[Inquiry] Re: Dynamics And Logic

[Jon Awbrey]

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DAL.  Note 7

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To broaden our experience with simple examples, let us
now contemplate the sixteen functions of concrete type
!P! x !Q! -> B and abstract type B x B -> B.  For ease
of future reference, I will set here a few tables that
specify the actions of E and D on the 16 functions and
allow us to view the results in several different ways.

By way of initial orientation, Table 7 lists equivalent expressions
for the sixteen functions in several different formalisms, indexing
systems, or languages for the propositional calculus, also known as
"zeroth order logic".

Table 7.  Propositional Forms on Two Variables
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| L_1 ` ` | L_2 ` ` | L_3 ` ` | L_4 ` ` `| L_5 ` ` ` ` ` ` `| L_6 ` ` `|
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` `| ` ` ` ` ` ` ` ` `| ` ` ` ` `|
| Decimal | Binary` | Vector` | Cactus` `| English ` ` ` ` `| Ordinary |
o---------o---------o---------o----------o------------------o----------o
| ` ` ` ` | ` ` ` p : 1 1 0 0 | ` ` ` ` `| ` ` ` ` ` ` ` ` `| ` ` ` ` `|
| ` ` ` ` | ` ` ` q : 1 0 1 0 | ` ` ` ` `| ` ` ` ` ` ` ` ` `| ` ` ` ` `|
o---------o---------o---------o----------o------------------o----------o
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` `| ` ` ` ` ` ` ` ` `| ` ` ` ` `|
| f_0 ` ` | f_0000` | 0 0 0 0 | ` `() ` `| false ` ` ` ` ` `| ` `0` ` `|
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` `| ` ` ` ` ` ` ` ` `| ` ` ` ` `|
| f_1 ` ` | f_0001` | 0 0 0 1 | `(p)(q) `| neither p nor q `| ~p & ~q `|
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` `| ` ` ` ` ` ` ` ` `| ` ` ` ` `|
| f_2 ` ` | f_0010` | 0 0 1 0 | `(p) q` `| q and not p ` ` `| ~p & `q `|
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` `| ` ` ` ` ` ` ` ` `| ` ` ` ` `|
| f_3 ` ` | f_0011` | 0 0 1 1 | `(p)` ` `| not p ` ` ` ` ` `| ~p` ` ` `|
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` `| ` ` ` ` ` ` ` ` `| ` ` ` ` `|
| f_4 ` ` | f_0100` | 0 1 0 0 | ` p (q) `| p and not q ` ` `| `p & ~q `|
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` `| ` ` ` ` ` ` ` ` `| ` ` ` ` `|
| f_5 ` ` | f_0101` | 0 1 0 1 | ` ` (q) `| not q ` ` ` ` ` `| ` ` `~q `|
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` `| ` ` ` ` ` ` ` ` `| ` ` ` ` `|
| f_6 ` ` | f_0110` | 0 1 1 0 | `(p, q) `| p not equal to q | `p + `q `|
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` `| ` ` ` ` ` ` ` ` `| ` ` ` ` `|
| f_7 ` ` | f_0111` | 0 1 1 1 | `(p `q) `| not both p and q | ~p v ~q `|
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` `| ` ` ` ` ` ` ` ` `| ` ` ` ` `|
| f_8 ` ` | f_1000` | 1 0 0 0 | ` p `q` `| p and q ` ` ` ` `| `p & `q `|
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` `| ` ` ` ` ` ` ` ` `| ` ` ` ` `|
| f_9 ` ` | f_1001` | 1 0 0 1 | ((p, q)) | p equal to q` ` `| `p = `q `|
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` `| ` ` ` ` ` ` ` ` `| ` ` ` ` `|
| f_10` ` | f_1010` | 1 0 1 0 | ` ` `q` `| q ` ` ` ` ` ` ` `| ` ` ` q `|
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` `| ` ` ` ` ` ` ` ` `| ` ` ` ` `|
| f_11` ` | f_1011` | 1 0 1 1 | `(p (q)) | not p without q `| `p => q `|
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` `| ` ` ` ` ` ` ` ` `| ` ` ` ` `|
| f_12` ` | f_1100` | 1 1 0 0 | ` p ` ` `| p ` ` ` ` ` ` ` `| `p` ` ` `|
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` `| ` ` ` ` ` ` ` ` `| ` ` ` ` `|
| f_13` ` | f_1101` | 1 1 0 1 | ((p) q) `| not q without p `| `p <= q `|
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` `| ` ` ` ` ` ` ` ` `| ` ` ` ` `|
| f_14` ` | f_1110` | 1 1 1 0 | ((p)(q)) | p or q` ` ` ` ` `| `p v `q `|
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` `| ` ` ` ` ` ` ` ` `| ` ` ` ` `|
| f_15` ` | f_1111` | 1 1 1 1 | ` (())` `| true` ` ` ` ` ` `| ` `1` ` `|
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` `| ` ` ` ` ` ` ` ` `| ` ` ` ` `|
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Jon Awbrey

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inquiry e-lab: http://stderr.org/pipermail/inquiry/
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