of the lattice of models (2^TLC, c), that extends from the null set {} to the full set TLC = <|a_1, ..., a_25|>, and its isomorph the lattice of propositions (TLC^, =>), that stretches from the constant function 0 : TLC -> B to the constant function 1 : TLC -> B. o-------------------------------------o-------------------------------------o | | | | TLC | | | | | | <|a_1, ..., a_25|> | 1 : TLC -> B | | | | | {<0000000000000000000000000>, | {<0000000000000000000000000> ~> 1, | | <0000000000000000000000001>, | <0000000000000000000000001> ~> 1, | | ... | ... | | <1111111111111111111111110>, | <1111111111111111111111110> ~> 1, | | <1111111111111111111111111>} | <1111111111111111111111111> ~> 1} | | | | | o | o | | / \ | / \ | | / \ | / \ | | / \ | / \ | | / \ | / \ | | / 2^TLC \ | / TLC^ \ | | / \ | / \ | | / Pow (TLC) \ | / (TLC -> B) \ | | ... ... | ... ... | | \ 2^(2^25) / | \ 2^(2^25) / | | \ subsets / | \ functions / | | \ / | \ / | | \ / | \ / | | \ / | \ / | | \ / | \ / | | \ / | \ / | | o | o | | | | | { } | ( ) | | | | | | 0 : TLC -> B | | | | | | {<0000000000000000000000000> ~> 0, | | | <0000000000000000000000001> ~> 0, | | | ... | | | <1111111111111111111111110> ~> 0, | | | <1111111111111111111111111> ~> 0} | | | | o-------------------------------------o-------------------------------------o Figure 4. Subset Lattice Pow(TLC) and Proposition Lattice TLC^ = (TLC -> B) Imagine that that I point an e-cam or web-cam through the window by my desk, having sent off to a secret address in Schenectady for a complete set of 25 TLC detectors through which to filter the input. The various entities that pass or that rest within the scope of my window, the bluebirds nesting in the crabapple tree, the mourning doves in the spruce, the whole motley crew at the bird feeder, the pedestrians and the bikers and the cars, the chipmunks, rabbits, and squirrels, the grass, the holly bushes, the rhododendron, the occasional crow, or duck, or canadian goose, the mailbox with our house number on it, or maybe it's more like a "numeral", or even just a plastic token thereof -- all these things, in all their variety, get codified as so many vectors of 25 bits. Now we all know that you can't play 20 questions with Nature and win, but apparently 5 more are trump enough. Still, if I now view Nature as speaking to me in code, beaming me elements of B^25 until I can sift Her plan, I eventually come to notice that it's apparently some species of error-correcting code, in other words, the codes that I receive over the longest time I know are but a sparse selection of all the codes that might be. To be continued ... Jon Awbrey o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at oakland.edu Thu Jun 19 06:36:04 2003 From: jawbrey at oakland.edu (Jon Awbrey) Date: Wed Jan 21 20:34:30 2004 Subject: [Inquiry] test Message-ID: <3EF1A024.F4D29500@oakland.edu> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o testing 1, 2, 3, ..., oo o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at oakland.edu Wed Jun 18 08:52:02 2003 From: jawbrey at oakland.edu (Jon Awbrey) Date: Wed Jan 21 20:34:30 2004 Subject: [Inquiry] Re: Examples! Examples! Examples! References: <3EE340F2.726DAFF7@oakland.edu> <3EE3616C.D082D47E@oakland.edu> <3EE3A2E4.5E4F29D4@oakland.edu> <3EE40644.BA8EA54D@oakland.edu> <3EE4A306.83620651@oakland.edu> <3EE4DED9.D3969CB8@oakland.edu> <3EE4FA33.EFD1998E@oakland.edu> <3EE52E97.598A699D@oakland.edu> <3EE5FDB2.94ABBDEF@oakland.edu> <3EE696AE.6A3BE29C@oakland.edu> <3EE74D5D.A734821@oakland.edu> <3EE79678.FBF11EA2@oakland.edu> <3EE80452.61D73F6F@oakland.edu> <3EE8CEBF.C5962085@oakland.edu> <3EE9402B.5C056F54@oakland.edu> <3EE9FC0F.4E7FC761@oakland.edu> <3EED521F.1057451D@oakland.edu> <3EEE8EDD.7FC1EB82@oakland.edu> Message-ID: <3EF06E82.D0E02636@oakland.edu> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o EEE. Note 19 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Aside from looking out my window every now and then, I am still staring at this stereoscopic latticework: o-------------------------------------o-------------------------------------o | | | | TLC | 1 : TLC -> B | | o | o | | / \ | / \ | | / \ | / \ | | / \ | / \ | | / \ | / \ | | / 2^TLC \ | / TLC^ \ | | / \ | / \ | | / Pow (TLC) \ | / (TLC -> B) \ | | ... ... | ... ... | | \ 2^(2^25) / | \ 2^(2^25) / | | \ subsets / | \ functions / | | \ / | \ / | | \ / | \ / | | \ / | \ / | | \ / | \ / | | \ / | \ / | | o | o | | { } | 0 : TLC -> B | | | | o-------------------------------------o-------------------------------------o Figure 4. Subset Lattice Pow(TLC) and Proposition Lattice TLC^ = (TLC -> B) I see that I need to clear up some points of terminology that arise through my loose use of elliptical constructs. For example, when I speak of TLC = <|a_1, ..., A_25|> as a "model space" and 2^TLC as a "lattice of models", what I really mean is that the elements of TLC are the models of each and every proposition in the language L(!TLC!) that employs the vocabulary !TLC! = {a_1, ..., a_25}. Of course, I do not mean to say that these elements are the formal models, much less the natural models, of the axiom !a!_1 for TLC, displayed in Table 1, that we are currently contemplating. In short, it would be more proper to refer to TLC and Pow(TLC) as a "space of interpretations" and a "lattice of interpretations", respectively. One issue that needs to be mentioned, and constantly kept in mind, but not fussed over too much at this early stage of investigation, is what has generally been called the "theory laden" character of observation, in this Example, the obvious fact that data as coded is nothing that merits being called "Nature's Own Language" (NOL), and if we chance to call it "raw data", no one should dream for a moment that it's a bit un-pre-processed or unrefined for all that. The upshot of this reflection is that an indivisual coding scheme, like !TLC!, is always an abductive construct, and always deployed subordinate to the total hypothesis that needs be implicated with any brand of experimental approach to Nature. As such, languages and their associated conceptual frameworks are eminently fallible. Jon Awbrey o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at oakland.edu Wed Jun 18 23:22:32 2003 From: jawbrey at oakland.edu (Jon Awbrey) Date: Wed Jan 21 20:34:30 2004 Subject: [Inquiry] Re: Examples! Examples! Examples! References: <3EE340F2.726DAFF7@oakland.edu> <3EE3616C.D082D47E@oakland.edu> <3EE3A2E4.5E4F29D4@oakland.edu> <3EE40644.BA8EA54D@oakland.edu> <3EE4A306.83620651@oakland.edu> <3EE4DED9.D3969CB8@oakland.edu> <3EE4FA33.EFD1998E@oakland.edu> <3EE52E97.598A699D@oakland.edu> <3EE5FDB2.94ABBDEF@oakland.edu> <3EE696AE.6A3BE29C@oakland.edu> <3EE74D5D.A734821@oakland.edu> <3EE79678.FBF11EA2@oakland.edu> <3EE80452.61D73F6F@oakland.edu> <3EE8CEBF.C5962085@oakland.edu> <3EE9402B.5C056F54@oakland.edu> <3EE9FC0F.4E7FC761@oakland.edu> <3EED521F.1057451D@oakland.edu> <3EEE8EDD.7FC1EB82@oakland.edu> Message-ID: <3EF13A88.B4D5BAAA@oakland.edu> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o EEE. Note 19 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Aside from looking out my window every now and then, I am still staring at this stereoscopic latticework: o-------------------------------------o-------------------------------------o | | | | TLC | 1 : TLC -> B | | o | o | | / \ | / \ | | / \ | / \ | | / \ | / \ | | / \ | / \ | | / 2^TLC \ | / TLC^ \ | | / \ | / \ | | / Pow (TLC) \ | / (TLC -> B) \ | | ... ... | ... ... | | \ 2^(2^25) / | \ 2^(2^25) / | | \ subsets / | \ functions / | | \ / | \ / | | \ / | \ / | | \ / | \ / | | \ / | \ / | | \ / | \ / | | o | o | | { } | 0 : TLC -> B | | | | o-------------------------------------o-------------------------------------o Figure 4. Subset Lattice Pow(TLC) and Proposition Lattice TLC^ = (TLC -> B) I see that I need to clear up some points of terminology that arise through my loose use of elliptical constructs. For example, when I speak of TLC = <|a_1, ..., A_25|> as a "model space" and 2^TLC as a "lattice of models", what I mean is that the elements of TLC collectively comprise the models of the various propositions in the language L(!TLC!) that employs the vocabulary !TLC! = {a_1, ..., a_25}. Of course, I do not mean to imply these elements are the formal models, much less the natural models, of the axiom !a!_1 for TLC that we are currently contemplating, as it was displayed in Table 1. In short, it would probably be more precise to refer to TLC and Pow(TLC) as a "space of interpretations" and a "lattice of interpretations", respectively. One issue that needs to be mentioned, and constantly kept in mind, but not fussed over too much at this early stage of investigation, is what has generally been called the "theory laden character" of observation, in this Example, the obvious fact that data as coded are nothing that merit being called "Nature's Own Language" (NOL), and if we chance to call it "raw data", no one should dream for a moment that it's a bit un-pre-processed or unrefined for all that. The upshot of this reflection is that an individual coding scheme, like !TLC!, is always an abductive construct, and always deployed subordinate to the total hypothesis that needs be implicated with any brand of experimental approach to Nature. As such, languages and their associated conceptual frameworks are eminently fallible. Jon Awbrey o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at oakland.edu Wed Jun 18 13:38:43 2003 From: jawbrey at oakland.edu (Jon Awbrey) Date: Wed Jan 21 20:34:30 2004 Subject: [Inquiry] Re: Examples! Examples! Examples! References: <3EE340F2.726DAFF7@oakland.edu> <3EE3616C.D082D47E@oakland.edu> <3EE3A2E4.5E4F29D4@oakland.edu> <3EE40644.BA8EA54D@oakland.edu> <3EE4A306.83620651@oakland.edu> <3EE4DED9.D3969CB8@oakland.edu> <3EE4FA33.EFD1998E@oakland.edu> <3EE52E97.598A699D@oakland.edu> <3EE5FDB2.94ABBDEF@oakland.edu> <3EE696AE.6A3BE29C@oakland.edu> <3EE74D5D.A734821@oakland.edu> <3EE79678.FBF11EA2@oakland.edu> <3EE80452.61D73F6F@oakland.edu> <3EE8CEBF.C5962085@oakland.edu> <3EE9402B.5C056F54@oakland.edu> <3EE9FC0F.4E7FC761@oakland.edu> <3EED521F.1057451D@oakland.edu> <3EEE8EDD.7FC1EB82@oakland.edu> <3EF06E82.D0E02636@oakland.edu> Message-ID: <3EF0B1B3.6B3BE624@oakland.edu> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o EEE. Note 20 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o In referring to an element x of the 25-dimensional B-valued model space, or more preciously the space of interpretations TLC = <|a_1, ..., a_25|>, we have the following two options: 1. Denote x by means of a coordinate sequence: x = in B^25. 2. Denote x by means of a conjunctive expression: x = Prod_j (j = 1 to 25) e_j where e_j = (a_j) if x_j = 0, and e_j = a_j if x_j = 1. Option (1) is called the "coordinate representation" and option (2) is called the "conjunctive representation" of the elements in TLC. The type of proposition that gets generated under the conjunctive representation is also called a "singular proposition", since it picks out a single point of the model space. The coordinate option is more compact, but its use depends on remembering the verbose lexical terms that are abbreviated in the forms of a_1, ..., a_25, and thus on what is described as an "ordered basis". Then again, a bunch of boolean codes, if you lose the key to their meanings, is a mess of useless bits. Jon Awbrey o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at oakland.edu Fri Jun 20 13:16:45 2003 From: jawbrey at oakland.edu (Jon Awbrey) Date: Wed Jan 21 20:34:30 2004 Subject: [Inquiry] Re: Examples! Examples! Examples! References: <3EE340F2.726DAFF7@oakland.edu> Message-ID: <3EF34F8D.360A27A@oakland.edu> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o EEE. Note 21 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o There are many ways of probing the space TLC = <|a_1, ..., a_25|> to find the "models" of a given proposition in TLC^ = (TLC -> B), that is, the assignments of {false, true} values or {0, 1} values to the 25 boolean variables that are associated with the alphabet !TLC! = {a_1, ..., a_25} that render the value of the proposition, say, q : TLC -> B, true, or equal to 1 in B. However one does it, doing this amounts to computing the truth table of the function q, a truth table that spans 2^25 values x = for the independent variable x in TLC ~=~ B^25, and for each one of these a determinate value of the dependent variable, given by q(x) in B. Obviously, we do not want to generate this truth table in a literal fashion if we can find a smarter, more virtual way to acquire the same information. As a manifestoly pertinent example of a proposition q : TLC -> B, let's go back to my initial version of a TLC axiom !a! = !a!_1 : TLC -> B that I got from poring over the Figure of the Top Level Category lattice here: http://www.jfsowa.com/ontology/toplevel.htm Now, having never received any feedback from the author as to whether my axiom !a!_1 is a faithful representation of the intended structure, I will simply go forward with it as one example of a proposition that is worth examining for its implications, its impliers, and its models. Here again is the expression of the proposition !a!_1. Table 1. TLC in Cactus Language (Version 1) o-----------------------------------------------------------------------o | | | (( Object ),( Process ),( Schema ),( Script ), | | ( Juncture ),( Participation ),( Description ),( History ), | | ( Structure ),( Situation ),( Reason ),( Purpose )) | | | | ( Independent ,( Actuality ),( Form )) | | ( Relative ,( Prehension ),( Proposition )) | | ( Mediating ,( Nexus ),( Intention )) | | | | ( Physical ,( Actuality ),( Prehension ),( Nexus )) | | ( Abstract ,( Form ),( Proposition ),( Intention )) | | | | ( Continuant ,( Object ),( Schema ),( Juncture ), | | ( Description ),( Structure ),( Reason )) | | | | ( Occurrent ,( Process ),( Script ),( Participation ), | | ( History ),( Situation ),( Purpose )) | | | | ( Actuality ,( Object ),( Process )) | | ( Form ,( Schema ),( Script )) | | ( Prehension ,( Juncture ),( Participation )) | | ( Proposition ,( Description ),( History )) | | ( Nexus ,( Structure ),( Situation )) | | ( Intention ,( Reason ),( Purpose )) | | | o-----------------------------------------------------------------------o A more symbolic, hopefully smarter way to tease out the models of a proposition q : TLC -> B, as denoted by and expressed in a particular syntactic formula !q!, for example, an expression in the cactus language, !q! in L(!TLC!) = !C!(!TLC!), is to compute the "disjunctive normal form" (DNF) of the given proposition q or its given expression !q!. Well, it's only smarter if one does it in a smarter way, otherwise one is pretty much doing exactly the same amount of work as it takes to generate the whole truth table, not to mention the overhead expense that is needed to maintain the whole motley, if dynamic array of symbolic indirections that ultimately refer to logical values. Jon Awbrey o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at oakland.edu Fri Jun 20 13:24:04 2003 From: jawbrey at oakland.edu (Jon Awbrey) Date: Wed Jan 21 20:34:30 2004 Subject: [Inquiry] Re: Examples! Examples! Examples! References: <3EE340F2.726DAFF7@oakland.edu> <3EF34F8D.360A27A@oakland.edu> Message-ID: <3EF35144.B3249A1C@oakland.edu> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o EEE. Note 22 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o It is useful to keep on using both the coordinate representation and the conjunctive representation of points in TLC = <|a_1, ..., a_25|>, taking up whichever form appears to be handiest for the task at hand, and transforming back and forth between them as the occasions demand. Just by way of a concrete example, here are equivalent expressions for a random -- I flipped a coin 25 times -- cell or point in TLC, the expressions being read by reading down the following columns: Table 10. Random Access of TLC o---------o---------o---------o-------------------o | j | x_j | e_j | e_j | o---------o---------o---------o-------------------o | | | | | | 1 | 1 | a_1 | Abstract | | | | | | | 2 | 0 | (a_2) | (Actuality) | | | | | | | 3 | 1 | a_3 | Continuant | | | | | | | 4 | 0 | (a_4) | (Description) | | | | | | | 5 | 1 | a_5 | Form | | | | | | | 6 | 0 | (a_6) | (History) | | | | | | | 7 | 0 | (a_7) | (Independent) | | | | | | | 8 | 1 | a_8 | Intention | | | | | | | 9 | 1 | a_9 | Juncture | | | | | | | 10 | 1 | a_10 | Mediating | | | | | | | 11 | 0 | (a_11) | (Nexus) | | | | | | | 12 | 0 | (a_12) | (Object) | | | | | | | 13 | 0 | (a_13) | (Occurrent) | | | | | | | 14 | 0 | (a_14) | (Participation) | | | | | | | 15 | 1 | a_15 | Physical | | | | | | | 16 | 1 | a_16 | Prehension | | | | | | | 17 | 1 | a_17 | Process | | | | | | | 18 | 1 | a_18 | Proposition | | | | | | | 19 | 1 | a_19 | Purpose | | | | | | | 20 | 1 | a_20 | Reason | | | | | | | 21 | 0 | (a_21) | (Relative) | | | | | | | 22 | 0 | (a_22) | (Schema) | | | | | | | 23 | 1 | a_23 | Script | | | | | | | 24 | 1 | a_24 | Situation | | | | | | | 25 | 0 | (a_25) | (Structure) | | | | | | o---------o---------o---------o-------------------o Jon Awbrey o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at oakland.edu Fri Jun 20 14:38:10 2003 From: jawbrey at oakland.edu (Jon Awbrey) Date: Wed Jan 21 20:34:30 2004 Subject: [Inquiry] Re: Examples! Examples! Examples! References: <3EE340F2.726DAFF7@oakland.edu> <3EF34F8D.360A27A@oakland.edu> <3EF35144.B3249A1C@oakland.edu> Message-ID: <3EF362A2.32E9F109@oakland.edu> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o EEE. Note 23 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Let !r! (rho) be the "random interpretation" or the randomly selected element of TLC that was generated last time. The question now arises: Does !r! constitute a model or a countermodel of the axiom !a! that I took to define the intended lattice of Top Level Categories in terms of its lexicon !TLC! = {a_1, ..., a_25}? A question like this is called a "propositional query". It can be addressed by conjoining to the axiom !a! the conjunctive representation of the query !r! and asking if the joint proposition has a model in the TLC domain. In this particular situation, since the query !r! is a singular proposition, there are only two possibilities: there is just one model or none at all. Putting the question in this way, it can be answered by the same method of computing a DNF, or some other form of canonical expression that is equivalent to a DNF in informational value, for the axiom plus query combination that is shown in Table 11. Table 11. Conjunction of the TLC Axiom !a! with the Random Query !r! o----------------------------------------------------------------------o | | | (( Object ),( Process ),( Schema ),( Script ), | | ( Juncture ),( Participation ),( Description ),( History ), | | ( Structure ),( Situation ),( Reason ),( Purpose )) | | | | (( | | | | ( Independent ,( Actuality ),( Form )) | | ( Relative ,( Prehension ),( Proposition )) | | ( Mediating ,( Nexus ),( Intention )) | | | | ( Physical ,( Actuality ),( Prehension ),( Nexus )) | | ( Abstract ,( Form ),( Proposition ),( Intention )) | | | | ( Continuant ,( Object ),( Schema ),( Juncture ), | | ( Description ),( Structure ),( Reason )) | | | | ( Occurrent ,( Process ),( Script ),( Participation ), | | ( History ),( Situation ),( Purpose )) | | | | ( Actuality ,( Object ),( Process )) | | ( Form ,( Schema ),( Script )) | | ( Prehension ,( Juncture ),( Participation )) | | ( Proposition ,( Description ),( History )) | | ( Nexus ,( Structure ),( Situation )) | | ( Intention ,( Reason ),( Purpose )) | | | | )) | | | | Abstract ( Actuality ) Continuant ( Description ) | | Form ( History ) ( Independent ) Intention | | Juncture Mediating ( Nexus ) ( Object ) | | ( Occurrent ) ( Participation ) Physical Prehension | | Process Proposition Purpose Reason | | ( Relative ) ( Schema ) Script Situation | | ( Structure ) | | | o----------------------------------------------------------------------o Jon Awbrey o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at oakland.edu Fri Jun 20 14:52:40 2003 From: jawbrey at oakland.edu (Jon Awbrey) Date: Wed Jan 21 20:34:31 2004 Subject: [Inquiry] Re: Examples! Examples! Examples! References: <3EE340F2.726DAFF7@oakland.edu> <3EF34F8D.360A27A@oakland.edu> <3EF35144.B3249A1C@oakland.edu> <3EF362A2.32E9F109@oakland.edu> Message-ID: <3EF36608.89954C0F@oakland.edu> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o EEE. Note 24 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o The Theme One Program that I wrote some years ago has a function called "Model" that takes a propositional expression !q! in the cactus syntax and outputs a propositional expression Model(!q!) in the same syntax. Model(!q!) serves as a canonical form that is more or less equivalent in informational utility to the DNF. Table 12 displays the actual output of this Model function when it is fed the propositional expression for the axiom plus query conjunction !a! !r!. Table 12. Output of the "Model" Function for the Proposition !a! !r! o----------------------------------------------------------------------o | | | Abstract | | Continuant | | Form | | Intention - | | (Intention ) - | | (Form ) - | | (Continuant ) - | | (Abstract ) - | | | o----------------------------------------------------------------------o This says that Model finds no models for the conjunction !a! !r!, indeed, it finds a contradiction between !a! and !r! on the path that takes the first four positive features of !r! and tests their consistency with the constraints of !a!. So, given this much of a hint, let's see if we can see for ourselves whether we believe this report. Can there be an "Abstract Continuant Form Intention" under the conditions that are stipulated by the TLC axiom !a!? 1. An Abstract must be exactly one of Form, Proposition, or Intention. 2. But the 4-fold conjunction above specifies both Form and Intention. Therefore we have a contradiction, and !r! does not satisfy !a!. Jon Awbrey o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at oakland.edu Sat Jun 21 22:00:10 2003 From: jawbrey at oakland.edu (Jon Awbrey) Date: Wed Jan 21 20:34:31 2004 Subject: [Inquiry] Re: Examples! Examples! Examples! References: <3EE340F2.726DAFF7@oakland.edu> Message-ID: <3EF51BBA.BD3384BB@oakland.edu> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o EEE. Note 25 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Example 1. John Sowa's "Top Level Categories" (cont.) One of the nice things about the conjunctive singular representation of the cells or points in TLC = <|a_1, ..., a_25|> is that one is no longer bound by the arbitrary ordering of the logical basis elements in !TLC! = {a_1, ..., a_25}, the way that one is forced to cleave to a fixed ordering of the a_j when using the coordinate representation. For example, our random interpretation !r! is represented in coordinate style by the following 25-tuple of boolean values: !r! = <1,0,1,0,1,0,0,1,1,1,0,0,0,0,1,1,1,1,1,1,0,0,1,1,0> Squeezing out the commas, !r! is denoted by this bit string: !r! = 1010100111000011111100110 In order to compare elements of TLC we have to maintain the same ordering of coordinate values x_j in every case. In the conjunctive representation, however, we can write the positive and negative literals in any order that is most convenient at the moment in question, for example, representing !r! in a form with all positive literals up front and all negative literals trailing along at the end of the conjunctive expression, like this: o---------------------------------------------------------------------o | | | Abstract Continuant Form Intention Juncture Mediating Physical | | Prehension Process Proposition Purpose Reason Script Situation | | ( Actuality )( Description )( History )( Independent ) | | ( Nexus )( Object )( Occurrent )( Participation ) | | ( Relative )( Schema )( Structure ) | | | o---------------------------------------------------------------------o Jon Awbrey o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at oakland.edu Sun Jun 22 22:31:05 2003 From: jawbrey at oakland.edu (Jon Awbrey) Date: Wed Jan 21 20:34:31 2004 Subject: [Inquiry] Re: Examples! Examples! Examples! References: <3EE340F2.726DAFF7@oakland.edu> <3EF51BBA.BD3384BB@oakland.edu> Message-ID: <3EF67479.2B17CE15@oakland.edu> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o EEE. Note 26 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Example 1. John Sowa's "Top Level Categories" (cont.) The axiom !a! that picks out a portion of the universe of discourse TLC = <|a_1, ..., a_25|> contains a conjunct of the following form that specifies a "universal partition" into twelve mutually exclusive and exhaustive categories: o----------------------------------------------------------------------o | | | (( Description ),( History ),( Juncture ),( Object ), | | ( Participation ),( Process ),( Purpose ),( Reason ), | | ( Schema ),( Script ),( Situation ),( Structure )) | | | o----------------------------------------------------------------------o As long as we are focusing solely on the points of the TLC model space that satisfy this partition property, then it is possible to exploit a huge redundancy in the description of these points, namely, as soon as we know that a given point x has any one of these 12 features, then we know right off that it does not have any one of the others. The component of my Theme One program that does logical modeling makes good use of this brand of information-theoretic redundancy, in the settings where it applies, and a good picture of how this all works out can be seen as we turn to the task of finding all of the models that the TLC axiom !a! has in the universe of TLC. One of the ultimate outputs of the program, when run on the TLC axiom !a!, is the following text, which lists in outline form, the maximal consistent sets of positive features that are allowed by its propositional constraints. Table 13. Summary of "Model" Output for the TLC Axiom !a! o---------o-------------------------------------------------o | | | | 1. | Object | | | Continuant | | | Actuality | | | Independent | | | Physical | | 2. | Process | | | Occurrent | | | Actuality | | | Independent | | | Physical | | 3. | Schema | | | Continuant | | | Form | | | Independent | | | Abstract | | 4. | Script | | | Occurrent | | | Form | | | Independent | | | Abstract | | 5. | Juncture | | | Continuant | | | Prehension | | | Relative | | | Physical | | 6. | Participation | | | Occurrent | | | Prehension | | | Relative | | | Physical | | 7. | Description | | | Continuant | | | Proposition | | | Relative | | | Abstract | | 8. | History | | | Occurrent | | | Proposition | | | Relative | | | Abstract | | 9. | Structure | | | Continuant | | | Nexus | | | Mediating | | | Physical | | 10. | Situation | | | Occurrent | | | Nexus | | | Mediating | | | Physical | | 11. | Reason | | | Continuant | | | Intention | | | Mediating | | | Abstract | | 12. | Purpose | | | Occurrent | | | Intention | | | Mediating | | | Abstract | | | | o---------o-------------------------------------------------o This output summarizes, in terms of positive features alone, all of the points of TLC = <|a_1, ..., a_25|> that satisfy the TLC axiom !a!, since the positive features listed here, in the presence of the given partition constraints, suffice to determine all of the other features. Jon Awbrey o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at oakland.edu Mon Jun 23 13:52:36 2003 From: jawbrey at oakland.edu (Jon Awbrey) Date: Wed Jan 21 20:34:31 2004 Subject: [Inquiry] Re: Examples! Examples! Examples! References: <3EE340F2.726DAFF7@oakland.edu> <3EF51BBA.BD3384BB@oakland.edu> <3EF67479.2B17CE15@oakland.edu> Message-ID: <3EF74C74.B8EB51E@oakland.edu> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o EEE. Note 27 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Example 1. John Sowa's "Top Level Categories" (cont.) As a practical matter of cognitive load factors, it will facilitate the next phase of our work on this Example to alphabetize the appearances of the TLC alphabet !TLC! in the expression of the TLC axiom !a!. For ease of recall, then, Table 14 presents a cosmetic amendation of Table 1, and Table 15 details the corresponding summary of models. Table 14. TLC in Cactus Language (Version 1, Alphabetized) o-----------------------------------------------------------------------o | | | (( Description ),( History ),( Juncture ),( Object ), | | ( Participation ),( Process ),( Purpose ),( Reason ), | | ( Schema ),( Script ),( Situation ),( Structure )) | | | | (( | | | | ( Independent ,( Actuality ),( Form )) | | ( Mediating ,( Intention ),( Nexus )) | | ( Relative ,( Prehension ),( Proposition )) | | | | ( Abstract ,( Form ),( Intention ),( Proposition )) | | ( Physical ,( Actuality ),( Nexus ),( Prehension )) | | | | ( Continuant ,( Description ),( Juncture ),( Object ), | | ( Reason ),( Schema ),( Structure )) | | | | ( Occurrent ,( History ),( Participation ),( Process ), | | ( Purpose ),( Script ),( Situation )) | | | | ( Actuality ,( Object ),( Process )) | | ( Form ,( Schema ),( Script )) | | ( Intention ,( Purpose ),( Reason )) | | ( Nexus ,( Situation ),( Structure )) | | ( Prehension ,( Juncture ),( Participation )) | | ( Proposition ,( Description ),( History )) | | | | )) | | | o-----------------------------------------------------------------------o Table 15. Summary of "Model" Output for the TLC Axiom !a! o---------o-------------------------------------------------o | | | | 1. | Description | | | Continuant | | | Proposition | | | Relative | | | Abstract | | 2. | History | | | Occurrent | | | Proposition | | | Relative | | | Abstract | | 3. | Juncture | | | Continuant | | | Prehension | | | Relative | | | Physical | | 4. | Object | | | Continuant | | | Actuality | | | Independent | | | Physical | | 5. | Participation | | | Occurrent | | | Prehension | | | Relative | | | Physical | | 6. | Process | | | Occurrent | | | Actuality | | | Independent | | | Physical | | 7. | Purpose | | | Occurrent | | | Intention | | | Mediating | | | Abstract | | 8. | Reason | | | Continuant | | | Intention | | | Mediating | | | Abstract | | 9. | Schema | | | Continuant | | | Form | | | Independent | | | Abstract | | 10. | Script | | | Occurrent | | | Form | | | Independent | | | Abstract | | 11. | Situation | | | Occurrent | | | Nexus | | | Mediating | | | Physical | | 12. | Structure | | | Continuant | | | Nexus | | | Mediating | | | Physical | | | | o---------o-------------------------------------------------o Jon Awbrey o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at oakland.edu Tue Jun 24 05:40:28 2003 From: jawbrey at oakland.edu (Jon Awbrey) Date: Wed Jan 21 20:34:31 2004 Subject: [Inquiry] Re: Examples! Examples! Examples! References: <3EE340F2.726DAFF7@oakland.edu> Message-ID: <3EF82A9C.AC13ED7D@oakland.edu> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o EEE. Note 28 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Example 1. John Sowa's "Top Level Categories" (cont.) I derived my first description of the TLC space, expressed in terms of the axiom !a!, purely from looking at the lattice picture, Figure 2.6 in the text of 'KR' or Figure 1 at either of the following webloci: http://www.jfsowa.com/ontology/toplevel.htm http://users.bestweb.net/~sowa/ontology/toplevel.htm Had I woken up in the other hemisphere of my brain on the day in question, I might have begun by reading the text first and looking at the pictures second, in which case I could have derived the proposition !b! (beta) that is displayed in Table 16, from the discussion surrounding the matrix table, Figure 2.7 in the text or Figure 2 on either of the above webpages. Table 16. TLC in Cactus Language: Proposition !b! o-----------------------------------------------------------o | | | (( Object , Independent Physical Continuant )) | | (( Process , Independent Physical Occurrent )) | | (( Schema , Independent Abstract Continuant )) | | (( Script , Independent Abstract Occurrent )) | | (( Juncture , Relative Physical Continuant )) | | (( Participation , Relative Physical Occurrent )) | | (( Description , Relative Abstract Continuant )) | | (( History , Relative Abstract Occurrent )) | | (( Structure , Mediating Physical Continuant )) | | (( Situation , Mediating Physical Occurrent )) | | (( Reason , Mediating Abstract Continuant )) | | (( Purpose , Mediating Abstract Occurrent )) | | | | (( Actuality , Independent Physical )) | | (( Form , Independent Abstract )) | | (( Prehension , Relative Physical )) | | (( Proposition , Relative Abstract )) | | (( Nexus , Mediating Physical )) | | (( Intention , Mediating Abstract )) | | | o-----------------------------------------------------------o Each of the clauses in proposition !b! has the form of a propositional equation, that is, a logical equivalence, between a single feature and a conjunction of several features. A clause of this form is naturally comprehended as a "definition" of its lead feature in terms of a class of more primitive stock features. But it does not seem to me, just at first glance, that proposition !a! and proposition !b! can be equivalent, inasmuch as !a! demands that everything in the quotient universe of discourse be partitioned into just one of 12 categories, while !b! only defines a set of 18 features in terms of a stock of 7 others. So let let next examine the relation between !a! and !b!. Reference: | John F. Sowa, |'Knowledge Representation: | Logical, Philosophical, and Computational Foundations', | Brooks/Cole, Pacific Grove, CA, 2000. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at oakland.edu Tue Jun 24 09:32:18 2003 From: jawbrey at oakland.edu (Jon Awbrey) Date: Wed Jan 21 20:34:31 2004 Subject: [Inquiry] Re: Examples! Examples! Examples! References: <3EE340F2.726DAFF7@oakland.edu> <3EF82A9C.AC13ED7D@oakland.edu> Message-ID: <3EF860F2.3CE13DB2@oakland.edu> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o EEE. Note 29 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Example 1. John Sowa's "Top Level Categories" (cont.) With a little more thought it seems clear that the complex feature definitions in proposition !b! need to be supplemented with a set of constraints that describe the partitions of the intended space along the three dimensions (1) {Independent, Relative, Mediating}, (2) {Physical, Abstract}, and (3) {Continuant, Occurrent} in order to have a chance of constituting a sole sufficient axiom for TLC. Table 17 presents one way of doing this, as expressed in Proposition !c! (gamma). Table 17. TLC in Cactus Language: Proposition !c! o-----------------------------------------------------------o | | | (( Independent ),( Relative ),( Mediating )) | | | | (( Physical ),( Abstract )) | | | | (( Continuant ),( Occurrent )) | | | | (( Actuality , Independent Physical )) | | (( Form , Independent Abstract )) | | (( Prehension , Relative Physical )) | | (( Proposition , Relative Abstract )) | | (( Nexus , Mediating Physical )) | | (( Intention , Mediating Abstract )) | | | | (( Object , Independent Physical Continuant )) | | (( Process , Independent Physical Occurrent )) | | (( Schema , Independent Abstract Continuant )) | | (( Script , Independent Abstract Occurrent )) | | (( Juncture , Relative Physical Continuant )) | | (( Participation , Relative Physical Occurrent )) | | (( Description , Relative Abstract Continuant )) | | (( History , Relative Abstract Occurrent )) | | (( Structure , Mediating Physical Continuant )) | | (( Situation , Mediating Physical Occurrent )) | | (( Reason , Mediating Abstract Continuant )) | | (( Purpose , Mediating Abstract Occurrent )) | | | o-----------------------------------------------------------o Generally speaking, a dichotomous division, exclusive disjunction, or logical inequivalence between two logical features X and Y can be expressed as "(X, Y)", but I used a more indirect form for the the distinctions {Physical, Abstract} and {Continuant, Occurrent} due to the leftmost-shallowest variable casing order of the Model function in Theme One, simply to give the program a better chance of displaying the logical features in the customary order of 'KR'. Here is the outline of models for Proposition !c!, expressed in terms of positive features, as often serves in these families of partitioned universes: Table 18. Summary of "Model" Output for the Proposition !c! o---------o-------------------------------------------------o | | | | I | Independent | | IP | Physical | | IPC | Continuant | | | Actuality | | | Object <1> | | IPO | Occurrent | | | Actuality | | | Process <2> | | IA | Abstract | | IAC | Continuant | | | Form | | | Schema <3> | | IAO | Occurrent | | | Form | | | Script <4> | | R | Relative | | RP | Physical | | RPC | Continuant | | | Prehension | | | Juncture <5> | | RPO | Occurrent | | | Prehension | | | Participation <6> | | RA | Abstract | | RAC | Continuant | | | Proposition | | | Description <7> | | RAO | Occurrent | | | Proposition | | | History <8> | | M | Mediating | | MP | Physical | | MPC | Continuant | | | Nexus | | | Structure <9> | | MPO | Occurrent | | | Nexus | | | Situation <10> | | MA | Abstract | | MAC | Continuant | | | Intention | | | Reason <11> | | MAO | Occurrent | | | Intention | | | Purpose <12> | | | | o---------o-------------------------------------------------o Jon Awbrey o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at oakland.edu Tue Jun 24 15:04:03 2003 From: jawbrey at oakland.edu (Jon Awbrey) Date: Wed Jan 21 20:34:31 2004 Subject: [Inquiry] Re: Examples! Examples! Examples! References: <3EE340F2.726DAFF7@oakland.edu> <3EF82A9C.AC13ED7D@oakland.edu> <3EF860F2.3CE13DB2@oakland.edu> Message-ID: <3EF8AEB3.6B946BD0@oakland.edu> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o EEE. Note 30 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Example 1. John Sowa's "Top Level Categories" (cont.) At this point, the little program whose last line of code I laid down in 1989, when Turbo Pascal could address only about 640 Kb of heap space, and a good chunk of that gets nibbled up by the ambient operating system, starts to run out of elbow room, but there is just enough space left to answer one or two more pressing questions. I started out accepting Proposition !a! as an axiom for the intended subset of TLC = <|a_1, ... a_25|>, and now I have found a Proposition !c! that appears to indicate the very same set of models. I cannot tell for sure at this point because the particular function of Theme One that creates the "outline summary" of models does not produce a fully equivalent canonical form, but only one that suffices in a common set of circumstances. Ideally, one would like to write out the proposition of the form (!a! , !c!), that expresses the difference or inequality between !a! and !c!, or the proposition ((!a! , !c!)), that expresses their equality, and then find the models of one of these propositions as a negative or a positive way, respectively, of testing the validity of the equality. I was not able to do that with the amount of memory space that I currently have, but I was able to test the two propositions !a! (!c!) and !c! (!a!) independently, and thereby verified the logical equivalence of !a! and !c!. The Model output for !a! (!c!) is too long to show here, but the output for !c! (!a!) looks like this: Independent Relative - (Relative ) Mediating - (Mediating ) Prehension - (Prehension ) Proposition - (Proposition ) Nexus - (Nexus ) Intention - (Intention ) Juncture - (Juncture ) Participation - (Participation ) Description - (Description ) History - (History ) Structure - (Structure ) Situation - (Situation ) Reason - (Reason ) Purpose - (Purpose ) Physical Abstract - (Abstract ) Form - (Form ) Schema - (Schema ) Script - (Script ) Continuant Occurrent - (Occurrent ) Process - (Process ) Actuality Object - (Object ) - (Actuality ) - (Continuant ) Object - (Object ) Occurrent Actuality Process - (Process ) - (Actuality ) - (Occurrent ) - (Physical ) Actuality - (Actuality ) Object - (Object ) Process - (Process ) Abstract Continuant Occurrent - (Occurrent ) Script - (Script ) Form Schema - (Schema ) - (Form ) - (Continuant ) Schema - (Schema ) Occurrent Form Script - (Script ) - (Form ) - (Occurrent ) - (Abstract ) - (Independent ) Actuality - (Actuality ) Form - (Form ) Object - (Object ) Process - (Process ) Schema - (Schema ) Script - (Script ) Relative Mediating - (Mediating ) Nexus - (Nexus ) Intention - (Intention ) Structure - (Structure ) Situation - (Situation ) Reason - (Reason ) Purpose - (Purpose ) Physical Abstract - (Abstract ) Proposition - (Proposition ) Description - (Description ) History - (History ) Continuant Occurrent - (Occurrent ) Participation - (Participation ) Prehension Juncture - (Juncture ) - (Prehension ) - (Continuant ) Juncture - (Juncture ) Occurrent Prehension Participation - (Participation ) - (Prehension ) - (Occurrent ) - (Physical ) Prehension - (Prehension ) Juncture - (Juncture ) Participation - (Participation ) Abstract Continuant Occurrent - (Occurrent ) History - (History ) Proposition Description - (Description ) - (Proposition ) - (Continuant ) Description - (Description ) Occurrent Proposition History - (History ) - (Proposition ) - (Occurrent ) - (Abstract ) - (Relative ) Prehension - (Prehension ) Proposition - (Proposition ) Juncture - (Juncture ) Participation - (Participation ) Description - (Description ) History - (History ) Mediating Physical Abstract - (Abstract ) Intention - (Intention ) Reason - (Reason ) Purpose - (Purpose ) Continuant Occurrent - (Occurrent ) Situation - (Situation ) Nexus Structure - (Structure ) - (Nexus ) - (Continuant ) Structure - (Structure ) Occurrent Nexus Situation - (Situation ) - (Nexus ) - (Occurrent ) - (Physical ) Nexus - (Nexus ) Structure - (Structure ) Situation - (Situation ) Abstract Continuant Occurrent - (Occurrent ) Purpose - (Purpose ) Intention Reason - (Reason ) - (Intention ) - (Continuant ) Reason - (Reason ) Occurrent Intention Purpose - (Purpose ) - (Intention ) - (Occurrent ) - (Abstract ) - (Mediating ) - Any models would have been marked with stars (*), and the absence of any models says that nothing in TLC satisfies !c! without satisfying !a!. Jon Awbrey o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at oakland.edu Tue Jun 24 23:18:50 2003 From: jawbrey at oakland.edu (Jon Awbrey) Date: Wed Jan 21 20:34:32 2004 Subject: [Inquiry] Re: Examples! Examples! Examples! References: <3EE340F2.726DAFF7@oakland.edu> <3EF82A9C.AC13ED7D@oakland.edu> <3EF860F2.3CE13DB2@oakland.edu> <3EF8AEB3.6B946BD0@oakland.edu> Message-ID: <3EF922AA.59221309@oakland.edu> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o EEE. Note 31 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Let us pause for a moment and look at what we've been doing from a semiotic point of view. A semiotic process is a transformation of signs. One very often views it by picking a representative sign and following its "curve", "orbit", or "trajectory", as it is often called, through a sequence of changes under the continuation or the iteration of the transformation in question. A very important type of semiotic process is one that takes place in a partitioned space of signs and where every orbit remains within a single part of the partition. In this case, we say that the semiosis "preserves" or "respects" the partition or its associated equivalence relation. Figure 19 shows how these ideas apply to the present Example. o-------------------------------o------------------------------------------o | Object Domain TLC^ ~=~ 2^TLC | Language Domain L(!TLC!) | o-------------------------------o------------------------------------------o | | | o---------------o | | /| (()) |\ | | 1 / | (!a! (!c!)) | \ | | o~~~~~~~~~~~~~~~~~~~~~~/~~| (!c! (!a!)) | \ | | / \ / | ((!a!),(!c!)) | \ | | / \ / | ... | \ | | / \ / o---------------o \ | | / \ / \ | | / \ / o---------------o | | / \ / | !a! | | | / \ / | !c! | | | / !a! o~~~~~~/~~~~~~~~~~~~~~~~~~| BE | | | / / / | BE | | | / / / | ... | | | / TLC^ / o---------------o o---------------o | | / / | (!a!) | / | | / / | (!c!) | / | | (!a!) o~~~~~~~~~~~~~~~/~~~~~~~~~| BE<(!a!)> | / | | \ / | BE<(!c!)> | / | | \ / | ... | / | | \ / o---------------o / | | \ / \ / | | \ / \ o---------------o / | | \ / \ | () | / | | \ / \ | !a! (!c!) | / | | o~~~~~~~~~~~~~~~~~~~~~~\~~| !c! (!a!) | / | | 0 \ | (!a! , !c!) | / | | \| ... |/ | | o---------------o | | | o--------------------------------------------------------------------------o Figure 19. Lattice of Propositions Inducing a Partition of Sentences The Figure shows a sample of four object elements in the lattice TLC^ and four logical equivalence classes of propositional expressions in the cactus language L(!TLC!). () and (()) are the usual appearances of the constant false and the constant true proposition, respectively. BE is a special kind of "boolean expansion" of the expression !q!. This is the type of normal form, akin to the disjunctive normal form, that we take as the canonical equivalent of the given expression !q!, and it constitutes the output of the Model function in Theme One. As usual in these types of situations, there are an infinite number of sentences in L that belong to each logical equivalence class and that correspond to each element of the object lattice. The process by which we passed from the TLC axiom !a! to its normal form BE was a graphical transformation that took the parse graph of !a! through a sequence of logical equivalents, finally arriving at BE, a traversal and a projection of which gave the corresponding outline of models. Likewise for !c! and BE. Finally, we tested the equivalence of !a! and !c! by computing the BE of !a! (!c!) and !c! (!a!), which were found to be in the logical equivalence class of (), that is, the proposition with no models. This means that their negations (!a! (!c!)) and (!c! (!a!)), respectively, are in the logical equivalence class of (()), that is, the valid proposition. But these latter expressions are the cactus forms for !a! => !c! and !c! => !a!, respectively, and so we have come to the conclusion that !a! <=> !c!. Jon Awbrey o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at oakland.edu Wed Jun 25 07:54:21 2003 From: jawbrey at oakland.edu (Jon Awbrey) Date: Wed Jan 21 20:34:32 2004 Subject: [Inquiry] Re: Examples! Examples! Examples! -- Correction References: <3EE340F2.726DAFF7@oakland.edu> <3EF82A9C.AC13ED7D@oakland.edu> <3EF860F2.3CE13DB2@oakland.edu> <3EF8AEB3.6B946BD0@oakland.edu> <3EF922AA.59221309@oakland.edu> Message-ID: <3EF99B7D.63664680@oakland.edu> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Correction. There was a mistake in the top right text box of Figure 19. The correct cactus expression for "!a! <=> !c!" is "((!a! , !c!))", and here is the corrected Figure: o-------------------------------o------------------------------------------o | Object Domain TLC^ ~=~ 2^TLC | Language Domain L(!TLC!) | o-------------------------------o------------------------------------------o | | | o---------------o | | /| (()) |\ | | 1 / | (!a! (!c!)) | \ | | o~~~~~~~~~~~~~~~~~~~~~~/~~| (!c! (!a!)) | \ | | / \ / | ((!a! , !c!)) | \ | | / \ / | ... | \ | | / \ / o---------------o \ | | / \ / \ | | / \ / o---------------o | | / \ / | !a! | | | / \ / | !c! | | | / !a! o~~~~~~/~~~~~~~~~~~~~~~~~~| BE | | | / / / | BE | | | / / / | ... | | | / TLC^ / o---------------o o---------------o | | / / | (!a!) | / | | / / | (!c!) | / | | (!a!) o~~~~~~~~~~~~~~~/~~~~~~~~~| BE<(!a!)> | / | | \ / | BE<(!c!)> | / | | \ / | ... | / | | \ / o---------------o / | | \ / \ / | | \ / \ o---------------o / | | \ / \ | () | / | | \ / \ | !a! (!c!) | / | | o~~~~~~~~~~~~~~~~~~~~~~\~~| !c! (!a!) | / | | 0 \ | (!a! , !c!) | / | | \| ... |/ | | o---------------o | | | o--------------------------------------------------------------------------o Figure 19. Lattice of Propositions Inducing a Partition of Sentences o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at oakland.edu Wed Jun 25 14:36:13 2003 From: jawbrey at oakland.edu (Jon Awbrey) Date: Wed Jan 21 20:34:32 2004 Subject: [Inquiry] Re: Examples! Examples! Examples! References: <3EE340F2.726DAFF7@oakland.edu> <3EF82A9C.AC13ED7D@oakland.edu> <3EF860F2.3CE13DB2@oakland.edu> <3EF8AEB3.6B946BD0@oakland.edu> <3EF922AA.59221309@oakland.edu> Message-ID: <3EF9F9AD.4F6710E2@oakland.edu> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o EEE. Note 32 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Semiotic Reflections (cont.) As I contemplate the recent fray of logical maneuvers from the safe and sound security of my semiotic study, a number of recurring, though not especially direct, questions come to mind. For one thing, why do we have so many ways of saying the same thing, anyway? Just by way of keeping a concrete example constantly in mind, here are some fresh copies of the two alternative TLC axioms that we have found so far: Table 20. TLC in Cactus Language: Axiom !a! o-----------------------------------------------------------------------o | | | (( Object ),( Process ),( Schema ),( Script ), | | ( Juncture ),( Participation ),( Description ),( History ), | | ( Structure ),( Situation ),( Reason ),( Purpose )) | | | | ( Independent ,( Actuality ),( Form )) | | ( Relative ,( Prehension ),( Proposition )) | | ( Mediating ,( Nexus ),( Intention )) | | | | ( Physical ,( Actuality ),( Prehension ),( Nexus )) | | ( Abstract ,( Form ),( Proposition ),( Intention )) | | | | ( Continuant ,( Object ),( Schema ),( Juncture ), | | ( Description ),( Structure ),( Reason )) | | | | ( Occurrent ,( Process ),( Script ),( Participation ), | | ( History ),( Situation ),( Purpose )) | | | | ( Actuality ,( Object ),( Process )) | | ( Form ,( Schema ),( Script )) | | ( Prehension ,( Juncture ),( Participation )) | | ( Proposition ,( Description ),( History )) | | ( Nexus ,( Structure ),( Situation )) | | ( Intention ,( Reason ),( Purpose )) | | | o-----------------------------------------------------------------------o Table 21. TLC in Cactus Language: Axiom !c! o-----------------------------------------------------------o | | | (( Independent ),( Relative ),( Mediating )) | | | | (( Physical ),( Abstract )) | | | | (( Continuant ),( Occurrent )) | | | | (( Actuality , Independent Physical )) | | (( Form , Independent Abstract )) | | (( Prehension , Relative Physical )) | | (( Proposition , Relative Abstract )) | | (( Nexus , Mediating Physical )) | | (( Intention , Mediating Abstract )) | | | | (( Object , Independent Physical Continuant )) | | (( Process , Independent Physical Occurrent )) | | (( Schema , Independent Abstract Continuant )) | | (( Script , Independent Abstract Occurrent )) | | (( Juncture , Relative Physical Continuant )) | | (( Participation , Relative Physical Occurrent )) | | (( Description , Relative Abstract Continuant )) | | (( History , Relative Abstract Occurrent )) | | (( Structure , Mediating Physical Continuant )) | | (( Situation , Mediating Physical Occurrent )) | | (( Reason , Mediating Abstract Continuant )) | | (( Purpose , Mediating Abstract Occurrent )) | | | o-----------------------------------------------------------o These two propositions look so different to me that I will probably go back and check my work a couple more times before I become moderately well convinced that they are really equivalent. NB. Actually, I just noticed that Figure 2.6 in my copy of 'KR' has "Situation" and "Execution" instead of "Structure" and "Situation", but the webpages appear to outvote it by a two-thirds majority. To add to the multiplicity of equivalent references, we also have the boolean expansions BE and BE, that, for all their service as "canonical forms", are strictly speaking different propositional expressions, though here their equivalence is usually a whole lot easier to recognize, which is why they serve as canons or norms in the first place. Next time, I will try to address the question of why expressive variety is necessary to the very utility of language. Jon Awbrey o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at oakland.edu Thu Jun 26 11:10:27 2003 From: jawbrey at oakland.edu (Jon Awbrey) Date: Wed Jan 21 20:34:32 2004 Subject: [Inquiry] Re: Examples! Examples! Examples! References: <3EE340F2.726DAFF7@oakland.edu> <3EF82A9C.AC13ED7D@oakland.edu> <3EF860F2.3CE13DB2@oakland.edu> <3EF8AEB3.6B946BD0@oakland.edu> <3EF922AA.59221309@oakland.edu> <3EF9F9AD.4F6710E2@oakland.edu> Message-ID: <3EFB1AF3.98968853@oakland.edu> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o EEE. Note 33 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Example 1. John Sowa's "Top Level Categories" (cont.) There are a number of advantages to taking the semiotic or the sign-relational view of logic. For one thing it becomes possible to consider our use of logical languages as a special case of more general, that is to say, far less specialized and structured cases of sign usage. This helps to place the normative science of logic within a larger context other normative sciences, and even further, within the overlapping frames of the descriptive sciences that are designed to study the actual course of adaptation, communication, information, learning, and thought. But for now I just want to focus on the concrete question of why it is useful to have many ways to say the same thing, given that we have to spend so much time after the fact trying to recognize the equivalence or the near equivalence of different expressions of the facts. I think that getting some insight into the nature of this phenomenon would take us a measured distance in the work to achieve the intercoordination of diverse ontologies and views. To get an object lesson in what this means, go back to the two different axioms for TLC that I gathered at the following site: http://suo.ieee.org/email/msg10094.html Consider the kinds of arguments that the champions of Axiom !a! and the champions of Axiom !c! might get into about each and every item of their nevertheless common discursive alphabet !TLC! = {Abstract, ..., Structure}, whether Actuality is a genus consisting of the species Object and Process, as the Alphists say, or whether Actuality is defined as the conjunction of Independent and Physical, as the Gammists say, when all the time there is nothing of substance behind the appearance of contention. Now, this is a dilbertly simple example, but I was taught that there is a lot to be learned from isolating the roots of complex phenomena in the _|_-most situations where something similar is found to arise. So I think that these zeroth order cases are worth our due attention. Jon Awbrey o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o