Monday, January 14, 2008

Deviant Propositional Logics and Existential Import

Previously I had noted a version of propositional logic in which implication becomes like conjunction; and Tom noted in the comments that in context this seemed analogous to taking A propositions to have existential import. I think this is right, and it started me thinking about the propositional analogues for different assumptions about existential import. As I've noted before, it's somewhat obscure to me what's going on in 'existential import', and what I'm writing here is merely some rough notes; but here is my thought about it.

Let's take four of the many assumptions you could make about existential import.

(1) Under the first assumption, which I will call ALL, all the main types of categorical proposition have existential import: A, E, I, O.

(2) Under the second assumption, which I will call NONE, none of the main types have existential import.

(3) Under the third, which I will call SOME, only particular propositions have existential import: I, O.

(4) Under the fourth, which I will call AFFIRM, only affirmative propositions have existential import: A, I.

So what properties would the analogous propositional logics have?

The first issue is that implications in propositional logic are essentially the same as A propositions. Now, whatever existential import may be, an A proposition with existential import is such it is legitimate to conclude from All S is P to There is some S. Propositionally, this is means concluding p from p → q. Both ALL and AFFIRM allow this inference. Thus for both of them, implication collapses into conjunction, or, to be more precise, p → q is only possible where (p & q) is also true.

The second issue is that disjunctions are essentially E propositions. Thus, a principle that allows E propositions to have existential import is one in which, from a disjunction, you can conclude the falsehood of either of its disjuncts. Thus (p v q) is No nonp is nonq, which allows the conclusion There is some nonp, which is ~p. Among the above four, only ALL allows this inference.

The third issue is that conjunctions are essentially particular propositions. Thus any assumption that denies existential import for these propositions makes conjunction elimination impossible. NONE, then, makes it impossible to conclude from (p & q) to p or to q. AFFIRM gives existential import to I propositions but not to O propositions. I'm not wholly sure what this means for its propositional analogue, but I think it means that it allows conjunction elimination but only if neither conjunct is negated. Thus, we can conclude from (p & q) to p, but not from (p & ~q) to p.

Thus we have:

ALL: Implication is only allowed when antecedent and consequent are both true; disjunction is only allowed when both disjuncts are false (i.e., every disjunction is false). But conjunction elimination is allowed.

NONE: Conjunction elimination is forbidden, but disjunction and implication seem to act normally.

SOME: I think this may be the assumption for standard propositional logic. It is certainly the only one of the four that gives the right answers on the above issues: conjunction elimination is allowed, from (p → q) you can't infer p, and disjunctions aren't always false.

AFFIRM: Implication collapses to conjunction, and conjunction elimination is impossible if one of the two conjuncts involves negation. However, you can have conjunction elimination if neither of the two conjuncts involves negation.

This is all very rough and vague; I'm sure that much more could be said about these four. And, of course, there are other schemes of existential import. It might be worth noting that SOME is the usual modern assumption for existential import; NONE is the assumption for Free Logic; AFFIRM is the assumption for which Lewis Carroll argued; Venn thinks that Jevons hints at ALL, and thinks it is implied by the traditional approach to immediate inferences and the square of opposition, but I think he himself accepts SOME, and this is what Keynes attributes to him. John Neville Keynes, who gives us perhaps the most thorough study of existential import, attributes ALL to Mill (at least for real propositions); he doesn't attribute NONE to anyone, but considers it. Keynes rules in favor of SOME.

Keynes also considers a view, which he attributes to De Morgan and Jevons, in which every categorical proposition imports existence for its subject, its predicate, and their contradictories (so that All S is P implies the existence of S, nonS, P, and nonP); that would yield a very odd propositional analogue indeed, since in such an analogue every proposition implies a contradiction. (I'm not sure it would have contradiction explosion, though. At first glance it seems that explosion would be limited by the fact that you can only conclude the terms and their negations. I'll have to think about this a bit more.)