Saturday, March 08, 2008

Truth and consequence

Over at Greg's blog, there was a post on which is prior, logical truth or consequence. They are, in many cases, interdefinable. For example, we might want something like: A |= B iff |= A→B. Of course, this will depend on having the appropriate expressive resources in the language. The left to right direction fails for any logic without a conditional, defined or primitive. An example of this would be the conjunction-disjunction fragment of classical propositional logic. This looks a little artificial though. A more natural example is the logical system of Aristotle's Prior Analytics. It has no conditional locutions, definable or primitive. This depends on using the interpretation found in Smith's introduction to the Prior Analytics, which i believe derives form Corcoran's work, rather than the axiomatic interpretation given by Lukasiewicz. There might be another counterexample coming from connectives defined using Kleene's strong matrix because the conditional defined on that (at least, the standard one) results in no tautologies. For example, p→p gets the value one-half when p is assigned the value one-half. I'm not sure about this because I'm not sure how consequence is defined for that system.

Does the schema fail in the right to left direction? I don't know of one and I'm doubtful there are any.

Why would one think that direction would always hold? The conditional is an object language expression that is supposed to capture the consequence relation. Sometimes the consequence relation can outstrip it. Things would be amiss if the object language outstripped the metalanguage. Here's an idea. There are fewer restrictions on the stuff that appears to the left of the turnstile, e.g. there can be infinitely many things on the left, they can be gathered using sets instead of conjunctions. If the right to left direction is going to fail, we'd need more restrictions on the stuff appearing on the left-hand side of the turnstile than the right. When put like that, it doesn't seem obvious that the right to left direction cannot fail, but the context would have to be very unusual if there is one. As a parting thought, it isn't clear that we could place restrictions like that on the left-hand side of the turnstile and still have something recognizable as a consequence relation, i.e. something satisfying Tarski's conditions.


Greg said...

Thanks for the links, and thanks also for the example of the Aristotelian syllogistic.

I share your sentiment that it seems weird for the metalanguage to outstrip the object language -- but as soon as you put it like that, I realized that there is an (important?) example of such a situation: one of the orthodox logical empiricist accounts of scientific theories. What I'm thinking of is basically the idea that the metalanguage is something that is completely understood (this might be something like a 'pure observation language' but need not be), whereas the object language is some scientific language with all of its arcane terms (like the sign for the quantum-mechanical wavefunction).

I don't see how this is going to generate a counterexample to
A |= B only if |= (If A then B),
but perhaps I'm not thinking right. So I guess the moral is: if a counterexample could be produced, then the object-language-outstrips-metalanguage condition you suggest in the post would be necessary but not sufficient.

For me, the big and interesting question about the consequence-before-truth issue is "So what?" The fervor and passion displayed by Etchemendy and Read make me think it's very important, but I can't figure out what really turns on this.

p.s. -- What is the source of your hesitation about consequence in many-valued logics? If we define consequence as truth-preservation, then p will always follow from p. Am I missing something you're seeing?

Shawn said...

I'm going to respond to the ps first. I think that definition of consequence for multi-valued logics works. I think I was being overly hesitant since I couldn't check the definition when I wrote it and then forgot to revise it.

I take it the logical empiricist view of metalanguages was supposed to encompass the consequence relation too right? I agree with you that it isn't clear how that generates a counterexample to the right to left direction. (Isn't it supposed to be "if" instead of "only if"?)

As for the "so what?" question, I'm not entirely sure. The consequence approach is more fine-grained right? You can have logics with different consequence relations but the same set of logical truths. That would likely influence how one studies logic. As for something more directly philosophical, not really sure. Etchemendy doesn't give any indications? I don't have my copy handy.

Aaron said...
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