From jawbrey at att.net Tue Jul 1 08:04:42 2003 From: jawbrey at att.net (Jon Awbrey) Date: Wed Jan 21 20:34:32 2004 Subject: [Inquiry] test Message-ID: <3F0186EA.D49D7498@att.net> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o testing 1, 2, 3, ..., oo o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at att.net Tue Jul 1 09:12:30 2003 From: jawbrey at att.net (Jon Awbrey) Date: Wed Jan 21 20:34:32 2004 Subject: [Inquiry] Intractatus Message-ID: <3F0196CE.7C217063@att.net> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o The thing is that we do not get to decide what the problems are, and we do not get to decide what the most fitting solutions are. Nature sets the problems and picks the solutions, and all we get a hand in is measuring the shadows that they cast on our grounds. It's the darndest thing about intractable and undecidable problems -- they won't go away just because we call them names. And if Nature tosses a problem our way, and says solve it or die, well, so be it. We'll just have to do our best. Now, what is that? Jon Awbrey o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at att.net Tue Jul 1 13:28:02 2003 From: jawbrey at att.net (Jon Awbrey) Date: Wed Jan 21 20:34:32 2004 Subject: [Inquiry] Re: Intractatus References: <3F0194DC.D1786EC9@att.net> Message-ID: <3F01D2B2.E6F536F3@att.net> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | 5.5. Half a loaf ... | | If we simply 'have' to solve an NP-complete problem, | then we are faced with a very long computation. | Is there anything that can be done to lighten | the load? In a number of cases various kinds | of probabilistic and approximate algorithms | have been developed, some very ingenious, | and these may often be quite serviceable, | as we have already seen in the case of | primality testing. Here are some | categories of "near" solutions | that have been developed. Et sic deinceps, he goes on from there, but most importantly, he does go on ... | Reference | | Wilf, Herbert S., |'Algorithms and Complexity', | Prentice-Hall, Englewood Cliffs, NJ, 1986. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at att.net Tue Jul 1 22:56:05 2003 From: jawbrey at att.net (Jon Awbrey) Date: Wed Jan 21 20:34:32 2004 Subject: [Inquiry] Re: Examples! Examples! Examples! Message-ID: <3F0257D5.7695755@att.net> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o EEE. Note 34 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Semiotic Reflections (cont.) There are a couple questions that come to mind when we look at pictures of the following sort. 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 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 Question 1. What is the reason for the internal diversity of languages? Question 2. What is the reason for the external diversity of languages? By "internal diversity" I mean the fact that there are generally so many ways to say the same thing in any language that addresses a non-trivial object domain. By "external diversity" I mean the fact that there are generally so many different languages that address the same object domain. I will hazard a guess at the first puzzle, putting off the second for the time being. To ask the question in a more concrete way: If languages are intended for effective communication and efficient expression, then why not just design them so that there is only one expression in each referential or semiotic equivalence class, in effect, making the whole language consist of canonical expressions? I take my first clue to the off-canonical mystery from the experiences that I used to have learning and teaching the "Art of the Story Problem" in basic mathematics courses. It had been discovered fairly early on in the pedagogy of mathematics that if students honed their formal skills on nothing but the formal manipulations that are permissible within a single formal language, then they would stare blankly at any realistic situation where those very formulas might apply. And I can take another hint from the character of the material that I was very often teaching, namely, the practice of transforming arbitrary matrices into canonical forms. The answer that suggests itself in the light of these reflections is this: Problems are posed in obscure forms from outside the sphere of our own convenience, and it is the job of language to cast a wide enough net to catch them in the form that they initially and inchoately appear. One observes that this is yet another variation on the cybernetic theme of regulating variety with variety, what Ashby called the "law of requisite variety". Jon Awbrey o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at att.net Wed Jul 2 07:22:04 2003 From: jawbrey at att.net (Jon Awbrey) Date: Wed Jan 21 20:34:32 2004 Subject: [Inquiry] Re: Examples! Examples! Examples! References: <3F0257D5.7695755@att.net> Message-ID: <3F02CE6C.ECEC895F@att.net> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o EEE. Note 35 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Semiotic Reflections (cont.) The hypothesis has suggested itself -- it is conventional to blame the hypothesis for suggesting itself in such risky cases, as it can hardly cry foul if one later steals the credit should it turn out to be apt -- that the internal diversity of languages is to be explained in part by their function in such cybernetic duties and governing offices as adaptation, control, homeostasis, information, regulation, all the ways that organisms, organizations, and other sorts of optimal control systems strive to stave off chaos and the hot breath of entropy by introjecting small models of the very same nemesystematic enmities within their own frame. An economic principle suggests -- again one tends to personify the principle -- that we might as well try to take two birds with one stone, if one stone is all we have, and so I will take up the hypothesis that linguistic variety in general, external or internal to a given language, is to be accounted, at least in some significant measure, to the life-saving utilities thereof. Pursuing the logical consequences and the practical consequences -- I do not say how they overlap -- of this hypothesis is the task that I take up next. Jon Awbrey o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at att.net Wed Jul 2 15:39:00 2003 From: jawbrey at att.net (Jon Awbrey) Date: Wed Jan 21 20:34:32 2004 Subject: [Inquiry] Re: Examples! Examples! Examples! References: <3F0257D5.7695755@att.net> <3F02CE6C.ECEC895F@att.net> Message-ID: <3F0342E4.F1820BC6@att.net> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o EEE. Note 36 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Semiotic Reflections (cont.) I continue to entertain the hypothesis -- and the hypothesis like no other patient object can hardly deny that it's being entertained -- that the good of linguistic diversity is near and dear to "life itself", to quoin a phrase already coursed, and thus contingent on your accepting that life itself is in some sense "practically indispensable", we may say that this form of variety, or variety of form, to cross the chiasma in all due course, is not to be dispensed with in practice, not, of course, while life yet lives. Therefore, let us trace the consequences, practical or theoretical, of the late-noted linguistic diversity, within or without a tongue -- not by this "without a tongue" to say "non-verbal", but then again, not to say not. The consequence that comes to my particular mind is this: Infinite diversity poses an obstacle to nominal thinking. Et sic deinceps ... Jon Awbrey o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at att.net Thu Jul 3 14:08:05 2003 From: jawbrey at att.net (Jon Awbrey) Date: Wed Jan 21 20:34:32 2004 Subject: [Inquiry] Re: Examples! Examples! Examples! References: <3F0257D5.7695755@att.net> <3F02CE6C.ECEC895F@att.net> <3F0342E4.F1820BC6@att.net> Message-ID: <3F047F15.A20F5442@att.net> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o EEE. Note 37 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Semiotic Reflections (cont.) I have kept before you a particular picture of the way that our significant languages relate to our objective realities -- there's a double meaning in that, that word "objective", that can also refer to an intentional object, a thing not yet, and yet to be, if our aim be ept -- it's just one of the dimensions of meaning that is now largely lost in the translation from the Greek grammed "pragma" to the Latin sieved and winnowed "object". 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 But for the present I ought to say what I intend by this: Infinite diversity poses an obstacle to nominal thinking. Let's look to the future contingent story of a prospectively many-wiled problem-solving siftware -- to give it a name and a local habituation, and thus personify the object narrative of our quest, let's call it "him", where he be "Peirce Eval". A problem about the relationship between signs and more signs and objects presents itself to our many-wiled hero. What is the quickest and easiest wile that he downlifts from off his pegboared of wiles? Well, one way to reduce the problem is to ignore the problematic domain of objects in toto, whether intentional or clear and present, and one way to get away with it is suggested by the very k-ptyches that we have staring at all this while. Here is how it just might work. Suppose that we had in hand a complete survey of all the signs that we needed to use for denoting the objects in our pragmatic domain. Suppose, also, that we had them all the signs nicely partitioned and sorted into the mutually exclusively exhaustive parts that refer to the several objects in our objective domain. Well, then, we would have a form-preserving copy of the object domain quite well in hand with the quotient object of the sign domain and the object domain should be a "world well wiled" out of mind. That might leave some people quite content with the 2-dim or 1-dim or 0-dim world that is weft behind, according to their taste and just how far they were wont to go, but it does not manage to master the possibilities, not if the real world is infinitely diverse, for example, if it just keeps on sending new objects, classifiable or otherwise, not to mention, till we make a sign to mention them under, entirely novel objects. So we find ourselves forced by the nature of reality to the point of standing on the standpoint that shall henceforward be called "rational thinking", at least, at the very least. Jon Awbrey o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at att.net Thu Jul 3 15:16:08 2003 From: jawbrey at att.net (Jon Awbrey) Date: Wed Jan 21 20:34:33 2004 Subject: [Inquiry] Re: Examples! Examples! Examples! References: <3F0257D5.7695755@att.net> <3F02CE6C.ECEC895F@att.net> <3F0342E4.F1820BC6@att.net> Message-ID: <3F048F08.83F3AD36@att.net> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o EEE. Note 37 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Semiotic Reflections (cont.) I have kept before you a particular picture of the way that our significant languages relate to our objective realities -- there's a double meaning in that, that word "objective", that can also refer to an intentional object, a thing not yet, and yet to be, if our aim be ept -- it's just one of the dimensions of meaning that is now largely lost in the translation from the Greek grammed "pragma" to the Latin sieved and winnowed "object". 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 But for the present I ought to say what I intend by this: Infinite diversity poses an obstacle to nominal thinking. Let's look to the future contingent story of a prospectively many-wiled problem-solving siftware -- to give it a name and a local habituation, and thus personify the object narrative of our quest, let's call it "him", where he be "Peirce Eval". A problem about the relationship between signs and more signs and objects presents itself to our many-wiled hero. What is the quickest and easiest wile that he down-lifts from off his pegboard of wiles? Well, one way to reduce the problem is to ignore the problematic domain of objects in toto, whether intentional or clear and present, and one way to get away with it is suggested by the very k-ptyches that we have staring at all this while. Here is how it just might work. Suppose that we had in hand a complete survey of all the signs that we needed to use for denoting the objects in our pragmatic domain. Suppose, also, that we had our gangs of signs nicely partitioned and sorted into the mutually exclusively exhaustive parts that refer to the several objects in our objective domain. Well, then, we would have a form-preserving copy of the object domain quite well in hand with the quotient object of the sign domain and the object domain should be a "world well wiled" out of mind. That might leave some people quite content with the 2-dim or 1-dim or 0-dim world that is weft behind, according to their taste and just how far they were wont to go, but it does not manage to master the possibilities, not if the real world is infinitely diverse, for example, if it just keeps on sending new objects, classifiable or otherwise, not to mention, till we make a sign to mention them under, entirely novel objects. So we find ourselves forced by the nature of reality to the point of standing on the standpoint that shall henceforward be called "rational thinking", at least, at the very least. Jon Awbrey o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at att.net Thu Jul 3 22:44:05 2003 From: jawbrey at att.net (Jon Awbrey) Date: Wed Jan 21 20:34:33 2004 Subject: [Inquiry] Re: Examples! Examples! Examples! References: <3F0257D5.7695755@att.net> <3F02CE6C.ECEC895F@att.net> <3F0342E4.F1820BC6@att.net> <3F048F08.83F3AD36@att.net> Message-ID: <3F04F805.64171D97@att.net> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o EEE. Note 38 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Semiotic Reflections (cont.) Last time we found a principle of thinking so manifestly indispensable as a lodestar of animated rationality that even thinkers as far apart as Descartes and Peirce could not detect a sliver of parallax in its apparent position. With respect, then, to the 3-adic being of sign relations, both angles of possible reduction to their complexity are now finally and utterly cut off. We cannot dispense with the thickness of the connotational fiber without quashing the requisite variety that a sign system needs to capture a world within its net. Nor can we keep the reticulum of signs to dispense with the objects that are its rationale. There is no escape from taking sign relations at the full. Jon Awbrey o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at att.net Wed Jul 9 15:32:01 2003 From: jawbrey at att.net (Jon Awbrey) Date: Wed Jan 21 20:34:33 2004 Subject: [Inquiry] Re: Category Theory References: <3EB3C21E.B2D6F03B@oakland.edu> <3EB3C5CC.7C6C962F@oakland.edu> <3EB3C939.C1366AF5@oakland.edu> <3EB3D36C.C67D095A@oakland.edu> <3EB3F8FF.6D3B4A76@oakland.edu> <3EB47D9C.40DD635C@oakland.edu> <3EB5B36C.3BCDD88@oakland.edu> <3EB5E8FB.40C613AE@oakland.edu> Message-ID: <3F0C7BC1.F1B42EF8@att.net> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o CAT. Note 9 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | 1.1. Axioms for Categories (cont.) | | A metacategory is to be any interpretation which satisfies all these axioms. | An example is the 'metacategory of sets', which has objects all sets and | arrows all functions, with the usual identity functions and the usual | composition of functions. Here "function" means a function with | specified domain and specified codomain. Thus a function | f : X -> Y consists of a set X, its domain, a set Y, | its codomain, and a rule x ~> fx (i.e., a suitable | set of ordered pairs ) which assigns, to | each element x in X, an element fx in Y. These | values will be written as fx, f_x, or f(x), as | may be convenient. For example, for any set S, | the assignment s ~> s for all s in S describes | the 'identity function' 1_S : S -> S; if S is a | subset of Y, the assignment s ~> s also describes | the 'inclusion' or 'insertion function' S -> Y; | these functions are 'different' unless S = Y. | Given functions f : X -> Y and g : Y -> Z, | the 'composite' function g o f : X -> Z is | defined by (g o f)x = g(fx) for all x in X. | Observe that g o f will mean first apply f, | then g -- in keeping with the practice of | writing each function f to the left of its | argument. Note, however, that many authors | use the opposite convention. | | To summarize, the metacatgory of all sets has as | objects, all sets, as arrows, all functions with the | usual composition. The metacategory of all groups is | described similarly: Objects are all groups G, H, K; | arrows are all those functions f from the set G to | the set H for which f : G -> H is a homomorphism | of groups. There are many other metacategories: | All topological spaces with continuous functions | as arrows; all compact Hausdorff spaces with the | same arrows; all ringed spaces with their morphisms, | etc. The arrows of any metacategory are often called | its 'morphisms'. | | Mac Lane, 'Cat Work Math', pp. 8-9. | | Saunders Mac Lane, |'Categories for the Working Mathematician', | 2nd edition, Springer, New York, NY, 1997. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at att.net Wed Jul 9 19:00:02 2003 From: jawbrey at att.net (Jon Awbrey) Date: Wed Jan 21 20:34:33 2004 Subject: [Inquiry] Re: Category Theory References: <3EB3C21E.B2D6F03B@oakland.edu> <3EB3C5CC.7C6C962F@oakland.edu> <3EB3C939.C1366AF5@oakland.edu> <3EB3D36C.C67D095A@oakland.edu> <3EB3F8FF.6D3B4A76@oakland.edu> <3EB47D9C.40DD635C@oakland.edu> <3EB5B36C.3BCDD88@oakland.edu> <3EB5E8FB.40C613AE@oakland.edu> <3F0C7BC1.F1B42EF8@att.net> Message-ID: <3F0CAC82.D8BD9D3@att.net> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o CAT. Note 10 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | 1.1. Axioms for Categories (concl.) | | Since the objects of a metacategory correspond exactly to | its identity arrows, it is technically possible to dispense | altogether with the objects and deal only with arrows. The | data for an 'arrows-only metacategory' C consist of arrows, | certain ordered pairs , called the composable pairs of | arrows, and an operation assigning to each composable pair | an arrow g o f, called their composite. We say | "g o f" is defined" for " is a composable pair". | | With these data one 'defines' an identity of C to be an arrow u | such that f o u = f whenever the composite f o u is defined and | u o g = g whenever u o g is defined. The data are then required | to satisfy the following three axioms: | | 1. The composite (k o g) o f is defined if and only if | the composite k o (g o f) is defined. When either is | defined, they are equal (and this 'triple composite' is | written as k o g o f). | | 2. The triple composite k o g o f is defined | whenever both composites k o g and g o f | are defined. | | 3. For each arrow g of C there exist identity arrows | u and u' of C such that u' o g and g o u are defined. | | In view of the explicit definition given above for | identity arrows, the last axiom is a quite powerful | one; it implies that u' and u are unique in (3), and | it gives for each arrow g a codomain u' and a domain u. | These axioms are equivalent to the preceding ones. More | explicitly, given a metacategory of objects and arrows, | its arrows, with the given composition, satisfy the | "arrows-only" axioms; conversely, an arrows-only | metacategory satisfies the objects-and-arrows | axioms when the identity arrows, defined as | above, are taken as the objects (Proof as | exercise). | | Mac Lane, 'Cat Work Math', p. 9. | | Saunders Mac Lane, |'Categories for the Working Mathematician', | 2nd edition, Springer, New York, NY, 1997. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at att.net Wed Jul 9 21:24:37 2003 From: jawbrey at att.net (Jon Awbrey) Date: Wed Jan 21 20:34:33 2004 Subject: [Inquiry] Re: Category Theory References: <3EB3C21E.B2D6F03B@oakland.edu> Message-ID: <3F0CCE65.456F735F@att.net> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o CAT. Note 11 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | 1.2. Categories | | A category (as distinguished from a metacategory) will | mean any interpretation of the category axioms within | set theory. Here are the details. A 'directed graph' | (also called a "diagram scheme") is a set O of objects, | a set A of arrows, and two functions: | | dom | -------> | A O (1) | -------> | cod | | In this graph, the set of composable pairs of arrows is the set: | | A x_O A = { : g, f in A and dom g = cod f}, | | called the "product over O". | | A 'category' is a graph with two additional functions: | | id o | 1. O ------> A, 2. A x_O A -----> A, | (2) | c ~~~~~~> id_c, ~~~~~> g o f, | | called identity and composition, | [the latter] also written as g f, | such that: | | dom(id_a) = a = cod(id_a), | | dom(g o f) = dom f, | | cod(g o f) = cod g, (3) | | for all objects a in O and all composable pairs | of arrows in A x_O A, and such that the | associativity and unit axioms (1.1) and (1.2) | hold. In treating a category C, we usually | drop the letters A and O, and write: | | c C, f in C (4) | | for "c is an object of C" and "f is an arrow of C", | respectively. We also write: | | hom(b, c) = {f : f in C, dom f = b, cod f = c} (5) | | for the set of arrows from b to c. Categories can | be defined directly in terms of composition acting | on these "hom-sets" (Section 8 below); we do not | follow this custom because we put the emphasis | not on sets (a rather special category), but | on axioms, arrows, and diagrams of arrows. | We will later observe that our definition | of a category amounts to saying that a | category is a monoid for the product | x_O, in the general sense described | in the introduction. | | Mac Lane, 'Cat Work Math', p. 10. | | Saunders Mac Lane, |'Categories for the Working Mathematician', | 2nd edition, Springer, New York, NY, 1997. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at att.net Thu Jul 10 08:24:05 2003 From: jawbrey at att.net (Jon Awbrey) Date: Wed Jan 21 20:34:33 2004 Subject: [Inquiry] Re: Category Theory References: <3EB3C21E.B2D6F03B@oakland.edu> <3F0CCE65.456F735F@att.net> Message-ID: <3F0D68F5.EC7C5725@att.net> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o CAT. Note 12 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o NB. Mac Lane uses a symbol for the one object and one (identity) arrow category that looks like a dot with a sling out of it and an arrow back into it. I will use a "Greek amphora" or "emphattic at" sign for this, like so "!@!". | 1.2. Categories (cont.) | | For the moment, we consider examples. | | $0$ is the empty category (no objects, no arrows). | | $1$ is the category !@! with one object and one (identity) arrow. | | $2$ is the category !@! -> !@! with two objects a, b, | and just one arrow a -> b not the identity. | | $3$ is the category with three objects whose non-identity arrows | are arranged as in the triangle [in the "transitive" manner]: | | o | ^ \ | / v | o---->o | | $||$ is the category with two objects a, b and just two | arrows a -> b not the identity arrows. We call two | such arrows 'parallel arrows'. | | In each of the cases above there is only | one possible definition of composition. | | Mac Lane, 'Cat Work Math', pp. 10-11. | | Saunders Mac Lane, |'Categories for the Working Mathematician', | 2nd edition, Springer, New York, NY, 1997. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at att.net Thu Jul 10 10:11:18 2003 From: jawbrey at att.net (Jon Awbrey) Date: Wed Jan 21 20:34:33 2004 Subject: [Inquiry] Re: Category Theory References: <3EB3C21E.B2D6F03B@oakland.edu> <3F0CCE65.456F735F@att.net> <3F0D68F5.EC7C5725@att.net> Message-ID: <3F0D8216.93FC0416@att.net> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o CAT. Note 13 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | 1.2. Categories (cont.) | | Discrete Categories. A category is 'discrete' when every arrow | is an identity. Every set X is the set of objects of a discrete | category (just add one identity arrow x -> x for each x in X), | and every discrete category is so determined by its set of | objects. Thus, discrete categories are sets. | | Monoids. A monoid is a category with one object. Each monoid is thus | determined by the set of all its arrows, by the identity arrow, and | by the rule for the composition of arrows. Since any two arrows | have a composite, a monoid may then be described as a set M with | a binary operation M x M -> M which is associative and has an | identity (= unit). Thus a monoid is exactly a semigroup with | identity element. For any category C and any object a in C, | the set hom(a, a) of all arrows a -> a is a monoid. | | Groups. A group is a category with one object in which | every arrow has a (two-sided) inverse under composition. | | Matrices. For each commutative ring K, the set Matr_K of | all rectangular matrices with entries in K is a category; | the objects are all positive integers m, n, ..., and each | m x n matrix A is regarded as an arrow A : n -> m, with | composition the usual matrix product. | | Sets. If V is any set of sets, we take Ens_V to be the category | with objects all sets X in V, arrows 'all' functions f : X -> Y, | with the usual composition of functions. By Ens we mean any one | of these categories. | | Mac Lane, 'Cat Work Math', p. 11. | | Saunders Mac Lane, |'Categories for the Working Mathematician', | 2nd edition, Springer, New York, NY, 1997. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at att.net Thu Jul 10 11:04:06 2003 From: jawbrey at att.net (Jon Awbrey) Date: Wed Jan 21 20:34:33 2004 Subject: [Inquiry] Re: Category Theory References: <3EB3C21E.B2D6F03B@oakland.edu> <3F0CCE65.456F735F@att.net> <3F0D68F5.EC7C5725@att.net> <3F0D8216.93FC0416@att.net> Message-ID: <3F0D8E76.25EB317E@att.net> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o CAT. Note 14 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | 1.2. Categories (cont.) | | Preorders. By a preorder we mean a category P in which, given objects | p and p', there is at most one arrow p -> p'. In any preorder P, define | a binary relation =< on the objects of P with p =< p' if and only if there | is an arrow p -> p' in P. This binary relation is reflexive (because there | is an identity arrow p -> p for each p) and transitive (because arrows can be | composed). Hence a preorder is a set (of objects) equipped with a reflexive | and transitive binary relation. Conversely, any set P with such a relation | determines a preorder, in which the arrows p -> p' are exactly those ordered | pairs for which p =< p'. Since the relation is transitive, there is | a unique way of composing these arrows; since it is reflexive, there are the | necessary identity arrows. | | Preorders include 'partial orders' (preorders with the added axiom that | p =< p' and p' =< p imply p = p') and 'linear orders' (partial orders | such that, given p and p', either p =< p' or p' =< p). | | Mac Lane, 'Cat Work Math', p. 11. | | Saunders Mac Lane, |'Categories for the Working Mathematician', | 2nd edition, Springer, New York, NY, 1997. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at att.net Thu Jul 10 12:50:55 2003 From: jawbrey at att.net (Jon Awbrey) Date: Wed Jan 21 20:34:33 2004 Subject: [Inquiry] Re: Category Theory References: <3EB3C21E.B2D6F03B@oakland.edu> <3F0CCE65.456F735F@att.net> <3F0D68F5.EC7C5725@att.net> <3F0D8216.93FC0416@att.net> <3F0D8E76.25EB317E@att.net> Message-ID: <3F0DA77F.C1086A28@att.net> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o CAT. Note 15 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | 1.2. Categories (cont.) | | Ordinal Numbers. We regard each ordinal number n as the linearly ordered | set of all the preceding ordinals n = {0, 1, ..., n-1}; in particular, 0 | is the empty set, while the first infinite ordinal is !w! = {0, 1, 2, ...}. | Each ordinal n is linearly ordered, and hence is a category (a preorder). | For example, the categories $1$, $2$, and $3$ listed above are the preorders | belonging to the (linearly ordered) ordinal numbers 1, 2, and 3. Another | example is the linear order !w! [omega]. As a category, it consists of | the arrows: | | 0 -> 1 -> 2 -> 3 -> ..., | | all their composites, and the identity arrows for each object. | | !D! is the category with objects all finite ordinals and arrows | f : m -> n all order-preserving functions (i =< j in m implies | f_i =< f_j in n). This category !D! [Delta], sometimes called | the 'simplicial category', plays a central role (Chapter 7). | | Finord = Set_!w! is the category with objects all finite ordinals n | and arrows f : m -> n all functions from m to n. This is essentially | the category of all finite sets, using just one finite set n for each | finite cardinal number n. | | Mac Lane, 'Cat Work Math', pp. 11-12. | | Saunders Mac Lane, |'Categories for the Working Mathematician', | 2nd edition, Springer, New York, NY, 1997. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at att.net Thu Jul 10 13:56:10 2003 From: jawbrey at att.net (Jon Awbrey) Date: Wed Jan 21 20:34:33 2004 Subject: [Inquiry] Re: Category Theory References: <3EB3C21E.B2D6F03B@oakland.edu> <3F0CCE65.456F735F@att.net> <3F0D68F5.EC7C5725@att.net> <3F0D8216.93FC0416@att.net> <3F0D8E76.25EB317E@att.net> <3F0DA77F.C1086A28@att.net> Message-ID: <3F0DB6CA.86EF1271@att.net> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o CAT. Note 16 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | 1.2. Categories (concl.) | | Large Categories. In addition to the metacategory of all sets -- | which is not a set -- we want an actual category Set, the category | of all 'small' sets. We shall assume that there is a big enough set | U, the "universe", then describe a set x as "small" if it is a member | of the universe, and take Set to be the category whose set U of objects | is the set of all small sets, with arrows all functions from one small set | to another. With this device (details in Section 7 below) we construct other | familiar large categories, as follows: | | Set. Objects, all small sets; | arrows, all functions between them. | | Set_*. Pointed sets: Objects, small sets each with a selected base point; | arrows, base-point-preserving functions. | | Ens. Category of all sets and functions within a (variable) set V. | | Cat. Objects, all small categories; | arrows, all functors (Section 3). | | Mon. Objects, all small monoids; | arrows, all morphisms of monoids. | | Grp. Objects, all small groups; | arrows, all morphisms of groups. | | Ab. Objects, all small (additive) abelian groups, | with morphisms of such. | | Rng. All small rings, with the ring homomorphisms | (preserving units) between them. | | CRng. All small commutative rings and their morphisms. | | R-Mod. All small left modules over the ring R, with linear maps. | | Mod-R. Small right R-modules. | | K-Mod. Small modules over the commutative ring K. | | Top. Small topological spaces and continuous maps. | | Toph. Topological spaces, with arrows homotopy classes of maps. | | Top_*. Spaces with selected base point, | base-point-preserving maps. | | Particular categories (like these) will always appear | in bold-face type [not shown here]. Script capitals | are used by many authors to denote categories. | | Mac Lane, 'Cat Work Math', p. 12. | | Saunders Mac Lane, |'Categories for the Working Mathematician', | 2nd edition, Springer, New York, NY, 1997. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at att.net Thu Jul 10 21:54:11 2003 From: jawbrey at att.net (Jon Awbrey) Date: Wed Jan 21 20:34:33 2004 Subject: [Inquiry] Re: Category Theory References: <3EB3C21E.B2D6F03B@oakland.edu> Message-ID: <3F0E26D3.A805943B@att.net> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o CAT. Note 17 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | 1.3. Functors | | A 'functor' is a morphism of categories. In detail, for | categories C and B a functor T : C -> B with domain C and | codomain B consists of two suitably related functions: The | 'object function' T, which assigns to each object c of C an | object Tc of B and the 'arrow function' (also written T) which | assigns to each arrow f : c -> c' of C an arrow Tf : Tc -> Tc' | of B, in such a way that: | | T(1_c) = 1_Tc, T(g o f) = Tg o Tf, (1) | | the latter whenever the composite g o f is defined in C. A functor, | like a category, can be described in the "arrows-only" fashion: It | is a function T from arrows f of C to arrows Tf of B, carrying each | identity of C to an identity of B and each composable pair | in C to a composable pair in B, with Tg o Tf = T(g o f). | | A simple example is the power set functor $P$ : Set -> Set. Its object | function assigns to each set X the usual power set $P$X, with elements | all subsets S c X; its arrow function assigns to each f : X -> Y that | map $P$f : $P$X -> $P$Y which sends each S c X to its image fS c Y. | Since both $P$(1_X) = 1_$P$X and $P$(g o f) = $P$g o $P$f, this | clearly defines a functor $P$ : Set -> Set. | | Mac Lane, 'Cat Work Math', p. 13. | | Saunders Mac Lane, |'Categories for the Working Mathematician', | 2nd edition, Springer, New York, NY, 1997. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at att.net Fri Jul 11 23:00:52 2003 From: jawbrey at att.net (Jon Awbrey) Date: Wed Jan 21 20:34:34 2004 Subject: [Inquiry] Re: Category Theory References: <3EB3C21E.B2D6F03B@oakland.edu> <3F0E26D3.A805943B@att.net> Message-ID: <3F0F87F4.7DA731A5@att.net> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o CAT. Note 18 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o NB. When necessary to embolden characters, I will use percent brackets, for example: %R% = the real numbers, %Z% = the integers. | 1.3. Functors (cont.) | | Functors were first explicitly recognized in algebraic topology, | where they arise naturally when geometric properties are described | by means of algebraic invariants. | | For example, singular homology in a given dimension n (n a natural number) | assigns to each topological space X an abelian group H_n (X), the n^th | homology group of X, and also to each continuous map f : X -> Y of | spaces a corresponding homomorphism H_n (f) : H_n (X) -> H_n (Y) | of groups, and this in such a way that H_n becomes a functor | Top -> Ab. | | For example, if X = Y = S^1 is the circle, H_1 (S^1) = %Z%, so | the group homomorphism H_1 (f) : %Z% -> %Z% is determined by | an integer d (the image of 1); this integer is the usual | "degree" of the continuous map f : S^1 -> S^1. In this | case and in general, homotopic maps f, g : X -> Y yield | the same homomorphism H_n (X) -> H_n (Y), so H_n can | actually be regarded as a functor Toph -> Grp, | defined on the homotopy category. | | The Eilenberg-Steenrod axioms for homology start with the axioms | that H_n, for each natural number n, is a functor on Toph, and | continue with certain additional properties of these functors. | The more recently developed extraordinary homology and | cohomology theories are also functors on Toph. | | The homotopy groups !p!_n (X) of a space X can also be regarded as | functors; since they depend on the choice of a base point in X, | they are functors Top_* -> Grp. | | The leading idea in the use of functors in topology is that H_n or !p!_n | gives an algebraic picture or image not just of the topological spaces, | but also of all the continuous maps between them. | | Mac Lane, 'Cat Work Math', p. 13. | | Saunders Mac Lane, |'Categories for the Working Mathematician', | 2nd edition, Springer, New York, NY, 1997. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at att.net Sat Jul 12 20:54:26 2003 From: jawbrey at att.net (Jon Awbrey) Date: Wed Jan 21 20:34:34 2004 Subject: [Inquiry] Quine -- Two Dogmas Of Empiricism Message-ID: <3F10BBD2.DD0B3EFE@att.net> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o TDOE. Note 1 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | Two Dogmas of Empiricism | | Modern empiricism has been conditioned in large part by two dogmas. | One is a belief in some fundamental cleavage between truths which | are 'analytic', or grounded in meanings independently of matters | of fact, and truths which are 'synthetic', or grounded in fact. | The other dogma is 'reductionism': the belief that each | meaningful statement is equivalent to some logical | construct upon terms which refer to immediate | experience. Both dogmas, I shall argue, are | ill-founded. One effect of abandoning them | is, as we shall see, a blurring of the | supposed boundary between speculative | metaphysics and natural science. | Another effect is a shift | toward pragmatism. | | Quine, "Two Dogmas", p. 20. | | W.V. Quine, |"Two Dogmas of Empiricism", 'Philosophical Review', January 1951. | Reprinted as pages 20-46 in 'From a Logical Point of View', | 2nd edition, Harvard University Press, Cambridge, MA, 1980. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at att.net Sat Jul 12 22:04:02 2003 From: jawbrey at att.net (Jon Awbrey) Date: Wed Jan 21 20:34:34 2004 Subject: [Inquiry] Re: Category Theory References: <3EB3C21E.B2D6F03B@oakland.edu> <3F0E26D3.A805943B@att.net> <3F0F87F4.7DA731A5@att.net> Message-ID: <3F10CC22.40898FC4@att.net> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o CAT. Note 19 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | 1.3. Functors (cont.) | | Functors arise naturally in algebra. | | To any commutative ring K the set of all non-singular | n x n matrices with entries in K is the usual general | linear group GL_n (K); moreover, each homomorphism | f : K -> K' of rings produces in the evident way a | homomorphism GL_n f : GL_n (K) -> GL_n (K') of groups. | These data define for each natural number n a functor | GL_n : CRng -> Grp. | | For any group G the set of all products of commutators x y x^(-1) y^(-1), | (x, y in G), is a normal subgroup [G, G] of G, called the 'commutator' | subgroup. Since any homomorphism G -> H of groups carries commutators | to commutators, the assignment G ~> [G, G] defines an evident functor | Grp -> Grp, while G ~> G/[G, G] defines a functor Grp -> Ab, the | factor-commutator functor. Observe, however, that the center Z(G) | of G (all a in G with ax = xa for all x) does not naturally define | a functor Grp -> Grp, because a homomorphism G -> H may carry an | element in the center of G to one not in the center of H. | | A functor which simply "forgets" some or all of the structure of an | algebraic object is commonly called a 'forgetful' functor (or, an | 'underlying' functor). Thus the forgetful functor U : Grp -> Set | assigns to each group G the set UG of its elements ("forgetting" | the multiplication and hence the group structure), and assigns | to each morphism f : G -> G' of groups the same function f, | regarded just as a function between sets. The forgetful | functor U : Rng -> Ab assigns to each ring R the additive | abelian group of R and to each morphism f : R -> R' of | rings the same function, regarded just as a morphism | of addition. | | Mac Lane, 'Cat Work Math', p. 14. | | Saunders Mac Lane, |'Categories for the Working Mathematician', | 2nd edition, Springer, New York, NY, 1997. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at att.net Sun Jul 13 10:08:04 2003 From: jawbrey at att.net (Jon Awbrey) Date: Wed Jan 21 20:34:34 2004 Subject: [Inquiry] Re: Category Theory References: <3EB3C21E.B2D6F03B@oakland.edu> <3F0E26D3.A805943B@att.net> <3F0F87F4.7DA731A5@att.net> <3F10CC22.40898FC4@att.net> Message-ID: <3F1175D4.A0BA1225@att.net> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o CAT. Note 20 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | 1.3. Functors (cont.) | | Functors may be composed. Explicitly, given functors: | | T S | C ---> B ---> A | | between categories A, B, and C, the composite functions: | | c ~> S(Tc), f ~> S(Tf) | | on objects c and arrows f of C define a functor S o T : C -> A, called the | 'composite' (in that order) of S with T. This composition is associative. | For each category B there is an identity functor I_B : B -> B, which acts as | an identity for this composition. Thus we may consider the metacategory of | all categories: its objects are all categories, its arrows are all functors | with the composition above. Similarly, we may form the category Cat of all | small categories -- but not the category of all categories. | | Mac Lane, 'Cat Work Math', p. 14. | | Saunders Mac Lane, |'Categories for the Working Mathematician', | 2nd edition, Springer, New York, NY, 1997. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at att.net Sun Jul 13 11:00:18 2003 From: jawbrey at att.net (Jon Awbrey) Date: Wed Jan 21 20:34:34 2004 Subject: [Inquiry] Re: Category Theory References: <3EB3C21E.B2D6F03B@oakland.edu> <3F0E26D3.A805943B@att.net> <3F0F87F4.7DA731A5@att.net> <3F10CC22.40898FC4@att.net> <3F1175D4.A0BA1225@att.net> Message-ID: <3F118212.FA8D80AE@att.net> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o CAT. Note 21 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | 1.3. Functors (cont.) | | An 'isomorphism' T : C -> B of categories is a functor | T from C to B which is a bijection, both on objects and | on arrows. Alternatively, but equivalently, a functor | T : C -> B is an isomorphism if and only if there is a | functor S : B -> C for which both composites S o T and | T o S are identity functors; then S is the 'two-sided | inverse' S = T^(-1). | | Certain properties much weaker than isomorphism will be useful. | | A functor T : C -> B is 'full' when to every pair c, c' of objects of C | and to every arrow g : Tc -> Tc' of B, there is an arrow f : c -> c' of C | with g = Tf. Clearly the composite of two full functors is a full functor. | | A functor T : C -> B is 'faithful' (or an embedding) when to every pair | c, c' of objects of C and to every pair f_1, f_2 : c -> c' of parallel | arrows of C the equality Tf_1 = Tf_2 : Tc -> Tc' implies f_1 = f_2. | Again, composites of faithful functors are faithful. For example, | the forgetful functor Grp -> Set is faithful but not full and | not a bijection on objects. | | These two properties may be visualized in terms of hom-sets (see (2.5)). | Given a pair of objects c, c' in C, the arrow function of T : C -> B | assigns to each f : c -> c' an arrow Tf : Tc -> Tc' and so defines | a function: | | T_c,c' : hom(c, c') -> hom(Tc, Tc'), f ~> Tf. | | Then T is full when every such function is surjective, and faithful | when every such function is injective. For a functor which is both | full and faithful (i.e., "fully faithful"), every such function is | a bijection, but this need not mean that the functor itself is an | isomorphism of categories, for there may be objects of B not in | the image of T. | | Mac Lane, 'Cat Work Math', pp. 14-15. | | Saunders Mac Lane, |'Categories for the Working Mathematician', | 2nd edition, Springer, New York, NY, 1997. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at att.net Sun Jul 13 16:54:22 2003 From: jawbrey at att.net (Jon Awbrey) Date: Wed Jan 21 20:34:34 2004 Subject: [Inquiry] Re: Category Theory References: <3EB3C21E.B2D6F03B@oakland.edu> <3F0E26D3.A805943B@att.net> <3F0F87F4.7DA731A5@att.net> <3F10CC22.40898FC4@att.net> <3F1175D4.A0BA1225@att.net> <3F118212.FA8D80AE@att.net> Message-ID: <3F11D50E.AFAD5B8A@att.net> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o CAT. Note 22 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | 1.3. Functors (concl.) | | A 'subcategory' S of a category C is a collection of | some of the objects and some of the arrows of C, which | includes with each arrow f both the object dom f and the | object cod f, with each object s its identity arrow 1_s, | and with each pair of composable arrows s -> s' -> s" | their composite. These conditions ensure that these | collections of objects and arrows themselves constitute | a category S. Moreover, the injection (inclusion) map | S -> C which sends each object and each arrow of S to | itself (in C) is a functor, the 'inclusion functor'. | This inclusion functor is automatically faithful. | | We say that S is a 'full subcategory' of C when the inclusion functor | S -> C is full. A full subcategory, given C, is thus determined by | giving just the set of its objects, since the arrows between any two | of these objects s, s' are all morphisms s -> s' in C. For example, | the category Set_f of all finite sets is a full subcategory of the | category Set. | | Mac Lane, 'Cat Work Math', p. 15. | | Saunders Mac Lane, |'Categories for the Working Mathematician', | 2nd edition, Springer, New York, NY, 1997. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at att.net Sun Jul 13 22:40:35 2003 From: jawbrey at att.net (Jon Awbrey) Date: Wed Jan 21 20:34:34 2004 Subject: [Inquiry] Re: Category Theory References: <3EB3C21E.B2D6F03B@oakland.edu> Message-ID: <3F122633.65F60F4@att.net> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o CAT. Note 23 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | 1.4. Natural Transformations | | Given two functors S, T : C -> B, a 'natural transformation' | !t! : S -> T is a function which assigns to each object c of C | an arrow !t!_c = !t!c : Sc -> Tc of B in such a way that every | arrow f : c -> c' in C yields a diagram: | | !t!c | c o Sc o------------>o Tc | | | | | | | | | f | Sf | | Tf (1) | | | | | v v v | c' o Sc' o------------>o Tc' | !t!c' | | which is commutative. When this holds, we also say that | !t!_c : Sc -> Tc is 'natural' in c. If we think of the | functor S as giving a picture in B of (all the objects | and arrows of) C, then a natural transformation !t! is | a set of arrows mapping (or, translating) the picture S | to the picture T, with all squares (and parallegrams!) | like that above commutative: | | a Sa !t!a Ta | o o---------------------->o | |\ |\ |\ | | \ f | \ Sf | \ Tf | | \ | \ | \ | | v | v Sb Th | v | h | o b Sh | o------------------|--->o Tb | | / | / !t!b | / | | / | / | / | | / g | / Sg | / Tg | vv vv vv | o o---------------------->o | c Sc !t!c Tc | | We call !t!a, !t!b, !t!c, ..., the 'components' | of the natural transformation !t!. | | A natural transformation is often called a 'morphism of functors'; | a natural transformation !t! with every component !t!c invertible in | B is called a 'natural equivalence' or better a 'natural isomorphism'; | in symbols, !t! : S ~=~ T. In this case, the inverses (!t!c)^(-1) in B | are the components of a natural isomorphism !t!^(-1) : T -> S. | | Mac Lane, 'Cat Work Math', p. 16. | | Saunders Mac Lane, |'Categories for the Working Mathematician', | 2nd edition, Springer, New York, NY, 1997. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at att.net Mon Jul 14 10:46:41 2003 From: jawbrey at att.net (Jon Awbrey) Date: Wed Jan 21 20:34:35 2004 Subject: [Inquiry] Notes On Categories Message-ID: <3F12D061.35E82758@att.net> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o NOC. Note 1 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Here I will document some of the computational approaches to category theory that I starting working on back in the 1980's, all of which work as yet remains in the "Schubert Category" of unfinished symphonies. It helps me a little bit to write the names of categories in the plural, so as not to confuse them with individuals. It also helps if I treat the arrows of Arr(C) as the primary entities in the category C, recovering the objects of Obj(C) as secondary entities by collecting all the entities that appear in s(f) = Source(f) and t(f) = Target(f) as one ranges over all of the arrows f in Arr(C). The last time that I tried to do "categories by computer", I was using data structures that had the following shapes: Category C o /|\ / | \ ... | ... | Arrow f o / \ s t / \ s(f) o o t(f) A functor, then, is something that I picture like this: Functor F o . | . . | . . | . . | . Category C o o o Category D = CF | ./ \. | | . / \ . | | . / \ . | | . / \ . | Arrow f o o o o Arrow fF / \ . . . . / \ / .\ . . /. \ s . t . . s . t /. \ . . / .\ o o o o x y xF yF This is a rough sketch of the actual data structures that I used to represent a functor F as a "matching" between the parallel items of categories C and D. NB. I will have to revert to the convention that I was accustomed to using then, where all operators are applied on the right of their arguments. What the picture says is that the functor F : C -> CF takes each arrow f in C to an arrow fF in CF, and each object x in C to an object xF in CF, in such a manner that (fs)F = (fF)s and (ft)F = (fF)t. To be a functor, F must satisfy the following two systems of equations: (1_x)F = 1_(xF), for all x in Obj(C). (f o g)F = fF o gF, for all composable f, g in Arr(C). That was just how I kept track of things on the computer. It is, of course, more usual to draw a "functor square" like this, where we get one such picture for each object x and arrow f in C. F x o-------->o xF | | | | f | | fF | | v v y o-------->o yF F Jon Awbrey o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at att.net Mon Jul 14 12:36:08 2003 From: jawbrey at att.net (Jon Awbrey) Date: Wed Jan 21 20:34:35 2004 Subject: [Inquiry] Re: Quine -- Two Dogmas Of Empiricism References: <3F10BBD2.DD0B3EFE@att.net> Message-ID: <3F12EA08.7C03B0C8@att.net> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o TDOE. Note 2 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | 1. Background for Analyticity | | Kant's cleavage between analytic and synthetic truths | was foreshadowed in Hume's distinction between relations | of ideas and matters of fact, and in Leibniz's distinction | between truths of reason and truths of fact. Leibniz spoke | of the truths of reason as true in all possible worlds. | Picturesqueness aside, this is to say that the truths | of reason are those which could not possibly be false. | In the same vein we hear analytic statements defined as | statements whose denials are self-contradictory. But this | definition has small explanatory value; for the notion of | self-contradictoriness, in the quite broad sense needed for | this definition of analyticity, stands in exactly the same | need of clarification as does the notion of analyticity | itself. The two notions are the two sides of a single | dubious coin. | | Kant conceived of an analytic statement as one that attributes to its | subject no more than is already conceptually contained in the subject. | This formulation has two shortcomings: it limits itself to statements of | subject-predicate form, and it appeals to a notion of containment which is | left at a metaphorical level. But Kant's intent, evident more from the use | he makes of the notion of analyticity than from his definition of it, can be | restated thus: a statement is analytic when it is true by virtue of meanings | and independently of fact. Pursuing this line, let us examine the concept of | 'meaning' which is presupposed. | | Quine, "Two Dogmas", pp. 20-21. | | W.V. Quine, |"Two Dogmas of Empiricism", 'Philosophical Review', January 1951. | Reprinted as pages 20-46 in 'From a Logical Point of View', | 2nd edition, Harvard University Press, Cambridge, MA, 1980. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at att.net Mon Jul 14 13:24:04 2003 From: jawbrey at att.net (Jon Awbrey) Date: Wed Jan 21 20:34:35 2004 Subject: [Inquiry] Re: Quine -- Two Dogmas Of Empiricism References: <3F10BBD2.DD0B3EFE@att.net> <3F12EA08.7C03B0C8@att.net> Message-ID: <3F12F544.B6713BE@att.net> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o TDOE. Note 3 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | 1. Background for Analyticity (cont.) | | Meaning, let us remember, is not to be identified with naming. | Frege's example of "Evening Star" and "Morning Star", and Russell's | of "Scott" and "the author of 'Waverley'", illustrate that terms can | name the same thing but differ in meaning. The distinction between | meaning and naming is no less important at the level of abstract | terms. The terms "9" and "the number of the planets" name one | and the same abstract entity but presumably must be regarded as | unlike in meaning; for astronomical observation was needed, and | not mere reflection on meanings, to determine the sameness of the | entity in question. | | The above examples consists of singular terms, concrete and | abstract. With general terms, or predicates, the situation | is somewhat different but parallel. Whereas a singular term | purports to name an entity, abstract or concrete, a general | term does not; but a general term is 'true of' an entity, | or of each of many, or of none. The class of all entities | of which a general term is true is called the 'extension' | of the term. Now paralleling the contrast between the | meaning of a singular term and the entity named, we | must distinguish equally between the meaning of a | general term and its extension. The general terms | "creature with a heart" and "creature with kidneys", | for example, are perhaps alike in extension but unlike | in meaning. | | Quine, "Two Dogmas", p. 21. | | W.V. Quine, |"Two Dogmas of Empiricism", 'Philosophical Review', January 1951. | Reprinted as pages 20-46 in 'From a Logical Point of View', | 2nd edition, Harvard University Press, Cambridge, MA, 1980. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at att.net Mon Jul 14 14:00:02 2003 From: jawbrey at att.net (Jon Awbrey) Date: Wed Jan 21 20:34:35 2004 Subject: [Inquiry] Re: Quine -- Two Dogmas Of Empiricism References: <3F10BBD2.DD0B3EFE@att.net> <3F12EA08.7C03B0C8@att.net> <3F12F544.B6713BE@att.net> Message-ID: <3F12FDB2.5876C69C@att.net> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o TDOE. Note 4 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | 1. Background for Analyticity (cont.) | | Confusion of meaning with extension, in the case of general terms, | is less common than confusion of meaning with naming in the case | of singular terms. It is indeed a commonplace in philosophy to | oppose intension (or meaning) to extension, or, in a variant | vocabulary, connotation to denotation. | | The Aristotelian notion of essence was the forerunner, no doubt, | of the modern notion of intension or meaning. For Aristotle it | was essential in men to be rational, accidental to be two-legged. | But there is an important difference between this attitude and the | doctrine of meaning. From the latter point of view it may indeed | be conceded (if only for the sake of argument) that rationality is | involved in the meaning of the word "man" while two-leggedness is | not; but two-leggedness may at the same time be viewed as involved | in the meaning of "biped" while rationality is not. Thus from the | point of view of the doctrine of meaning it makes no sense to say | of the actual individual, who is at once a man and a biped, that | his rationality is essential and his two-leggedness accidental | or vice versa. Things had essences, for Aristotle, but only | linguistic forms have meanings. Meaning is what essence | becomes when it is divorced from the object of reference | and wedded to the word. | | Quine, "Two Dogmas", pp. 21-22. | | W.V. Quine, |"Two Dogmas of Empiricism", 'Philosophical Review', January 1951. | Reprinted as pages 20-46 in 'From a Logical Point of View', | 2nd edition, Harvard University Press, Cambridge, MA, 1980. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at att.net Mon Jul 14 15:04:08 2003 From: jawbrey at att.net (Jon Awbrey) Date: Wed Jan 21 20:34:35 2004 Subject: [Inquiry] Re: Quine -- Two Dogmas Of Empiricism References: <3F10BBD2.DD0B3EFE@att.net> <3F12EA08.7C03B0C8@att.net> <3F12F544.B6713BE@att.net> <3F12FDB2.5876C69C@att.net> Message-ID: <3F130CB8.9DC7D02B@att.net> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o TDOE. Note 5 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | 1. Background for Analyticity (cont.) | | For the theory of meaning a conspicuous question is the nature | of its objects: what sort of things are meanings? A felt need | for meant entities may derive from an earlier failure to appreciate | that meaning and reference are distinct. Once the theory of meaning | is sharply separated from the theory of reference, it is a short step | to recognizing as the primary business of the theory of meaning simply | the synonymy of linguistic forms and the analyticity of statements; | meanings themselves, as obscure intermediary entities, may well be | abandoned. | | The problem of analyticity then confronts us anew. Statements which are | analytic by general philosophical acclaim are not, indeed, far to seek. | They fall into two classes. Those of the first class, which may be | called 'logically true', are typified by: | | (1) No unmarried man is married. | | The relevant feature of this example is that it not merely | is true as it stands, but remains true under any and all | reinterpretations of "man" and "married". If we suppose | a prior inventory of 'logical' particles, comprising "no", | "un-", "not", "if", "then", "and", etc., then in general | a logical truth is a statement which is true and remains | true under all reinterpretations of its components than | than the logical particles. | | But there is also a second class of analytic statements, | typified by: | | (2) No bachelor is married. | | The characteristic of such a statement is that it can be | turned into a logical truth by putting synonyms for synonyms; | thus (2) can be turned into (1) by putting "unmarried man" for | its synonym "bachelor". We still lack a proper characterization | of this second class of analytic statements, and therewith of | analyticity generally, inasmuch as we have had in the above | description to lean on a notion of "synonymy" which is no | less in need of clarification than analyticity itself. | | Quine, "Two Dogmas", pp. 22-23. | | W.V. Quine, |"Two Dogmas of Empiricism", 'Philosophical Review', January 1951. | Reprinted as pages 20-46 in 'From a Logical Point of View', | 2nd edition, Harvard University Press, Cambridge, MA, 1980. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at att.net Mon Jul 14 15:56:06 2003 From: jawbrey at att.net (Jon Awbrey) Date: Wed Jan 21 20:34:35 2004 Subject: [Inquiry] Re: Quine -- Two Dogmas Of Empiricism References: <3F10BBD2.DD0B3EFE@att.net> <3F12EA08.7C03B0C8@att.net> <3F12F544.B6713BE@att.net> <3F12FDB2.5876C69C@att.net> <3F130CB8.9DC7D02B@att.net> Message-ID: <3F1318E6.507C3EA5@att.net> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o TDOE. Note 6 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | 1. Background for Analyticity (concl.) | | In recent years Carnap has tended to explain analyticity by appeal to | what he calls state-descriptions. A state-description is any exhaustive | assignment of truth values to the atomic, or noncompound, statements of | the language. All other statements of the language are, Carnap assumes, | built up of their component clauses by means of familiar logical devices, | in such a way that the truth value of any complex statement is fixed for | each state-description by specifiable logical laws. A statement is then | explained as analytic when it comes out true under every state-description. | This account is an adaptation of Leibniz's "true in all possible worlds". | But note that this version of analyticity serves its purpose only if the | atomic statements of the language are, unlike "John is a bachelor" and | "John is married", mutually independent. Otherwise there would be a | state-description which assigned truth to "John is a bachelor" and to | "John is married", and consequently "No bachelors are married" would | turn out synthetic rather than analytic under the proposed criterion. | Thus the criterion of analyticity in terms of state-descriptions | serves only for languages devoid of extralogical synonym-pairs, | such as "bachelor" and "unmarried man" -- synonym-pairs of the | type which give rise to the "second class" of analytic statements. | The criterion in terms of state-descriptions is a reconstruction | at best of logical truth, not of analyticity. | | I do not mean to suggest that Carnap is under any illusions on this | point. His simplified model language with its state-descriptions | is aimed primarily not at the general problem of analyticity but | at another purpose, the clarification of probability and induction. | Our problem, however, is analyticity; and here the major difficulty | lies not in the first class of analytic statements, the logical truths, | but rather in the second class, which depends on the notion of synonymy. | | Quine, "Two Dogmas", pp. 23-24. | | W.V. Quine, |"Two Dogmas of Empiricism", 'Philosophical Review', January 1951. | Reprinted as pages 20-46 in 'From a Logical Point of View', | 2nd edition, Harvard University Press, Cambridge, MA, 1980. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at att.net Mon Jul 14 16:28:06 2003 From: jawbrey at att.net (Jon Awbrey) Date: Wed Jan 21 20:34:35 2004 Subject: [Inquiry] Re: Quine -- Two Dogmas Of Empiricism References: <3F10BBD2.DD0B3EFE@att.net> <3F12EA08.7C03B0C8@att.net> <3F12F544.B6713BE@att.net> <3F12FDB2.5876C69C@att.net> <3F130CB8.9DC7D02B@att.net> <3F1318E6.507C3EA5@att.net> Message-ID: <3F132066.F84190BF@att.net> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o TDOE. Note 7 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | 2. Definition | | There are those who find it soothing to say that the analytic statements | of the second class reduce to those of the first class, the logical truths, | by 'definition'; "bachelor", for example, is 'defined' as "unmarried man". | But how do we find that "bachelor" is defined as "unmarried man"? Who | defined it thus, and when? Are we to appeal to the nearest dictionary, | and accept the lexicographer's formulation as law? Clearly this would | be to put the cart before the horse. The lexicographer is an empirical | scientist, whose business is the recording of antecedent facts; and if | he glosses "bachelor" as "unmarried man" it is because of his belief that | there is a relation of synonymy between those forms, implicit in general or | preferred usage prior to his own work. The notion of synonymy presupposed | here has still to be clarified, presumably in terms relating to linguistic | behavior. Certainly the "definition" which is the lexicographer's report | of an observed synonymy cannot be taken as the ground of the synonymy. | | Quine, "Two Dogmas", p. 24. | | W.V. Quine, |"Two Dogmas of Empiricism", 'Philosophical Review', January 1951. | Reprinted as pages 20-46 in 'From a Logical Point of View', | 2nd edition, Harvard University Press, Cambridge, MA, 1980. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at att.net Mon Jul 14 22:10:03 2003 From: jawbrey at att.net (Jon Awbrey) Date: Wed Jan 21 20:34:35 2004 Subject: [Inquiry] Re: Quine -- Two Dogmas Of Empiricism References: <3F10BBD2.DD0B3EFE@att.net> <3F12EA08.7C03B0C8@att.net> <3F12F544.B6713BE@att.net> <3F12FDB2.5876C69C@att.net> <3F130CB8.9DC7D02B@att.net> <3F1318E6.507C3EA5@att.net> <3F132066.F84190BF@att.net> Message-ID: <3F13708B.A806CA59@att.net> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o TDOE. Note 8 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | 2. Definition (cont.) | | Definition is not, indeed, an activity exclusively of philologists. | Philosophers and scientists frequently have occasion to "define" | a recondite term by paraphrasing it into terms of a more familiar | vocabulary. But ordinarily such a definition, like the philologist's, | is pure lexicography, affirming a relation of synonymy antecedent to | the exposition in hand. | | Just what it means to affirm synonymy, just what the interconnections | may be which are necessary and sufficient in order that two linguistic | forms be properly describable as synonymous, is far from clear; but, | whatever these interconnections may be, ordinarily they are grounded | in usage. Definitions reporting selected instances of synonymy come | then as reports upon usage. | | There is also, however, a variant type of definitional activity which does | not limit itself to the reporting of pre-existing synonymies. I have in | mind what Carnap calls 'explication' -- an activity to which philosophers | are given, and scientists also in their more philosophical moments. In | explication the purpose is not merely to paraphrase the definiendum into | an outright synonym, but actually to improve upon the definiendum by | refining or supplementing its meaning. But even explication, though | not merely reporting a pre-existing synonymy between definiendum and | definiens, does rest nevertheless on 'other' pre-existing synonymies. | The matter might be viewed as follows. Any word worth explicating | has some contexts which, as wholes, are clear and precise enough | to be useful; and the purpose of explication is to preserve the | usage of these favored contexts while sharpening the usage of | other contexts. In order that a given definition be suitable | for purposes of explication, therefore, what is required is not | that the definiendum in its antecedent usage be synonymous with | the definiens, but just that each of these favored contexts of | the definiendum, taken as a whole in its antecedent usage, be | synonymous with the corrsponding context of the definiens. | | Two alternative definientia may be equally appropriate for the purposes | of a given task of explication and yet not be synonymous with each other; | for they may serve interchangeably within the favored contexts but diverge | elsewhere. By cleaving to one of these definientia rather than the other, | a definition of explicative kind generates, by fiat, a relation of synonymy | between definiendum and definiens which did not hold before. But such a | definition still owes its explicative function, as seen, to pre-existing | synonymies. | | Quine, "Two Dogmas", pp. 24-25. | | W.V. Quine, |"Two Dogmas of Empiricism", 'Philosophical Review', January 1951. | Reprinted as pages 20-46 in 'From a Logical Point of View', | 2nd edition, Harvard University Press, Cambridge, MA, 1980. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at att.net Mon Jul 14 22:32:12 2003 From: jawbrey at att.net (Jon Awbrey) Date: Wed Jan 21 20:34:36 2004 Subject: [Inquiry] Re: Quine -- Two Dogmas Of Empiricism References: <3F10BBD2.DD0B3EFE@att.net> <3F12EA08.7C03B0C8@att.net> <3F12F544.B6713BE@att.net> <3F12FDB2.5876C69C@att.net> <3F130CB8.9DC7D02B@att.net> <3F1318E6.507C3EA5@att.net> <3F132066.F84190BF@att.net> <3F13708B.A806CA59@att.net> Message-ID: <3F1375BC.1A1CE45@att.net> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o TDOE. Note 9 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | 2. Definition (cont.) | | There does, however, remain still an extreme sort of definition | which does not hark back to prior synonymies at all: namely, | the explicitly conventional introduction of novel notations | for purposes of sheer abbreviation. Here the definiendum | becomes synonymous with the definiens simply because it | has been created expressly for the purpose of being | synonymous with the definiens. Here we have a | really transparent case of synonymy created | by definition; would that all species of | synonymy were as intelligible. For the | rest, definition rests on synonymy | rather than explaining it. | | Quine, "Two Dogmas", pp. 25-26. | | W.V. Quine, |"Two Dogmas of Empiricism", 'Philosophical Review', January 1951. | Reprinted as pages 20-46 in 'From a Logical Point of View', | 2nd edition, Harvard University Press, Cambridge, MA, 1980. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at att.net Mon Jul 14 23:48:07 2003 From: jawbrey at att.net (Jon Awbrey) Date: Wed Jan 21 20:34:36 2004 Subject: [Inquiry] Re: Quine -- Two Dogmas Of Empiricism References: <3F10BBD2.DD0B3EFE@att.net> <3F12EA08.7C03B0C8@att.net> <3F12F544.B6713BE@att.net> <3F12FDB2.5876C69C@att.net> <3F130CB8.9DC7D02B@att.net> <3F1318E6.507C3EA5@att.net> <3F132066.F84190BF@att.net> <3F13708B.A806CA59@att.net> <3F1375BC.1A1CE45@att.net> Message-ID: <3F138787.8B3F1D00@att.net> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o TDOE. Note 10 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | 2. Definition (concl.) | | The word "definition" has come to have a dangerously reassuring sound, | owing no doubt to its frequent occurrence in logical and mathematical | writings. We shall do well to digress now into a brief appraisal of | the role of definition in formal work. | | In logical and mathematical systems either of two mutually antagonistic | types of economy may be striven for, and each has its peculiar practical | utility. On the one hand we may seek economy of practical expression -- | ease and brevity in the statement of multifarious relations. This sort | of economy calls usually for distinctive concise notations for a wealth | of concepts. Second, however, and oppositely, we may seek economy in | grammar and vocabulary; we may try to find a minimum of basic concepts | such that, once a distinctive notation has been appropriated to each of | them, it becomes possible to express any desired further concept by mere | combination and iteration of our basic notations. This second sort of | economy is impractical in one way, since a poverty in basic idioms tends | to a necessary lengthening of discourse. But it is practical in another | way: it greatly simplifies theoretical discourse 'about' the language, | through minimizing the terms and the forms of construction wherein the | language consists. | | Both sorts of economy, though prima facie incompatible, are valuable in | their separate ways. The custom has consequently arisen of combining | both sorts of economy by forging in effect two langauges, the one | a part of the other. The inclsuive language, though redundant | in grammar and vocabulary, is economical in message lengths, | while the part, called primitive notation, is economical in | grammar and vocabulary. Whole and part are correlated by | rules of translation whereby each idiom not in primitive | notation is equated to some complex built up of primitive | notation. These rules of translation are the so-called | 'definitions' which appear in formalized systems. They | are best viewed not as adjuncts to one language but as | correlations between two languages, the one a part of | the other. | | But these correlations are not arbitrary. They are supposed | to show how the primitive notations can accomplish all purposes, | save brevity and convenience, of the redundant language. Hence | the definiendum and its definiens may be expected, in each case, | to be related in one or another of the three ways lately noted. | The definiens may be a faithful paraphrase of the definiendum | into the narrower notation, preseving a direct synonymy* as | of antecedent usage; or the definiens may, in the spirit | of explication, improve upon the antecedent usage of the | definiendum; or finally, the definiendum may be a newly | created notation, newly endowed with meaning here and now. | | In formal and informal work alike, thus, we find | that definition -- except in the extreme case of the | explicitly conventional introduction of new notations -- | hinges on prior relations of synonymy. Recognizing then | that the notion of definition does not hold the key to | synonymy and analyticity, let us look further into | synonymy and say no more of definition. | |*According to an important variant sense of "definition", the relation | preserved may be the weaker relation of mere agreement in reference; | see below, p. 132. But definition in this sense is better ignored in | the present connection, being irrelevant to the question of synonymy. | | Quine, "Two Dogmas", pp. 26-27. | | W.V. Quine, |"Two Dogmas of Empiricism", 'Philosophical Review', January 1951. | Reprinted as pages 20-46 in 'From a Logical Point of View', | 2nd edition, Harvard University Press, Cambridge, MA, 1980. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at att.net Tue Jul 15 07:32:08 2003 From: jawbrey at att.net (Jon Awbrey) Date: Wed Jan 21 20:34:36 2004 Subject: [Inquiry] Re: Quine -- Two Dogmas Of Empiricism References: <3F10BBD2.DD0B3EFE@att.net> Message-ID: <3F13F448.A9AC247B@att.net> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o TDOE. Note 11 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | 3. Interchangeability | | A natural suggestion, deserving close examination, is that the synonymy | of two linguistic forms consists simply in their interchangeability in | all contexts without change of truth value -- interchangeability, in | Leibniz's phrase 'salva veritate'. Note that synonyms so conceived | need not even be free from vagueness, as long as the vaguenesses | match. | | But it is not quite true that the synonyms "bachelor" and "unmarried man" | are everywhere interchangeable 'salva veritate'. Truths which become false | under substitution of "unmarried man" for "bachelor" are easily constructed | with the help of "bachelor of arts" or "bachelor's buttons"; also with the | help of quotation, thus: | | "Bachelor" has less than ten letters. | | Such counterinstances can, however, be set aside by treating | the phrases "bachelor of arts" and "bachelor's buttons" and the | quotation '"bachelor"' each as a single indivisible word and then | stipulating that the interchangeability 'salva veritate' which | is to be the touchstone of synonymy is not supposed to apply | to fragmentary occurrences inside of a word. This account of | synonymy, supposing it acceptable on other counts, has indeed | the drawback of appealing to a prior conception of "word" which | can be counted on to present difficulties of formulation in its | turn. Nevertheless some progress might be claimed in having | reduced the problem of synonymy to a problem of wordhood. | Let us pursue this line a bit, taking "word" for granted. | | Quine, "Two Dogmas", pp. 27-28. | | W.V. Quine, |"Two Dogmas of Empiricism", 'Philosophical Review', January 1951. | Reprinted as pages 20-46 in 'From a Logical Point of View', | 2nd edition, Harvard University Press, Cambridge, MA, 1980. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at att.net Tue Jul 15 08:30:17 2003 From: jawbrey at att.net (Jon Awbrey) Date: Wed Jan 21 20:34:36 2004 Subject: [Inquiry] Re: Quine -- Two Dogmas Of Empiricism References: <3F10BBD2.DD0B3EFE@att.net> <3F13F448.A9AC247B@att.net> Message-ID: <3F1401E9.BB6398BC@att.net> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o TDOE. Note 12 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | 3. Interchangeability (cont.) | | The question remains whether interchangeability | 'salva veritate' (apart from occurrences within words) | is a strong enough condition for synonymy, or whether, | on the contrary, some heteronymous expressions might be thus | interchangeable. Now let us be clear that we are not concerned | here with synonymy in the sense of complete identity in psychological | associations or poetic quality; indeed no two expressions are synonymous | in such a sense. We are concerned only with what may be called 'cognitive' | synonymy. Just what this is cannot be said without successfully finishing the | present study; but we know something about it from the need which arose for | it in connection with analyticity in Section 1. The sort of synonymy needed | there was merely such that any analytic statement could be turned into a | logical truth by putting synonyms for synonyms. Turning the tables and | assuming analyticity, indeed, we could explain cognitive synonymy of | terms as follows (keeping to the familiar example): to say that | "bachelor" and "unmarried man" are cognitively synonymous is | to say no more or less than that the statement: | | (3) All and only bachelors are unmarried men | | is analytic.* | |*This is cognitive synonymy in a primary, broad sense. Carnap ([3], | pp. 56ff) and Lewis ([2], pp. 83ff) have suggested how, once this | notion is at hand, a narrower sense of cognitive synonymy which | is preferable for some purposes can in turn be derived. But | this special ramification of concept-building lies aside | from the present purposes and must not be confused with | the broad sort of cognitive synonymy here concerned. | | Quine, "Two Dogmas", pp. 28-29. | | W.V. Quine, |"Two Dogmas of Empiricism", 'Philosophical Review', January 1951. | Reprinted as pages 20-46 in 'From a Logical Point of View', | 2nd edition, Harvard University Press, Cambridge, MA, 1980. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at att.net Tue Jul 15 09:34:19 2003 From: jawbrey at att.net (Jon Awbrey) Date: Wed Jan 21 20:34:36 2004 Subject: [Inquiry] Re: Quine -- Two Dogmas Of Empiricism References: <3F10BBD2.DD0B3EFE@att.net> <3F13F448.A9AC247B@att.net> <3F1401E9.BB6398BC@att.net> Message-ID: <3F1410EB.209325ED@att.net> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o TDOE. Note 13 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | 3. Interchangeability (cont.) | | What we need is an account of cognitive synonymy | not presupposing analyticity -- if we are to explain | analyticity conversely with help of cognitive synonymy | as undertaken in Section 1. And indeed such an independent | account of cognitive synonymy is at present up for consideration, | namely, interchangeability 'salva veritate' everywhere except within | words. The question before us, to resume the thread at last, is whether | such interchangeability is a sufficient condition for cognitive synonymy. | We can quickly assure ourselves that it is, by examples of the following | sort. The statement: | | (4) Necessarily all and only bachelors are bachelors | | is evidently true, even supposing "necessarily" so narrowly construed as | to be truly applicable only to analytic statements. Then, if "bachelor" | and "unmarried man" are interchangeable 'salva veritate', the result: | | (5) Necessarily all and only bachelors are unmarried men | | of putting "unmarried man" for an occurrence of "bachelor" in (4) must, | like (4), be true. But to say that (5) is true is to say that (3) is | analytic, and hence that "bachelor" and "unmarried man" are cognitively | synonymous. | | Let us see what there is about the above argument that gives it its air | of hocus-pocus. The condition of interchangeability 'salva veritate' | varies in its force with variations in the richness of the language | at hand. The above argument supposes we are working with a language | rich enough to contain the adverb "necessarily", this adverb being so | construed as to yield truth when and only when applied to an analytic | statement. But can we condone a language which contains such an adverb? | Does the adverb really make sense? To suppose that it does is to suppose | that we have already made satisfactory sense of "analytic". Then what are | we so hard at work on right now? | | Our argument is not flatly circular, but something like it. | It has the form, figuratively speaking, of a closed curve | in space. | | Quine, "Two Dogmas", pp. 29-30. | | W.V. Quine, |"Two Dogmas of Empiricism", 'Philosophical Review', January 1951. | Reprinted as pages 20-46 in 'From a Logical Point of View', | 2nd edition, Harvard University Press, Cambridge, MA, 1980. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at att.net Tue Jul 15 10:15:29 2003 From: jawbrey at att.net (Jon Awbrey) Date: Wed Jan 21 20:34:36 2004 Subject: [Inquiry] Re: Quine -- Two Dogmas Of Empiricism References: <3F10BBD2.DD0B3EFE@att.net> <3F13F448.A9AC247B@att.net> <3F1401E9.BB6398BC@att.net> <3F1410EB.209325ED@att.net> Message-ID: <3F141A91.25EDDA7D@att.net> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o TDOE. Note 14 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | 3. Interchangeability (cont.) | | Interchangeability 'salva veritate' is meaningless until relativized to | a language whose extent is specified in relevant respects. Suppose now | we consider a language containing just the following materials. There | is an indefinitely large stock of one-place predicates, (for example, | "F" where "Fx" means that x is a man) and many-place predicates (for | example, "G" where "Gxy" means that x loves y), mostly having to | do with extralogical subject matter. The rest of the language | is logical. The atomic sentences consist each of a predicate | followed by one or more variables "x", "y", etc.; and the | complex sentences are built up of the atomic ones by truth | functions ("not", "and", "or", etc.) and quantification. | In effect such a language enjoys the benefits also of | descriptions and indeed singular terms generally, | these being contextually definable in known ways. | Even abstract singular terms naming classes, | classes of classes, etc., are contextually | definable in case the assumed stock of | predicates includes the two-place | predicate of class membership. | Such a language can be adequate | to classical mathematics and | indeed to scientific discourse | generally, except in so far as | the latter involves debatable | devices such as contrary-to-fact | conditionals or modal adverbs like | "necessarily". Now a language of this | type is extensional, in this sense: any | two predicates which agree extensionally | (that is, are true of the same objects) | are interchangeable 'salva veritate'. | | Quine, "Two Dogmas", p. 30. | | W.V. Quine, |"Two Dogmas of Empiricism", 'Philosophical Review', January 1951. | Reprinted as pages 20-46 in 'From a Logical Point of View', | 2nd edition, Harvard University Press, Cambridge, MA, 1980. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at att.net Tue Jul 15 11:00:03 2003 From: jawbrey at att.net (Jon Awbrey) Date: Wed Jan 21 20:34:36 2004 Subject: [Inquiry] Re: Quine -- Two Dogmas Of Empiricism References: <3F10BBD2.DD0B3EFE@att.net> <3F13F448.A9AC247B@att.net> <3F1401E9.BB6398BC@att.net> <3F1410EB.209325ED@att.net> <3F141A91.25EDDA7D@att.net> Message-ID: <3F142503.66115F8E@att.net> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o TDOE. Note 15 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | 3. Interchangeability (cont.) | | In an extensional language, therefore, interchangeability | 'salva veritate' is no assurance of cognitive synonymy of | the desired type. That "bachelor" and "unmarried man" are | interchangeable 'salva veritate' in an extensional language | assures us of no more than that (3) is true. There is no | assurance here that the extensional agreement of "bachelor" | and "unmarried man" rests on meaning rather than merely on | accidental matters of fact, as does the extensional agreement | of "creature with a heart" and "creature with kidneys". | | For most purposes extensional agreement is the nearest approximation | to synonymy we need care about. But the fact remains that extensional | agreement falls far short of cognitive synonymy of the type required for | explaining analyticity in the manner of Section 1. The type of cognitive | synonymy required there is such as to equate the synonymy of "bachelor" | and "unmarried man" with the analyticity of (3), not merely with the | truth of (3). | | So we must recognize that interchangeability 'salva veritate', | if construed in relation to an extensional language, is not | a sufficient condition of cognitive synonymy in the sense | needed for deriving analyticity in the manner of Section 1. | If a language contains an intensional adverb "necessarily" in | the sense lately noted, or other particles to the same effect, | then interchangeability 'salva veritate' in such a language | does afford a sufficient condition of cognitive synonymy; | but such a language is intelligible only in so far as the | notion of analyticity is already understood in advance. | | Quine, "Two Dogmas", p. 31. | | W.V. Quine, |"Two Dogmas of Empiricism", 'Philosophical Review', January 1951. | Reprinted as pages 20-46 in 'From a Logical Point of View', | 2nd edition, Harvard University Press, Cambridge, MA, 1980. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at att.net Tue Jul 15 12:14:23 2003 From: jawbrey at att.net (Jon Awbrey) Date: Wed Jan 21 20:34:36 2004 Subject: [Inquiry] Re: Quine -- Two Dogmas Of Empiricism References: <3F10BBD2.DD0B3EFE@att.net> <3F13F448.A9AC247B@att.net> <3F1401E9.BB6398BC@att.net> <3F1410EB.209325ED@att.net> <3F141A91.25EDDA7D@att.net> <3F142503.66115F8E@att.net> Message-ID: <3F14366F.E6329E1F@att.net> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o TDOE. Note 16 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | 3. Interchangeability (concl.) | | The effort to explain cognitive synonymy first, for the sake | of deriving analyticity from it afterward as in Section 1, is | perhaps the wrong approach. Instead we might try explaining | analyticity somehow without appeal to cognitive synonymy. | Afterward we could doubtless derive cognitive synonymy from | analyticity satisfactorily enough if desired. We have seen | that cognitive synonymy of "bachelor" and "unmarried man" can | be explained as analyticity of (3). The same explanation works | for any pair of one-place predicates, of course, and it can | be extended in obvious fashion to many-place predicates. | Other syntactical categories can also be accommodated in | fairly parallel fashion. Singular terms may be said to be | cognitively synonymous when the statement of identity formed | by putting "=" between them is analytic. Statements may be said | simply to be cognitively synonymous when their biconditional (the | result of joining them by "if and only if") is analytic. If we | care to lump all categories into a single formulation, at the | expense of assuming again the notion of "word" which was | appealed to early in this section, we can describe any two | linguistic forms as cognitively synonymous when the two forms | are interchangeable (apart from occurrences within "words") | 'salva' (no longer 'veritate' but) 'analyticitate'. Certain | technical questions arise, indeed, over cases of ambiguity | or homonymy; let us not pause for them, however, for we | are already digressing. Let us rather turn our backs | on the problem of synonymy and address ourselves | anew to that of analyticity. | | Quine, "Two Dogmas", pp. 31-32. | | W.V. Quine, |"Two Dogmas of Empiricism", 'Philosophical Review', January 1951. | Reprinted as pages 20-46 in 'From a Logical Point of View', | 2nd edition, Harvard University Press, Cambridge, MA, 1980. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at att.net Tue Jul 15 14:15:34 2003 From: jawbrey at att.net (Jon Awbrey) Date: Wed Jan 21 20:34:36 2004 Subject: [Inquiry] Re: Quine -- Two Dogmas Of Empiricism References: <3F10BBD2.DD0B3EFE@att.net> Message-ID: <3F1452D6.B01EE33E@att.net> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o TDOE. Note 17 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | 4. Semantical Rules | | Analyticity at first seemed most naturally definable by appeal | to a realm of meanings. On refinement, the appeal to meanings | gave way to an appeal to synonymy or definition. But definition | turned out to be a will-o'-the-wisp, and synonymy turned out to be | best understood only by dint of a prior appeal to analyticity itself. | So we are back at the problem of analyticity. | | I do not know whether the statement "Everything green is extended" | is analytic. Now does my indecision over this example really betray | an incomplete understanding, an incomplete grasp of the "meanings", | of "green" and "extended"? I think not. The trouble is not with | "green" or "extended", but with "analytic". | | It is often hinted that the difficulty in separating analytic | statements from synthetic ones in ordinary language is due to | the vagueness of ordinary language and that the distinction is | clear when we have a precise artificial language with explicit | "semantical rules". This, however, as I shall now attempt to | show, is a confusion. | | Quine, "Two Dogmas", p. 32. | | W.V. Quine, |"Two Dogmas of Empiricism", 'Philosophical Review', January 1951. | Reprinted as pages 20-46 in 'From a Logical Point of View', | 2nd edition, Harvard University Press, Cambridge, MA, 1980. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at att.net Tue Jul 15 15:14:04 2003 From: jawbrey at att.net (Jon Awbrey) Date: Wed Jan 21 20:34:36 2004 Subject: [Inquiry] Re: Quine -- Two Dogmas Of Empiricism References: <3F10BBD2.DD0B3EFE@att.net> <3F1452D6.B01EE33E@att.net> Message-ID: <3F14608C.CC04AC36@att.net> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o TDOE. Note 18 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | 4. Semantical Rules (cont.) | | The notion of analyticity about which we are worrying is a purported | relation between statements and languages: a statement S is said to | be 'analytic for' a language L, and the problem is to make sense of | this relation generally, that is, for variable "S" and "L". The | gravity of this problem is not perceptibly less for artificial | languages than for natural ones. The problem of making sense | of the idiom "S is analytic for L", with variable "S" and "L", | retains its stubbornness even if we limit the range of the | variable "L" to artificial languages. Let me now try to | make this point evident. | | For artificial languages and semantical rules we look naturally | to the writings of Carnap. His semantical rules take various forms, | and to make my point I shall have to distinguish certain of the forms. | Let us suppose, to begin with, an artificial language L_0 whose semantical | rules have the form explicitly of a specification, by recursion or otherwise, | of all the analytic statements of L_0. The rules tell us that such and such | statements, and only those, are the analytic statements of L_0. Now here | the difficulty is simply that the rules contain the word "analytic", | which we do not understand! We understand what expressions the | rules attribute analyticity to, but we do not understand what | the rules attribute to those expressions. In short, before | we can understand a rule which begins "A statement S is | analytic for language L_0 if and only if ...", we must | understand the general relative term "analytic for"; | we must understand "S is analytic for L" where "S" | and "L" are variables. | | Alternatively we may, indeed, view the so-called rule as a conventional | definition of a new simple symbol "analytic-for-L_0", which might better | be written untendentiously as "K" so as not to seem to throw light on the | interesting word "analytic". Obviously any number of classes K, M, N, etc. | of statements of L_0 can be specified for various purposes or for no purpose; | what does it mean to say that K, as against M, N, etc., is the class of the | "analytic" statements of L_0? | | By saying what statements are analytic for L_0 we explain | "analytic-for-L_0" but not "analytic", not "analytic for". | We do not begin to explain the idiom "S is analytic for L" | with variable "S" and "L", even if we are content to limit | the range of "L" to the realm of artificial languages. | | Quine, "Two Dogmas", pp. 33-34. | | W.V. Quine, |"Two Dogmas of Empiricism", 'Philosophical Review', January 1951. | Reprinted as pages 20-46 in 'From a Logical Point of View', | 2nd edition, Harvard University Press, Cambridge, MA, 1980. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at att.net Tue Jul 15 16:15:01 2003 From: jawbrey at att.net (Jon Awbrey) Date: Wed Jan 21 20:34:36 2004 Subject: [Inquiry] Re: Quine -- Two Dogmas Of Empiricism References: <3F10BBD2.DD0B3EFE@att.net> <3F1452D6.B01EE33E@att.net> <3F14608C.CC04AC36@att.net> Message-ID: <3F146ED5.3E9CDC0E@att.net> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o TDOE. Note 19 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | 4. Semantical Rules (cont.) | | Actually we do know enough about the intended significance of | "analytic" to know that analytic statements are supposed to | be true. Let us then turn to a second form of semantical | rule, which says not that such and such statements are | analytic but simply that such and such statements are | included among the truths. Such a rule is not subject | to the criticism of containing the un-understood word | "analytic"; and we may grant for the sake of argument | that there is no difficulty over the broader term "true". | A semantical rule of this second type, a rule of truth, | is not supposed to specify all the truths of the language; | it merely stipulates, recursively or otherwise, a certain | multitude of statements which, along with others unspecified, | are to count as true. Such a rule may be conceded to be quite | clear. Derivatively, afterward, analyticity can be demarcated | thus: a statement is analytic if it is (not merely true but) | true according to the semantical rule. | | Still there is really no progress. Instead of appealing to an unexplained | word "analytic", we are now appealing to an unexplained phrase "semantical | rule". Not every true statement which says that the statements of some | class are true can count as a semantical rule -- otherwise 'all' truths | would be "analytic" in the sense of being true according to semantical | rules. Semantical rules are distinguishable, apparently, only by the | fact of appearing on a page under the heading "Semantical Rules"; | and this heading is itself then meaningless. | | We can say indeed that a statement is 'analytic-for-L_0' if and | only if it is true according to such and such specifically appended | "semantical rules", but then we find ourselves back at essentially the | same case which was originally discussed: "S is analytic-for-L_0" if and | only if ...". Once we seek to explain "S is analytic for L" generally for | variable "L" (even allowing limitation of "L" to artificial languages), | the explanation "true according to the semantical rules of L" is | unavailing; for the relative term "semantical rule of" is as | much in need of clarification, at least, as "analytic for". | | Quine, "Two Dogmas", p. 34. | | W.V. Quine, |"Two Dogmas of Empiricism", 'Philosophical Review', January 1951. | Reprinted as pages 20-46 in 'From a Logical Point of View', | 2nd edition, Harvard University Press, Cambridge, MA, 1980. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at att.net Wed Jul 16 05:56:59 2003 From: jawbrey at att.net (Jon Awbrey) Date: Wed Jan 21 20:34:37 2004 Subject: [Inquiry] Re: Quine -- Two Dogmas Of Empiricism References: <3F10BBD2.DD0B3EFE@att.net> <3F1452D6.B01EE33E@att.net> <3F14608C.CC04AC36@att.net> <3F146ED5.3E9CDC0E@att.net> Message-ID: <3F152F7B.91C745DB@att.net> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o TDOE. Note 20 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | 4. Semantical Rules (cont.) | | It may be instructive to compare the notion of semantical rule with that | of postulate. Relative to a given set of postulates, it is easy to say | what a postulate is: it is a member of the set. Relative to a given | set of semantical rules, it is equally easy to say what a semantical | rule is. But given simply a notation, mathematical or otherwise, | and indeed as thoroughly understood a notation as you please in | point of the translations or truth conditions of its statements, | who can say which of its true statements rank as postulates? | Obviously the question is meaningless -- as meaningless as | asking which points in Ohio are starting points. Any finite | (or effectively specifiable infinite) selection of statements | (preferably true ones, perhaps) is as much 'a' set of postulates | as any other. The word "postulate" is significant only relative | to an act of inquiry; we apply the word to a set of statements just | in so far as we happen, for the year or the moment, to be thinking of | those statements in relation to the statements which can be reached from | them by some set of transformations to which we have seen fit to direct our | attention. Now the notion of semantical rule is as sensible and meaningful as | that of postulate, if conceived in a similarly relative spirit -- relative, this | time, to one or another particular enterprise of schooling unconversant persons | in sufficient conditions for truth of statements of some natural or artificial | language L. But from this point of view no one signalization of a subclass | of the truths of L is intrinsically more a semantical rule than another; | and, if "analytic" means "true by semantical rules", no one truth of L | is analytic to the exclusion of another.* | |*The foregoing paragraph was not part of the present essay as | originally published. It was prompted by Martin [R.M. Martin, | "On 'Analytic'", 'Philosophical Studies', vol. 3 (1952), 42-47], | as was the end of Essay 7. | | Quine, "Two Dogmas", p. 35. | | W.V. Quine, |"Two Dogmas of Empiricism", 'Philosophical Review', January 1951. | Reprinted as pages 20-46 in 'From a Logical Point of View', | 2nd edition, Harvard University Press, Cambridge, MA, 1980. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at att.net Wed Jul 16 07:24:01 2003 From: jawbrey at att.net (Jon Awbrey) Date: Wed Jan 21 20:34:37 2004 Subject: [Inquiry] Re: Quine -- Two Dogmas Of Empiricism References: <3F10BBD2.DD0B3EFE@att.net> <3F1452D6.B01EE33E@att.net> <3F14608C.CC04AC36@att.net> <3F146ED5.3E9CDC0E@att.net> <3F152F7B.91C745DB@att.net> Message-ID: <3F1543E1.E5419A8C@att.net> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o TDOE. Note 21 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | 4. Semantical Rules (concl.) | | It might conceivably be protested that an artificial language L | (unlike a natural one) is a language in the ordinary sense 'plus' | a set of explicit semantical rules -- the whole constituting, let | us say, an ordered pair; and that the semantical rules of L then | are specifiable simply as the second component of the pair L. But, | by the same token and more simply, we might construe an artificial | language L outright as an ordered pair whose second component is the | class of its analytic statements; and then the analytic statements of L | become specifiable simply as the statements in the second component of L. | Or better still, we might just stop tugging at our bootstraps altogether. | | Not all the explanations of analyticity known to Carnap | and his readers have been covered explicitly in the above | considerations, but the extension to other forms is not hard | to see. Just one additional factor should be mentioned which | sometimes enters: sometimes the semantical rules are in effect | rules of translation into ordinary language, in which case the | analytic statements of the artificial language are in effect | recognized as such from the analyticity of their specified | translations in ordinary language. Here certainly there | can be no thought of an illumination of the problem of | analyticity from the side of the artificial language. | | From the point of view of the problem of analyticity the notion of an | artificial language with semantical rules is a 'feu follet par excellence'. | Semantical rules determining the analytic statements of an artificial language | are of interest only in so far as we already understand the notion of analyticity; | they are of no help in gaining this understanding. | | Appeal to hypothetical languages of an artificially simple | kind could conceivably be useful in clarifying analyticity, | if the mental or behavioral or cultural factors relevant to | analyticity -- whatever they may be -- were somehow sketched | into the simplified model. But a model which takes analyticity | merely as an irreducible character is unlikely to throw light on | the problem of explicating analyticity. | | It is obvious that truth in general depends on both language and extralinguistic | fact. The statement "Brutus killed Caesar" would be false if the world had | been different in certain ways, but it would also be false if the word | "killed" happened rather to have the sense of "begat". Thus one is | tempted to suppose in general that the truth of a statement is | somehow analyzable into a linguistic component and a factual | component. Given this supposition, it next seems reasonable | that in some statements the factual component should be null; | and these are the analytic statements. But, for all its | a priori reasonableness, a boundary between analytic | and synthetic statements simply has not been drawn. | That there is such a distinction to be drawn at | all is an unempirical dogma of empiricists, | a metaphysical article of faith. | | Quine, "Two Dogmas", pp. 35-37. | | W.V. Quine, |"Two Dogmas of Empiricism", 'Philosophical Review', January 1951. | Reprinted as pages 20-46 in 'From a Logical Point of View', | 2nd edition, Harvard University Press, Cambridge, MA, 1980. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at att.net Wed Jul 16 09:02:01 2003 From: jawbrey at att.net (Jon Awbrey) Date: Wed Jan 21 20:34:37 2004 Subject: [Inquiry] Re: Quine -- Two Dogmas Of Empiricism References: <3F10BBD2.DD0B3EFE@att.net> <3F1452D6.B01EE33E@att.net> <3F14608C.CC04AC36@att.net> <3F146ED5.3E9CDC0E@att.net> <3F152F7B.91C745DB@att.net> <3F1543E1.E5419A8C@att.net> Message-ID: <3F155AD9.345633D6@att.net> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o TDOE. Note 22 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | 5. The Verification Theory and Reductionism | | In the course of these somber reflections we have taken a dim view first | of the notion of meaning, then of the notion of cognitive synonymy, and | finally of the notion of analyticity. But what, it may be asked, of | the verification theory of meaning? This phrase has established | itself so firmly as a catchword of empiricism that we should be | very unscientific indeed not to look beneath it for a possible | key to the problem of meaning and the associated problems. | | The verification theory of meaning, which has been conspicuous in the | literature from Peirce onward, is that the meaning of a statement is | the method of empirically confirming or infirming it. An analytic | statement is that limiting case which is confirmed no matter what. | | As urged in Section 1, we can as well pass over the question of | meanings as entities and move straight to sameness of meaning, | or synonymy. Then what the verification theory says is that | statements are synonymous if and only if they are alike in | point of method of empirical confirmation or infirmation. | | This is an account of cognitive synonymy not of linguistic forms generally, | but of statements.* However, from the concept of synonymy of statements | we could derive the concept of synonymy for other linguistic forms, by | considerations somewhat similar to those at the end of Section 3. | Assuming the notion of "word", indeed, we could explain any | two forms as synonymous when the putting of one form for | an occurrence of the other in any statement (apart from | occurrences within "words") yields a synonymous statement. | Finally, given the concept of synonymy thus for linguistic | forms generally, we could define analyticity in terms of | synonymy and logical truth as in Section 1. For that | matter, we could define analyticity more simply in | terms of just synonymy of statements together with | logical truth; it is not necessary to appeal to | synonymy of linguistic forms other than statements. | For a statement may be described as analytic simply | when it is synonymous with a logically true statement. | |*The doctrine can indeed be formulated with terms rather than statements as the | units. Thus Lewis describes the meaning of a term as "'a criterion in mind', | by reference to which one is able to apply or refuse to apply the expression | in question in the case of presented, or imagined, things or situations" | [C.I. Lewis, 'An Analysis of Knowledge and Valuation', Open Court, LaSalle, | IL, 1946, p. 133]. -- For an instructive account of the vicissitudes of | the verification theory of meaning, centered however on the question | of meaning'fulness' rather than synonymy and analyticity, see Hempel. | | Quine, "Two Dogmas", pp. 37-38. | | W.V. Quine, |"Two Dogmas of Empiricism", 'Philosophical Review', January 1951. | Reprinted as pages 20-46 in 'From a Logical Point of View', | 2nd edition, Harvard University Press, Cambridge, MA, 1980. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at att.net Wed Jul 16 11:22:14 2003 From: jawbrey at att.net (Jon Awbrey) Date: Wed Jan 21 20:34:37 2004 Subject: [Inquiry] Re: Quine -- Two Dogmas Of Empiricism References: <3F10BBD2.DD0B3EFE@att.net> <3F1452D6.B01EE33E@att.net> <3F14608C.CC04AC36@att.net> <3F146ED5.3E9CDC0E@att.net> <3F152F7B.91C745DB@att.net> <3F1543E1.E5419A8C@att.net> <3F155AD9.345633D6@att.net> Message-ID: <3F157BB6.295216E3@att.net> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o TDOE. Note 23 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | 5. The Verification Theory and Reductionism (cont.) | | So, if the verification theory can be accepted as an adequate account | of statement synonymy, the notion of analyticity is saved after all. | However, let us reflect. Statement synonymy is said to be likeness | of method of empirical confirmation or infirmation. Just what are | these methods which are to be compared for likeness? What, in | other words, is the nature of the relation between a statement | and the experiences which contribute to or detract from its | confirmation? | | The most naive view of the relation is that it is one of direct report. | This is 'radical reductionism'. Every meaningful statement is held to be | translatable into a statement (true or false) about immediate experience. | Radical reductionism, in one form or another, well antedates the verification | theory of meaning explicitly so called. Thus Locke and Hume held that every | idea must either originate directly in sense experience or else be compounded | of ideas thus originating; and taking a hint from Tooke we might rephrase | this doctrine in semantical jargon by saying that a term, to be significant | at all, must be either a name of a sense datum or a compound of such names or | an abbreviation of such a compound. So stated, the doctrine remains ambiguous | as between sense data as sensory events and sense data as sensory qualities; | and it remains vague as to the admissible ways of compounding. Moreover, the | doctrine is unnecessarily and intolerably restrictive in the term-by-term | critique which it imposes. More reasonably, and without yet exceeding | the limits of what I have called radical reductionism, we may take full | statements as our significant units -- thus demanding that our statements | as wholes be translatable into sense-datum language, but not that they be | translatable term by term. | | This emendation would unquestionably have been welcome to Locke and Hume | and Tooke, but historically it had to await an important reorientation in | semantics -- the reorientation whereby the primary vehicle of meaning came | to be seen no longer in the term but in the statement. This reorientation, | seen in Bentham and Frege, underlies Russell's concept of incomplete symbols | defined in use; also it is implicit in the verification theory of meaning, | since the objects of verification are statements. | | Quine, "Two Dogmas", pp. 38-39. | | W.V. Quine, |"Two Dogmas of Empiricism", 'Philosophical Review', January 1951. | Reprinted as pages 20-46 in 'From a Logical Point of View', | 2nd edition, Harvard University Press, Cambridge, MA, 1980. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at att.net Wed Jul 16 13:18:49 2003 From: jawbrey at att.net (Jon Awbrey) Date: Wed Jan 21 20:34:37 2004 Subject: [Inquiry] Re: Quine -- Two Dogmas Of Empiricism References: <3F10BBD2.DD0B3EFE@att.net> <3F1452D6.B01EE33E@att.net> <3F14608C.CC04AC36@att.net> <3F146ED5.3E9CDC0E@att.net> <3F152F7B.91C745DB@att.net> <3F1543E1.E5419A8C@att.net> <3F155AD9.345633D6@att.net> <3F157BB6.295216E3@att.net> Message-ID: <3F159709.8195F63@att.net> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o TDOE. Note 24 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | 5. The Verification Theory and Reductionism (cont.) | | Radical reductionism, conceived now with statements as units, | set itself the task of specifying a sense-datum language and | showing how to translate the rest of significant discourse, | statement by statement, into it. Carnap embarked on this | project in the 'Aufbau'. | | The language which Carnap adopted as his starting point was not | a sense-datum language in the narrowest conceivable sense, for | it included also the notations of logic, up through higher set | theory. In effect it included the whole language of pure | mathematics. The ontology implicit in it (that is, the | range of values of its variables) embraced not only | sensory events but classes, classes of classes, and | so on. Empiricists there are who would boggle at | such prodigality. Carnap's starting point is | very parsimonious, however, in its extralogical | or sensory part. In a series of constructions in | which he exploits the resources of modern logic with | much ingenuity, Carnap succeeds in defining a wide array | of important additional sensory concepts which, but for his | constructions, one would not have dreamed were definable on | so slender a basis. He was the first empiricist who, not | content with asserting the reducibility of science to | terms of immediate experience, took serious steps | toward carrying out the reduction. | | Quine, "Two Dogmas", p. 39. | | W.V. Quine, |"Two Dogmas of Empiricism", 'Philosophical Review', January 1951. | Reprinted as pages 20-46 in 'From a Logical Point of View', | 2nd edition, Harvard University Press, Cambridge, MA, 1980. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at att.net Wed Jul 16 14:18:11 2003 From: jawbrey at att.net (Jon Awbrey) Date: Wed Jan 21 20:34:37 2004 Subject: [Inquiry] Re: Quine -- Two Dogmas Of Empiricism References: <3F10BBD2.DD0B3EFE@att.net> <3F1452D6.B01EE33E@att.net> <3F14608C.CC04AC36@att.net> <3F146ED5.3E9CDC0E@att.net> <3F152F7B.91C745DB@att.net> <3F1543E1.E5419A8C@att.net> <3F155AD9.345633D6@att.net> <3F157BB6.295216E3@att.net> <3F159709.8195F63@att.net> Message-ID: <3F15A4F3.826206FC@att.net> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o TDOE. Note 25 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | 5. The Verification Theory and Reductionism (cont.) | | If Carnap's starting point is satisfactory, | still his constructions were, as he himself | stressed, only a fragment of the full program. | The construction of even the simplest statements | about the physical world was left in a sketchy state. | Carnap's suggestions on this subject were, despite their | sketchiness, very suggestive. He explained spatio-temporal | point-instants as quadruples of real numbers and envisaged | assignment of sense qualities to point-instants according | to certain canons. Roughly summarized, the plan was that | qualities should be assigned to point-instants in such a | way as to achieve the laziest world compatible with our | experience. The principle of least action was to be | our guide in constructing a world from experience. | | Carnap did not seem to recognize, however, that his treatment | of physical objects fell short of reduction not merely through | sketchiness, but in principle. Statements of the form "Quality | q is at point-instant x;y;z;t" were, according to his canons, | to be apportioned truth vakues in such a way as to maximize | and minimize certain over-all features, and with growth of | experience the truth values were to be progressively revised | in the same spirit. I think that this is a good schematization | (deliberately oversimplified, to be sure) of what science really | does; but it provides no indication, not even the sketchiest, of | how a statement of the form "Quality q is at x;y;z;t" could ever | be translated into Carnap's initial language of sense data and | logic. The connective "is at" remains an added undefined | connective; the canons counsel us in its use but not | in its elimination. | | Carnap seems to have appreciated this point afterward; | for in his later writings he abandoned all notion of | the translatability of statements about the physical | world into statements about immediate experience. | Reductionism in its radical form has long since | ceased to figure in Carnap's philosophy. | | Quine, "Two Dogmas", pp. 39-40. | | W.V. Quine, |"Two Dogmas of Empiricism", 'Philosophical Review', January 1951. | Reprinted as pages 20-46 in 'From a Logical Point of View', | 2nd edition, Harvard University Press, Cambridge, MA, 1980. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at att.net Wed Jul 16 15:44:07 2003 From: jawbrey at att.net (Jon Awbrey) Date: Wed Jan 21 20:34:37 2004 Subject: [Inquiry] Re: Quine -- Two Dogmas Of Empiricism References: <3F10BBD2.DD0B3EFE@att.net> <3F1452D6.B01EE33E@att.net> <3F14608C.CC04AC36@att.net> <3F146ED5.3E9CDC0E@att.net> <3F152F7B.91C745DB@att.net> <3F1543E1.E5419A8C@att.net> <3F155AD9.345633D6@att.net> <3F157BB6.295216E3@att.net> <3F159709.8195F63@att.net> <3F15A4F3.826206FC@att.net> Message-ID: <3F15B917.26AC0A3B@att.net> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o TDOE. Note 26 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | 5. The Verification Theory and Reductionism (cont.) | | But the dogma of reductionism has, in a subtler and more tenuous form, | continued to influence the thought of empiricists. The notion lingers | that to each statement, or each synthetic statement, there is associated | a unique range of possible sensory events such that the occurrence of any | of them would add to the likelihood of truth of the statement, and that | there is associated also another unique range of possible sensory events | whose occurrence would detract from that likelihood. This notion is of | course implicit in the verification theory of meaning. | | The dogma of reductionism survives in the supposition that each statement, | taken in isolation from its fellows, can admit of confirmation or infirmation | at all. My countersuggestion, issuing essentially from Carnap's doctrine of | the physical world in the 'Aufbau', is that our statements about the external | world face the tribunal of sense experience not individually but only as a | corporate body.* | |*This doctrine was well argued by Duhem [Pierre Duhem, 'La Theorie Physique: | Son Object et Sa Structure', Paris, 1906, pp. 303-328]. Or see Lowinger | Armand Lowinger, 'The Methodology of Pierre Duhem', Columbia University | Press, New York, NY, 1941, pp. 132-140]. | | Quine, "Two Dogmas", pp. 40-41. | | W.V. Quine, |"Two Dogmas of Empiricism", 'Philosophical Review', January 1951. | Reprinted as pages 20-46 in 'From a Logical Point of View', | 2nd edition, Harvard University Press, Cambridge, MA, 1980. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at att.net Wed Jul 16 19:26:23 2003 From: jawbrey at att.net (Jon Awbrey) Date: Wed Jan 21 20:34:37 2004 Subject: [Inquiry] Re: Quine -- Two Dogmas Of Empiricism References: <3F10BBD2.DD0B3EFE@att.net> <3F1452D6.B01EE33E@att.net> <3F14608C.CC04AC36@att.net> <3F146ED5.3E9CDC0E@att.net> <3F152F7B.91C745DB@att.net> <3F1543E1.E5419A8C@att.net> <3F155AD9.345633D6@att.net> <3F157BB6.295216E3@att.net> <3F159709.8195F63@att.net> <3F15A4F3.826206FC@att.net> <3F15B917.26AC0A3B@att.net> Message-ID: <3F15ED2F.7C082BD3@att.net> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o TDOE. Note 27 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | 5. The Verification Theory and Reductionism (concl.) | | The dogma of reductionism, even in its attenuated form, is intimately | connected with the other dogma -- that there is a cleavage between | the analytic and the synthetic. We have found ourselves led, | indeed, from the latter problem to the former through the | verification theory of meaning. More directly, the one | dogma clearly supports the other in this way: as long | as it is taken to be significant in general to speak | of the confirmation and infirmation of a statement, | it seems significant to speak also of a limiting | kind of statement which is vacuously confirmed, | 'ipso facto', come what may; and such | a statement is analytic. | | The two dogmas are, indeed, at root identical. We lately reflected | that in general the truth of statements does obviously depend both | upon language and upon extralinguistic fact; and we noted that | this obvious circumstance carries in its train, not logically | but all too naturally, a feeling that the truth of a statement | is somehow analyzable into a linguistic component and a factual | component. The factual component must, if we are empiricists, | boil down to a range of confirmatory experiences. In the | extreme case where the linguistic component is all that | matters, a true statement is analytic. But I hope we are | now impressed with how stubbornly the distinction between | analytic and synthetic has resisted any straightforward | drawing. I am impressed also, apart from prefabricated | examples of black and white balls in an urn, with how | baffling the problem has always been of arriving at | any explicit theory of the empirical confirmation of | a synthetic statement. My present suggestion is that | it is nonsense, and the root of much nonsense, to speak | of a linguistic component and a factual component in the | truth of any individual statement. Taken collectively, | science has its double dependence upon language and | experience; but this duality is not significantly | traceable into the statements of science taken | one by one. | | The idea of defining a symbol in use was, as remarked, an advance | over the impossible term-by-term empiricism of Locke and Hume. | The statement, rather than the term, came with Bentham to be | recognized as the unit accountable to an empiricist critique. | But what I am now urging is that even in taking the statement | as unit we have drawn our grid too finely. The unit of empirical | significance is the whole of science. | | Quine, "Two Dogmas", pp. 41-42. | | W.V. Quine, |"Two Dogmas of Empiricism", 'Philosophical Review', January 1951. | Reprinted as pages 20-46 in 'From a Logical Point of View', | 2nd edition, Harvard University Press, Cambridge, MA, 1980. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at att.net Thu Jul 17 11:40:23 2003 From: jawbrey at att.net (Jon Awbrey) Date: Wed Jan 21 20:34:37 2004 Subject: [Inquiry] Re: Quine -- Two Dogmas Of Empiricism References: <3F10BBD2.DD0B3EFE@att.net> Message-ID: <3F16D177.822FBA3D@att.net> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o TDOE. Note 28 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | 6. Empiricism without the Dogmas | | The totality of our so-called knowledge or beliefs, from the most | casual matters of geography and history to the profoundest laws of | atomic physics or even of pure mathematics and logic, is a man-made | fabric which impinges on experience only along the edges. Or, to | change the figure, total science is like a field of force whose | boundary conditions are experience. A conflict with experience at | the periphery occasions readjustments in the interior of the field. | Truth values have to be redistributed over some of our statements. | Re-evaluation of some statements entails re-evaluation of others, | because of their logical interconnections -- the logical laws | being in turn simply certain further statements of the system, | certain further elements of the field. Having re-evaluated one | statement we must re-evaluate some others, which may be statements | logically connected with the first or may be the statements of logical | connections themselves. But the total field is so underdetermined by | its boundary conditions, experience, that there is much latitude of | choice as to what statements to re-evaluate in the light of any | single contrary experience. No particular experiences are | linked with any particular statements in the interior of | the field, except indirectly through considerations | of equilibrium affecting the field as a whole. | | Quine, "Two Dogmas", pp. 42-43. | | W.V. Quine, |"Two Dogmas of Empiricism", 'Philosophical Review', January 1951. | Reprinted as pages 20-46 in 'From a Logical Point of View', | 2nd edition, Harvard University Press, Cambridge, MA, 1980. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at att.net Thu Jul 17 13:28:23 2003 From: jawbrey at att.net (Jon Awbrey) Date: Wed Jan 21 20:34:37 2004 Subject: [Inquiry] Re: Quine -- Two Dogmas Of Empiricism References: <3F10BBD2.DD0B3EFE@att.net> <3F16D177.822FBA3D@att.net> Message-ID: <3F16EAC7.33EE41F6@att.net> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o TDOE. Note 29 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | 6. Empiricism without the Dogmas (cont.) | | If this view is right, it is misleading to speak of the empirical content of | an individual statement -- especially if it is a statement at all remote from | the experiential periphery of the field. Furthermore it becomes folly to seek | a boundary between synthetic statements, which hold contingently on experience, | and analytic statements, which hold come what may. Any statement can be held | true come what may, if we make drastic enough adjustments elsewhere in the | system. Even a statement very close to the periphery can be held true in | the face of recalcitrant experience by pleading hallucination or by amending | certain statements of the kind called logical laws. Conversely, by the same | token, no statement is immune to revision. Revision even of the logical law | of the excluded middle has been proposed as a means of simplifying quantum | mechanics; and what difference is there in principle between such a shift | and the shift whereby Kepler superseded Ptolemy, or Einstein Newton, or | Darwin Aristotle? | | For vividness I have been speaking in terms of varying distances | from a sensory periphery. Let me try now to clarify this notion | without metaphor. Certain statements, though 'about' physical | objects and not sense experience, seem peculiarly germane to | sense experience -- and in a selective way: some statements to | some experiences, others to others. Such statements, especially | germane to particular experiences, I picture as near the periphery. | But in this relation of "germaneness" I envisage nothing more than a | loose association reflecting the relative likelihood, in practice, of | our choosing one statement rather than another for revision in the event | of recalcitrant experience. For example, we can imagine recalcitrant | experiences to which we would surely be inclined to accommodate our | system by re-evaluating just the statement that there are brick | houses on Elm Street, together with related statements on the | same topic. We can imagine other recalcitrant experiences | to which we would be inclined to accommodate our system by | re-evaluating just the statement that there are no centaurs, | along with kindred statemnts. A recalcitrant experience can, | I have urged, be accommodated by any of various alternative | re-evaluations in various alternative quarters of the total | system; but, in the cases which we are now imagining, our | natural tendency to disturb the total system as little as | possible would lead us to focus our revisions upon these | specific statements concerning brick houses or centaurs. | These statements are felt, therefore, to have a sharper | empirical reference than highly theoretical statements | of physics or logic or ontology. The latter statements | may be thought of as relatively centrally located within | the total network, meaning merely that little preferential | connection with any particular sense data obtrudes itself. | | Quine, "Two Dogmas", pp. 43-44. | | W.V. Quine, |"Two Dogmas of Empiricism", 'Philosophical Review', January 1951. | Reprinted as pages 20-46 in 'From a Logical Point of View', | 2nd edition, Harvard University Press, Cambridge, MA, 1980. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at att.net Thu Jul 17 16:02:34 2003 From: jawbrey at att.net (Jon Awbrey) Date: Wed Jan 21 20:34:37 2004 Subject: [Inquiry] Re: Quine -- Two Dogmas Of Empiricism References: <3F10BBD2.DD0B3EFE@att.net> <3F16D177.822FBA3D@att.net> <3F16EAC7.33EE41F6@att.net> Message-ID: <3F170EEA.8BD392C3@att.net> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o TDOE. Note 30 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | 6. Empiricism without the Dogmas (cont.) | | As an empiricist I continue to think of the conceptual scheme of science as | a tool, ultimately, for predicting future experience in the light of past | experience. Physical objects are conceptually imported into the situation | as convenient intermediaries -- not by definition in terms of experience, | but simply as irreducible posits comparable, epistemologically, to the | gods of Homer. For my part I do, qua lay physicist, believe in physical | objects and not in Homer's gods; and I consider it a scientific error | to believe otherwise. But in point of epistemological footing the | physical objects and the gods differ only in degree and not in kind. | Both sorts of entities enter our conception only as cultural posits. | The myth of physical objects is epistemologically superior to most | in that it has proved more efficacious than other myths as a device | for working a manageable structure into the flux of experience. | | Positing does not stop with macroscopic physical objects. | Objects at the atomic level are posited to make the laws of | macroscopic objects, and ultimately the laws of experience, | simpler and more manageable; and we need not expect or demand | full definition of atomic and subatomic entities in terms of | macroscopic ones, any more than definition of macroscopic things | in terms of sense data. Science is a continuation of common sense, | and it continues the common-sense expedient of swelling ontology to | simplify theory. | | Physical objects, small and large, are not the only posits. | Forces are another example; and indeed we are told nowadays that | the boundary between energy and matter is obsolete. Moreover, the | abstract entities which are the substance of mathematics -- ultimately | classes and classes of classes and so on up -- are another posit in the | same spirit. Epistemologically these are myths on the same footing with | physical objects and gods, neither better nor worse except for differences | in the degree to which they expedite our dealings with sense experiences. | | The over-all algebra of rational and irrational numbers is | underdetermined by the algebra of rational numbers, but is | smoother and more convenient; and it includes the algebra | of rational numbers as a jagged or gerrymandered part. | Total science, mathematical and natural and human, | is similarly but more extremely underdetermined | by experience. The edge of the system must be | kept squared with experience; the rest, with | all its elaborate myths or fictions, has as | its objective the simplicty of laws. | | Quine, "Two Dogmas", pp. 44-45. | | W.V. Quine, |"Two Dogmas of Empiricism", 'Philosophical Review', January 1951. | Reprinted as pages 20-46 in 'From a Logical Point of View', | 2nd edition, Harvard University Press, Cambridge, MA, 1980. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o From jawbrey at att.net Thu Jul 17 18:48:05 2003 From: jawbrey at att.net (Jon Awbrey) Date: Wed Jan 21 20:34:38 2004 Subject: [Inquiry] Re: Quine -- Two Dogmas Of Empiricism References: <3F10BBD2.DD0B3EFE@att.net> <3F16D177.822FBA3D@att.net> <3F16EAC7.33EE41F6@att.net> <3F170EEA.8BD392C3@att.net> Message-ID: <3F1735B5.F0A1EC70@att.net> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o TDOE. Note 31 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | 6. Empiricism without the Dogmas (concl.) | | Ontological questions, under this view, are on a par with questions | of natural science. Consider the question whether to countenance | classes as entities. This, as I have argued elsewhere, is the | question whether to quantify with respect to variables which | take classes as values. Now Carnap [*] has maintained that | this is a question not of matters of fact but of choosing | a convenient language form, a convenient conceptual scheme | or framework for science. With this I agree, but only on the | proviso that the same be conceded regarding scientific hypotheses | generally. Carnap ([*], p. 32n) has recognized that he is able to | preserve a double standard for ontological questions and scientific | hypotheses only by assuming an absolute distinction between the | analytic and the synthetic; and I need not say again that | this is a distinction which I reject. | | The issue over there being classes seems more a question of convenient | conceptual scheme; the issue over there being centaurs, or brick houses | on Elm street, seems more a question of fact. But I have been urging that | this difference is only one of degree, and that it turns upon our vaguely | pragmatic inclination to adjust one strand of the fabric of science rather | than another in accommodating some particular recalcitrant experience. | Conservatism figures in such choices, and so does the quest for | simplicity. | | Carnap, Lewis, and others take a pragmatic stand on the question of choosing | between language forms, scientific frameworks; but their pragmatism leaves | off at the imagined boundary between the analytic and the synthetic. In | repudiating such a boundary I espouse a more thorough pragmatism. Each | man is given a scientific heritage plus a continuing barrage of sensory | stimulation; and the considerations which guide him in warping his | scientific heritage to fit his continuing sensory promptings are, | where rational, pragmatic. | |*Rudolf Carnap, "Empiricism, Semantics, and Ontology", |'Revue Internationale de Philosphie', vol. 4 (1950), pp. 20-40. | Reprinted in Leonard Linsky (ed.), 'Semantics and the Philosophy | of Language', University of Illinois Press, Urbana, IL, 1952. | | Quine, "Two Dogmas", pp. 45-46. | | W.V. Quine, |"Two Dogmas of Empiricism", 'Philosophical Review', January 1951. | Reprinted as pages 20-46 in 'From a Logical Point of View', | 2nd edition, Harvard University Press, Cambridge, MA, 1980. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o