Deviant logic

In the chapter on deviant logic in his philosophy of logic book, Quine makes the claim that:

1. The deviant logician changes the subject rather than disagreeing with classical logic.

Thus, the logician who denies excluded middle is not using the words "or" and "not" to indicate disjunction and negation. She is, perhaps with good reason (though Quine is sceptical of that), using some other connectives. Thus, when she denies "not not p entails p", she is not disagreeing with us when we assert "not not p entails p". The basic thought running behind this is that:

2. The rules of classical logic are grounded in the meanings of the logical connectives (using "connectives" very widely to include negation, quantifiers, etc.)

and so any departure from the rules is a change of subject.

There is a powerful kind of argument against deviant logic here. Claims (1) and (2) seem to tell us that it is not really possible to disagree with classical logic without self-contradiction. I am either using my words in the sense that they have in classical logic, in which case I had better not disagree with classical logic on pain of contradiction, or else I am using the words in a different sense and hence not disagreeing.

I now want to describe a class of apparently non-classical logics that do not change the subject. Thus, either a deviant logician doesn't always change the subject, or else these logics are not actually deviant. The idea is this. We have rules like:

• You can infer p from p.
• If r is a conjunction of p with q, then you can infer r from p and q, p from r and q from r (conjunction introduction and elimination).
• If r is a negation of a negation of p, then you can infer p from r.
• If you can infer p and a negation of p from r, and s is a negation of r, then you can infer s from r.

And so on. The interesting thing is the second rule does not tell us that p and q have a conjunction. And indeed that is how I am imagining the system deviating from classical logic. We simply disallow certain conjunctions, negations, etc.—there will be sentences that perhaps have no negation, and pairs of sentences that perhaps have no conjunction. If we represent the language along the lines of First Order Logic, there may be cases where "A" is a sentence and "B" is a sentence but "A and B" counts as malformed. The rules of disallowing combinations may take all sorts of forms. For instance, we might simply prohibit any sentences that contain a double negation. This would result in a severe intuitionist-type limitation on what can be proved.

The logic, thus, has standard classical rules in an important sense. The rules are correct whenever they can be applied—whenever there are output sentences that work. The subject is not changed—"or" means or, "and" means and, and "not" means not—but it can be a substantive claim whether for a pair of sentences A and B, there is a sentence that we might wish to denote "A or B".

This restriction does not count as a change of subject. Indeed, Quine himself notes that there can be languages which are incapable of translating all the English truthfunctionally and quantificationally connected sentences, and he seems to think that these languages do have connectives that mean the same thing as English ones. In fact, English itself has restrictions on the formation of sentences. Past several levels of embedding, there just is no way to make distinctions. You probably can't express "(A or (A and not (B or (B and (C or D) and E) or F) and not A))" in English. Yet English does not have a deviant logic. It's just that English's logic is likely incomplete.

There are two ways of looking at this. One way is to say that what I have offered is a family of genuinely deviant logics that don't change the subject, and hence that Quine's argument against deviant logics fails. The other way—and it is what I prefer—is to say that what I have given is in an important sense a family of non-deviant, and even classical, logics, but one that differs from First Order Logic.

I think it could be a good thing to define the connectives in terms of valid inference (perhaps understood in terms of entailment). For instance, one might say that:

3. A partially-defined functor C that takes a pair of sentences p and q into a new sentence C(p,q) is a conjunction if and only if you can validly infer p as well as q from C(p,q) and C(p,q) from the pair of premises p and q whenever C(p,q) is defined.

(We also need an extension to wffs.) If we do this, then excluded middle is true by definition in the following sense:

4. Whenever p is a disjunction of q with a negation of q, then p is true.

But no claim is made that every sentence has a negation or that every pair of sentences has a disjunction. That would be a substantive claim. But whenever a sentence has a negation and can be disjoined with that negation, the result of the latter disjunction is true. That is a claim that is true by definition of "negation" and "disjunction".

This also lets one stipulate into place new connectives like tonk. Tonk is a connective such that one can infer q from "p tonk q" and "p tonk q" from p. The problem with tonk is that once one has the connective, it seems one derive anything (e.g., 1+1=2, so 1+1=2 tonk q, so q, for any q). But not quite. One can only derive everything with tonk if one adds the additional thesis that sufficiently many pairs of sentences have tonks. For instance, if we grammatically restrict tonking so that one is only allowed to tonk a sentence with itself, we can continue to have a sound logic.

Why care about such logics? Well, they might be helpful with the Liar Paradox. They might provide a way of doing the sort of thing that Field does to resolve the Liar by invoking a deviant logic but within a logic that has all the classical rules of inference.

I think Sorensen's "The Metaphysics of Words" [PDF] is very relevant to the above.