## 2/22/2008

### Which came first: logical truth or consequence?

This term, I am teaching a philosophy of logic class. We've twice run across the following sentiment:

(CPT) Logical consequence is prior to logical truth.

This sentiment is also expressed as 'The real subject matter of logic is the notion of consequence, not a special body of truths.' (References: We've seen this in Stephen Read's Thinking about Logic Ch.2, and in John Etchemendy's work (1988, p.74) too.)

Seeing (CPT) surprised me, since logical truth is (in most cases -- see below) definable in terms of logical consequence, and vice versa: If C is a consequence of P1 ... Pn, then 'If P1 and ... and Pn, then C' is a logical truth. And if T is a logical truth, then T is a consequence of the null set of premises. This is well-known: Beall and Restall, in their recent Logical Pluralism, make exactly this point.

So, in light of the interdefinability of logical truth and consequence, what would prompt someone to say consequence is somehow prior to logical truth? Stephen Read appeals to valid arguments that ineliminably use infinitely many premisses: A(0), A(1), ... Therefore, ∀x Ax. We can't turn this into a logical truth ('If A(0) and A(1) and ..., then ∀x Ax') in standard languages, because standard languages don't allow for infinitely long sentences. This seems like a fair point in favor of (CPT), but it does assume that (i) you accept arguments with infinitely many premises, and (ii) reject languages with infinitely long expressions. [Edit: as Shawn correctly notes in the comments, these two assumptions are fairly widely held. But I have always been a bit suspicious (perhaps for no good reason) about the idea of an argument with infinitely many premises.]

Here's another argument for (CPT), from extremely weak languages. Imagine we have a propositional language with sentence letters p, q, ..., and only two sentential connectives: 'and' and 'or' specified in the usual way. In this language, there are no logical truths (because we don't have 'If... then...' or anything equivalent), but there are still logical consequences: A is still a logical consequence of 'A and B', and 'A or B' is a logical consequence of A. So here is a case where we have logical consequence without logical truth.

But both of these arguments (Read's and mine) rely on somewhat unusual cases. Are there other reasons to accept (CPT) that do not appeal to unusual circumstances? Is there a big literature out there that I don't know about? And does anything really hinge upon whether we think logical truth is prior to consequence, vice versa, or neither?

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At 22/2/08,  Shawn said...

I don't know if there is a large literature on this subject. I'm curious to know if there is. Dummett also says some things on this topic. I believe he discusses it in his Logical Basis of Metaphysics.

One consideration that, I believe, Dummett points to is that you can get systems that have the same set of logical truths but different consequence relations. Consequence is then a finer grained notion than logical truth. I'm not sure if there are instances of the converse. I have a feeling that there aren't, although I've got no proof of this.

Read's point seems pretty good. The assumptions seem fairly reasonable since they are both made in the use of standard first-order logic. Allowing infinitely many premises doesn't matter all that much since FOL is compact, but it does allow that many premises. Things get wonky if you allow infinitary connectives, but that doesn't seem like a strong reason for not allowing them. It certainly seems like a reason for not using them, although it isn't clear if that constitutes a rejection of them.

I'm very curious what ends up turning on this question of priority. It seems like the view one takes to logic will shift. There is something that, to me, seems a bit stifling of viewing logic as being concerned with a body of truths as its subject matter.

One possibility is whether a logic is axiomatizable. There are systems whose logical truths are axiomatizable but whose consequence relations aren't. FOL is like this I think. At least, PA is like that. I think I read that relevance logic is like that, but I don't have any citations handy. I'm not sure what other considerations tell in favor of one or the other side though.

At 24/2/08,  Marcus Rossberg said...

Whether this interdefinability holds depends, more generally, on whether the deduction theorem holds for the logic in question, as well as whether the consequence relation is defined proof-theoretically or model-theoretically. I have the suspicion that some prejudice for classical first-order logic might be responsible for the thought that it does not matter where one starts. In classical FOL, owing to the deduction theorem and completeness, it does not matter which is taken as primary, from the technical point of view.

Nevertheless, taking logical consequence to be prior seems to get things right for various non-technical reasons. Etchemendy's motives and the reasons spelled out by Shawn above are fairly heavy weight. And, in addition, it is worth thinking about that logic is supposed to codify correct reasoning. It seems *conceptually* right to take logical consequence as primary, for that reason. The class of logical truths (at least if you stay at first order), on the other hand, is a pretty uninteresting field of study.

At 25/2/08,  Robbie said...

I agree with Marcus that the "good reasoning" connection makes consequence prior in one good sense (unless, as Harman et al argue, there's no such connection).

Of course, it's not obvious that's the only sense of "priority" you might be interested in. You might think that the *point* of logic was to get something that codified inferences, i.e. to reach a definition of validity, but that logical truth was somehow less philosophically problematic, so that defining consequence in terms of truth was a philosophical advance.

Of course, if the deduction theorem fails consequence is pretty clearly the thing to go for. It arguably fails in a supervaluational logic, for example. (standard example: p|-Dp, but not|-p>Dp).

At 26/2/08,  Greg said...

Hi all --

It does seem to me that the failure of the deduction thm for 'serious' logics is a pretty good reason for considering consequence somehow prior to truth. Unless (and I guess this is basically one of Shawn's points) the converse direction of the deduction thm fails for some logics; and I don't know of any cases in which that happens (but as this post makes manifest, I'm pretty clueless in this area).

One last question: is there is any characteristic(s) shared by all or most logics for which the deduction theorem fails -- that might account for the failure? Or is it just case-by-case?

Thanks again for the help,
Greg

At 28/2/08,  Shawn said...

The deduction theorem fails for relevance logic. Well, it fails in a sense. You don't get the deduction theorem for premises combined with conjunction, e.g. A&B, but you do get it for premises which are fused, AoB, where 'o' is the fusion operator. It is like a conjunction, but intensional. The main difference, I believe, is that it isn't commutative. You lose the deduction theorem when you lose certain structural rules, such as connecting commutation of premise combination with conditionals, or, I believe, weakening/monotonicity. I said some vague things about fusion in some recent posts.

This is an excellent topic though.

I think Marcus has nailed it with his suspicion that classical FOL biases are driving the thought that you can take either truth or consequence as primitive.

At 29/2/08,  Greg said...

A couple of points:

First: I just realized that I think the deduction theorem is NOT what's key here; I wasn't keeping the single and double turnstiles separate in my head. The deduction thm says
(for any A, B) if A |- B, then |- 'A → B'.
But I'm asking about logical TRUTH and (semantic) CONSEQUENCE, not about proofs and theorems. (Duh.) So what I am really looking for is cases where the following fails:
(for any A, B) if A |= B, then |= 'A → B'
And that's what Read's ω-rule case is a counterexample to. (Also, I'm interested in any case where the converse fails, but like Shawn I'm skeptical that there are any such cases in 'real' logics -- or even just in any logic where '→' is supposed to model the/ a conditional.)

Second point: Why point to FOL bias as the motivating factor? The people who get tagged by Read, Etchemendy et al. as the 'logical-truth-prior-to-consequence' camp are Frege and Russell -- and they certainly don't suffer from FOL bias. That's just a historical observation; but conceptually I don't see it either:
(for any A, B) if A |= B, then |= 'A → B'
does not seem the peculiar provenance of classical FOL.

At 3/3/08,  Shawn said...

Ah, right. Good historical point. I keep (mistakenly) thinking of Frege as a classical FOL proponent. I was also thinking of Quine as an FOL proponent as well as a truth is prior to consequence person. This may not be right as I haven't read Quine on this for a while. It is possible that I'm simply making a mistake.

At 5/3/08,  Shawn said...

On a completely different note, what do you think of Read's book as an introduction to the philosophy of logic? Would you recommend it to grad students?

At 5/3/08,  Greg said...

Quine is, I'm almost certain, a Truth Before Consequence person (at least in rhetoric if not substance). I suspect that may have something to do with his [middle period] desire to have logical claims on the same epistemological footing as empirical ones -- the similarity is easier to see/imagine [depending on your perspective] if logic is not about this thing Entailment relating sentences to sentences, but rather just a set of sentences, like any other scientific theory.

And re: Read's book, I'm not an expert on this matter, but I can report my own experience coming to Read. There was a bunch of stuff in Read that I had picked up/ heard about in one way or another either in logic and/or language-related classes, but I still got a lot out of the book, because (i) it helped me connect the dots somewhat, and (ii) it filled in the holes left in the stuff I picked up haphazardly. But to give you a sense of the level of the text, my undergraduates (almost all philosophy majors) usually basically understand most of the material.

A couple more points: It is occasionally idiosyncratic (Read has peculiar views on what's wrong with Disjunctive Syllogism); it reflects a point of view instead of pretending to neutrality.

It will NOT really get you up to the state of the art in any of the fields covered (not even the state of the art in 1994, when it was published). But showing you the cutting edge is not what textbooks are really meant to do.

I hope this helps.

At 6/3/08,  Greg said...

I just prepared the section on Free Logic for my class. It was basically fine until Read came to his criticism of van Fraassen's supervaluational technique. I'm not sure what combination of misunderstanding and/or misrepresentation led Read to say that supervaluations force you to accept that Universal Instantiation ([For all x, Fx] --> Fa) and Existential Generalization (Fa --> [There exists an x s.t. Fx]) are both valid. This is just wrong. If the name a is non-denoting, then there are cases where 'For all x, Fx' is (super)true, but 'Fa' not true (specifically, truth-valueless). And there can be cases where EG fails too: 'Pegasus=Pegasus' is supertrue, but 'There exists an x s.t. x=Pegasus' will come out false.

So that's one part of Thinking about Logic that I think is misleading.

At 16/3/08,  Anonymous said...

At 20/3/08,  Shawn said...

Today one reason for claiming consequence is prior to logical truth occurred to me. If you are an inferentialist, about logic or about meaning generally, you might want consequence to be prior to truth. I expect that you'd want to define consequence based on primitive inferential rules rather than get inference out of consequence. Whether or not you go ahead and define logical truth from consequence will depend on the status of the notion of truth in your view. Granted, this wouldn't apply to many philosophers, but it is a possibility. I'm much less sure why someone who is not an inferentialist would care about which is prior. I expect that Dummett says something about this in Frege: Philosophy of Language cause there is a section where he criticizes Frege for the"retrograde" move from seeing logic as concerned primarily with consequence to seeing it as primarily concerned with truth.

At 28/3/08,  Robbie said...

I guess one thing that's weird about trying to think about these issues in non-classical settings is that what counts as a "conditional" will often be up for grabs. E.g. in LP, (A&~A)>B will be a logical truth, where ">" is the material conditional (P>Q=~PvQ). But of course B won't follow from A&~A. So there's a candidate for a logically true conditional that doesn't correspond to consequence.

But of course modus ponens fails for this connective in LP. It's a "real" logic, but it's at least questionable whether it's a "real" conditional.

So it looks to me before we can even ask questions we got to sort out what conditions something has to meet to be a real conditional. If modus ponens is a key factor, then if A>B is a logical truth, then very roughly, it looks we can start from A, cite the logical truth A>B, and then by modus ponens get to B, and thus get A|=B. (I guess this argument relies on completeness to allow us to "cite" the logical truth A>B in a proof, and then soundness to convert the proof A|-B into A|=B).

Just on the Read interpretation issue. I don't have his book to hand, but might he be referring to the restricted result that for atomic A, A(c) |= Ex(x=c) (cf. p.492 of van Fraassen's paper)? That does hold supervaluationistically, and is kind of interesting, since the corresponding conditional obviously isn't logical true---so a failure of the connection we were originally thinking about. Greg's example illustrates that you won't get it in full generality.

At 28/3/08,  Robbie said...

Sorry, that last paragraph should read (with typos removed):

Just on the Read interpretation issue. I don't have his book to hand, but might he be referring to the restricted result that for atomic A, A(c) |= Ex(Ax) (cf. p.492 of van Fraassen's paper)? That does hold supervaluationistically, and is kind of interesting, since the corresponding conditional obviously isn't logically true---so is a failure of the connection we were originally thinking about.

Greg's example illustrates that you won't get existential generalization in full, but you do get this.