Wolfgang Schwarz :: Are all truths entailed by logical truths?

Are all truths entailed by logical truths?

Posted on Saturday, 02 Dec 2006.

Sorry, the server has been down quite a lot recently. Hope it's back to normal now.

Here's the talk I gave at Kioloa. It's partly identical to the talk I gave at GAP.6 in Berlin, but with more speculative ideas towards the end and less missionary appeals in between.

Are all truths entailed by logical truths? Depends on what we mean by "all truths" and "entailed" and "logical".

Let's understand a truth to be a true sentence of English, possibly enriched by logical vocabulary. As for entailment, let's distinguish metaphysical entailment (necessarily, if P then Q) from analytical (or conceptual or a priori) entailment. The precise definition of these notions, and the differences between them, won't matter.

And let's distinguish two kinds of logical truths: truths that are in some sense true by virtue of their logical form (F-logical truths, for short), and truths that only contain logical vocabulary (V-logical truths). So, "if it rains, it rains" is F-logically true, but not V-logically true. On the other hand, " $$m[1]$ ' is V-logically true, but not F-logically true. Again, the precise definition of the two notions won't matter, except that I will be fairly tolerant with what counts as logical vocabulary.

Now we have four questions.

1. Are all truths metaphysically entailed by F-logical truths? -- Yes.

Proof: Consider the sentence 1) $$m[1]$ . "ACT" is the "actually" operator. Its main effect is to break the scope of modal operators: whenever S is true, it is possibly actually true, and whenever S is possibly actually true, then S is true: $$m[1]$ . Outside the scope of modal operators, "ACT" does nothing. So (1) is true. (It says, roughly, that something x has F if and only if x actually has F.) Moreover, its truth is guaranteed by the meaning of its logical terms, so it is an F-logical truth.

And (1) metaphysically entails everything: whenever (1) is true at some world w, every actual truth is also true at w. Suppose for reductio that (1) is true at some world w where some other sentence, say Pa, is true that isn't true at the actual world. So at w, both $$m[1]$ and $$m[1]$ are true. It follows that at w, $$m[1]$ is true as well. So $$m[1]$ . But then $$m[1]$ must be true, by the logic of "ACT". Contradiction.

2. Are all truths metaphysically entailed by V-logical truths? -- Yes.

Proof: (1) is also V-logically true. And it still metaphysically entails everything.

Sidenote: So all truths supervene on logical truths. (Or at any rate, if you want to be more strict on what counts as "logical", on analytic truths.) This shows, I believe, that "metaphysical entailment" is a kind of misnomer: metaphysics shouldn't be characterized as a search for truths upon which everything supervenes (as e.g. Frank Jackson does in From Metaphysics to Ethics). If that was the project, (1) would qualify as an interesting metaphysical truth. But in fact, (1) is entirely trivial and uninteresting.

3. Are all truths analytically entailed by F-logical truths? -- No.

Proof: Well, I don't really have a proof of this. It's just obvious. Whatever analytically (or a priori) follows from an analytic (or a priori) truth is itself an analytic (or a priori) truth. But not all truths are analytic truths. (Counterexample: "there are 18 species of penguins.")

Sidenote: So all truths follow metaphysically, but not analytically from (1). This shows that there must be something wrong with Jackson's argument in ch.3 of From Metaphysics to Ethics, which purports to show that everyone who is committed to a general thesis of metaphysical entailment is thereby also committed to a thesis of analytic entailment. (I've discussed this a long time ago here on the blog.)

4. Are all truths analytically entailed by V-logical truths? -- Perhaps.

This is the really interesting one. There are several routes that might lead to this conclusion. Here is one.

Suppose David Lewis is right about everything. Then in particular, all truths are metaphysically entailed by the distribution of fundamental, intrinsic properties in our world.

Moreover, there are reasons to believe that this entailment is not only metaphysical, but analytic, or a priori:

"all of us are committed to the a priori deducibility of the manifest way things are from the fundamental way things are." (Lewis in "Tharp's Third Theorem")

Finally, if Lewis is right, then we do not have access to the intrinsic natures of fundamental properties. Thus none of our words analytically track a property's intrinsic nature. "Charge -1" may rigidly pick out the property P that plays the charge role in our world, but it isn't analytic that something is charge -1 iff it is P. Rather, it is analytic that something is charge -1 iff it plays the charge role -- no matter its intrinsic nature.

Now consider the Humean worldbook that gives us the instantiation pattern of all fundamental properties:

$$m[1]$ .

The non-logical predicates in there do no real work: it wouldn't make a difference for us if we replaced, say, $$m[1]$ and $$m[1]$ throughout. So arguably, the Humean worldbook is analytically equivalent to its Ramsey sentence

$$m[1]$ ,

where the second-order quantifiers range over fundamental properties, rather than arbitrary classes.

So whatever the original worldbook entails (viz. everything) is also entailed by the Ramsey sentence. But the Ramsey sentence is a V-logical truth.

Comments

# Benjamin Schnieder on 04 December 2006, 13:04

Nice. However, some questions immediately come to mind (perhaps, if we spent some more time, the fairly obvious answers would also pop up, but right now, they do not ...):

(i) You assume - without argument - that

Ax AF(Fx <-> ACT Fx).

is true in virtue of its form. We do not quite see why. In particular, the relevant contrast between this sentence and the existential sentence

Ex Ey(~ x = y)

deserves to be clarified; what notion of form makes the former true in virtue of form but not the latter? You say about the former that its truth is guaranteed by the meaning of its logical terms, and that therefore it is an F-logical truth. On the one hand, it seems to us that in some sense the truth of the second sentence is in fact guaranteed by the meaning of its logical terms. On the other hand, the truth of "I am here now" is in some sense guaranteed by the meaning of its parts - do you think the cases are parallel?

(ii) On

Ax AF(Fx <-> ACT Fx).

again: The sentence does not express a necessary truth. But the majority (we guess) of people working on logical truth would say that being necessary is a necessary condition of being logical - which also seems reasonable to us.

(iii) Is your argument not rather an argument to the effect that "actually" should not be considered as a logical constant?

(iv) Given your notion of V-logical truth, your question 2 seems to be interesting only if there is a good notion of what it is to be a logical constant - if everything can be counted as logical constant, 2 is trivial.

As we said, these were the questions that came to our minds immediately. What are the obvious answers, then?

Benjamin & Miguel.

# wo on 05 December 2006, 09:27

Hey Benjamin and Miguel, nice to hear from you!

I agree that the distinction between logical and non-logical truths in either sense isn't clearcut. One reason to think "Ax AF(Fx <-> ACT Fx)" is true in virtue of its form is that it is true in every model of the relevant (2D modal) logic. Another reason is that arguably anyone who claims that it is false misunderstands what it says. In this sense, its truth is guaranteed by the meaning of its constituents; no matter what the world is like, given that the terms mean what they do, the sentence is true. All this is quite vague, I know. But it seems to me that none of that applies to the claim that there are two objects, "Ex Ey (x != y)": it is false in many models; one can reasonably deny it without misunderstanding; and if the world turned out to contain only one object, it would actually be false.

You're right that "I am here now" is probably in the same boat as "Ax AF(Fx <-> ACT Fx)", and I wouldn't mind counting it as F-logically (and V-logically) true.

I think "ACT" has the same claim for logicality as the alethic box: it is after all just a formal device for breaking the scope of the box. I don't have a good definition of logicality, but it seems to me that there is an interesting distinction between terms like "not", "necessarily" and "actually" on the one hand, and "violin", "orange" and "Hamburg" on the other.

But if you insist that "actually" is non-logical and that all logical truths must be necessary, maybe I could just as well drop the use of "logical", and instead put my claims as follows:

1 & 2. All truths are metaphysically entailed by analytic truths containing only vocabulary from modal predicate logic plus second order quantifiers plus "actually".

4. Perhaps all truths are analytically entailed by sentences containing only vocabulary from modal predicate logic plus second order quantifiers plus "here" and "I".

# Miguel on 05 December 2006, 14:09

Just some quick thoughts on what you write here:

"none of that applies to the claim that there are two objects, "Ex Ey (x != y)": it is false in many models; one can reasonably deny it without misunderstanding; and if the world turned out to contain only one object, it would actually be false"

(i) It seems plausible that "ExEy (~ x=y)" should be in the same boat as "Ex(x=x)". After all, given that its *not* a logical truth that there are two objects, why should it be a logical truth that there is one object? And if it *is* a logical truth that there is one object, what should keep us from regarding it to be logical truth that there are two (three, four...) objects? The existence of two (three, four...) objects is as much apriori and necessary as the existence of one; thus these notions cannot be used to justify treating the two claims differently. [The existence of one particulary *logical* object could, but I do not see that forthcoming...]

(ii) But if this is correct, one cannot argue that a sentence is a logical truth purely on the grounds that it is true in all models. In standard model-theory, "Ex (x=x)" is, while "ExEy (~ x=y)" is not.

(iii) You say that one can understand "ExEy (~ x=y)" and still deny it. I guess this is right (though I tend to think that your "reasonably" goes a little too far...). But I am also inclined to think that this holds for many sentences that one would like to see classified as logical truths...

(iv) It is necessary that there are infinitely many numbers. Thus the world could not turn out to contain only one object. If thats correct, we cannot exclude "ExEy (~ x=y)" from the logical truths by pointing out that it could have been false.

(v) If I understand you correctly, you say that a logical truth should be such that "its truth is guaranteed by the meaning of its constituents" - and that does seem to be a reasonable constraint to me. But you go on to spell this out by saying that "no matter what the world is like, given that the terms mean what they do, the sentence is true" - and *this* seems to hold for "ExEy (~ x=y)" too (see iv). I think that in order to get at the idea expressed by your "guarantee of truth by meaning"-constraint we need something stronger than modal notions.

Hope you are having fun down-under,
Miguel.

# wo on 06 December 2006, 01:39

Well, one can draw many lines between logical truths and non-logical truths, and I don't think it's worth arguing about which one is the *real* dividing line.

"Ex(x=x)" is a good candidate for landing on either side, depending on how exactly the line is drawn. After all, many people (e.g. Hodges) have tried to come up with alternative model theories for predicate logic precisely because they intuit that this shouldn't come out as a logical truth.

Likewise for the existence of numbers. I use "it could have turned out" as roughly equivalent to "it is not a priori that" or "it is not analytic that". (So it could have turned out that water is XYZ.) Now, could it have turned out that there aren't infinitely many numbers? On the one hand, it seems that it really could not. On the other hand, this has seemed almost incomprehensible to many philosophers (e.g. Boolos): how could it be analytic that there are infinitely many things?

You're right that in order to make precise the "guarantee of truth by meaning" constraint we probably need something stronger than modal notions. And unlike with "logical" truths, I think it is really worthwhile to get clear about this class of truths (the "analytic" truths). But at least for the purpose of this post, all I need is some rough and intuitive distinction, not a precise definition. I hope.

Wolfgang Schwarz