Wednesday, January 02, 2008

On Tom's Algebra of Logic

I have noted previously that propositional logic can be seen as a special case of categorical syllogistics. That many non-categorical propositions (like conditionals) can be handled categorically has been recognized for a very long time, although systematic investigation of this approach has been rare. To handle all of propositional logic as a special case of categorical syllogistics you just need categorical syllogistics and two assumptions. The two assumptions are:

(1) The universe of discourse is a singleton universe.

(2) Propositions may be treated as terms.

The reason for the second should be obvious. The reason for the first may be less obvious. Strictly speaking you could do without it, but the propositional logic that would result would be nonstandard and difficult to interpret; implication, in particular, would begin to have some bizarre properties. Categorical syllogistics makes heavy use of logical quantity; to do standard propositional logic you have to make an assumption that makes it OK to conclude a universal conclusion from a particular premise. The simplest way to do this is to take the whole universe for propositions as singular. (It's a good question, and one to which I don't know the answer, whether there are alternative assumptions that could be made to get the same result.)

In any case, given these assumptions you can easily handle all of standard truth-functional propositional logic: conditionals turn out to be A propositions, etc. Likewise, you can diagram propositional logic with any diagrams that diagram categorical syllogisms. For instance, my adaptation of Welton's adaptation of Lambert's lines gives us the following diagrams:

P & Q
|   | X | X | X |

P v Q
|   |   |   | X |

P → Q
|   | X |   |   |

The difference from the ordinary diagrams is due to assumption (1): given that the universe of discourse is a singleton universe, there is no need to indicate presence: presence is just whatever is left over when all the absences are determined. (If all the spaces are X'd then the statement is inconsistent.) The way to think of it is in terms of the world: P & Q tells us that the world is such that P and Q; P v Q tells us that the world is such that P or Q or both; etc.

I was thinking of this recently when thinking about Tom's system for handling categorical propositions. I previously diagrammed Tom's eight types; but there I assumed that they were working as standard categorical propositions. I've recently come to wonder, though, if perhaps Tom's system is really more like a propositional logic. The major reason for this is that Tom's types act like cases of propositional logic in categorical form. [ ] is most easily understood as disjunction, while ( ) is conjunction. Thus:

[+a+b] : a v b
[+a-b] : a v ~b
[-a+b] : ~a v b
[-a-b] : ~a v ~b

(+a+b) : a & b
(+a-b) : a & ~b
(-a+b) : ~a & b
(-a-b) : ~a & ~b

Thus Tom's approach seems to be a way of handling all categorical propositions as if they were complex propositions in propositional logic as represented in categorical syllogistics. Yes, I realize that that's complicated. The idea is this: each categorical proposition is treated as if it were a disjunction or conjunction of propositionalized versions of its terms. All A is B is understood as being (in effect) if a thing's A, it's B, or The world is a B-or-not-A world; and so forth. This means that in Tom's approach all categorical propositions are handled under the assumptions that make it possible for us to treat standard propositional logic as a special case of categorical syllogistics. (This is one reason why, for instance, Tom doesn't have to worry about the quantity of singular propositions, and why he regularly prefers to interpret his symbolism in propositional terms, and why he so easily handles multiple quantification.) But the version of propositional logic Tom uses is the propositional logic that is the special case of categorical syllogistics.

That one can handle ordinary categorical syllogisms as a special case (Tom's approach) [or is it perhaps merely an analogue?] of a special case (propositional logic) of ordinary categorical syllogisms is an interesting feature of the syllogistic approach that I hadn't quite appreciated before.

That's the way things seem to me, at least, on reading some of Tom's more recent work over at Blogicum.