Really-Really-On-Topic Theories of Truth

7 11 2007

A T-equivalence for a sentence x is an instance of ‘S is true iff p’ in which ‘S’ is replaced with a quotational designator of x and ‘p’ with a sentence synonymous with x. A truth theory for L is a theory that has, for every sentence of L, a T-equivalence as a theorem. A theory of meaning for L is a finite theory knowledge of which could suffice for knowing the meaning of every sentence of L.

Theories of truth are not theories of meaning (this is widely acknowledged, though Davidson seems to go back and forth). Proof: Let L be a toy language comprising only the sentence ‘Snow is white’. Let T be a theory comprising the axiom ‘“Snow is white” is true iff snow is white’. T is a truth theory for L. But knowing that ‘Snow is white’ is true iff snow is white cannot suffice for knowing that ‘Snow is white’ means that snow is white, since a sentence can be true iff snow is white without meaning that snow is white. ‘Blood is red’ is such a sentence. In general, the information that a sentence S is true iff p is not sufficient to infer that S means that p, since every sentence that has the same truth value as ‘p’ will be true iff p, regardless of whether it means that p. Call this the Very Simple Coextensionality Problem (VSCP). A theory suffers from VSCP if the knowledge that it provides about a sentence S could be had about any sentence that is coextensional with S. No theory suffering from VSCP can be a theory of meaning.

Proper truth theories will not have axioms for whole sentences, but axioms for sub-sentential expressions. Such theories do not suffer from VSCP. Since they provide information about what, say, the predicate of an atomic sentence S is true of, they provide knowledge about S that one cannot have about every sentence that is coextensional with S. But these theories suffer from the Simple Coextensionality Problem (SCP): the knowledge they provide about a sentence S could be had about any sentence that is word-by-word coextensional with S. No theory suffering from SCP can be a theory of meaning.

How can we give a theory that avoids SCP? Let S and S* be two non-synonymous sentences that are word-by-word coextensional. Let S mean that p and S* mean that q. Everything that a truth theory T tells us about S also holds for S* (this is SCP). E.g., T will tell us – via the T-equivalence for S – that S is true iff p, while S* is also true iff p. But T will typically not tell us that S* is true iff p. Rather, it will tell us – via the T-equivalence for S* – that S* is true iff q. So maybe by going meta-theoretical we can avoid SCP. Let MT be a theory that tells us, for every sentence S of L meaning that p, that some truth theory for L has a theorem stating that S is true iff p. For MT to escape SCP there should be two word-by-word coextensional but non-synonymous sentences S (meaning that p) and S* (meaning that q) such that no truth theory has a theorem stating that S* is true iff p. Unfortunately, for any two such sentences, there are many such theories. I will try to show how to restrict the set of truth theories such that the meta-theory MT can avoid SCP.

(1) Let S and S* be two non-synonymous sentences that are word-by-word coextensional. Let S mean that p and S* mean that q. Let T be {‘S is true iff p’, ‘S* is true iff q’ ‘S* is true iff p’}. T is a truth theory for L. But T has ‘S* is true iff p’ as a theorem. Hence, MT does not avoid SCP. But T is a strange truth theory. It contains information that is strictly unnecessary for its being a truth theory. Let us restrict our attention to minimal truth theories, where T is minimal iff you cannot get a truth theory from T by removing some of its axioms.

(2) As before, but let T be {‘S is true iff p’, ‘S* is true iff (p&q)’, ‘p&q’}. T is a minimal truth theory for L. But T has ‘S* is true iff p’ as a theorem. Hence, MT does not avoid SCP. But T is a strange truth theory. It contains an axiom that provides no information whatsoever about L. Let us restrict our attention to on-topic truth theories, where T is on-topic iff every axiom of T is about L (either by referring to or by quantifying over expressions of L).

(3) As before, but let T be {‘S is true iff p’, ‘(S* is true iff (p&q)) & (p&q)’}. T is a minimal on-topic truth theory for L. But T has ‘S* is true iff p’ as a theorem. Hence, MT does not avoid SCP. But T is a strange truth theory. It contains an axiom one of whose conjuncts is not on-topic. Let us restrict our attention to really-on-topic truth theories, where T a really-on-topic iff the conjunction of axioms of T does not contain a conjunct that is not on-topic.

(4) As before, but let T be {‘S is true iff p’, ‘S* is true iff (p&q)’, ‘∀e(e≠e) v q’}. T is a minimal really-on-topic truth theory for L. But T has ‘S* is true iff p’ as a theorem. Hence, MT does not avoid SCP. But T is a strange truth theory. It contains an axiom which is logically equivalent to an axiom that is not on-topic. Hence, let us restrict our attention to really-really-on-topic truth theories, where T is really-really-on-topic iff the conjunction of axioms of T does not contain a conjunct that is logically equivalent to a sentence that is not on-topic.

So my guess is: A theory which provides for every sentence S (meaning that p) the information that there is a minimal really-really-on-topic truth theory for L which has a theorem that states that S is true iff p, avoids SCP. (Of course, due to some unmentioned Slightly More Complex Coextensionality Problems, it still doesn’t stand a chance as a theory of meaning.) Admittedly, a somewhat roundabout way of making a minor point which, considering the big picture, is quite insignificant. Still fun, though.

Posted by Miguel.

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5 responses

7 11 2007
Iggy

Interesting proposal. But the following might be a problem:

(i) A biconditional is, one might think, nothing but a conjunction of two conditionals. Let us, for the moment, work with this assumption. A theory of truth for some language which contains the sentence ‘grass is grass’ may legitimately contain the axiom

(G) “grass is grass” is true iff grass is grass,

which is a conjunction of two conditionals:

(G*) (“grass is grass” is true if grass is grass) & (if grass is grass then “grass is grass” is true).

The first of the two conjuncts is logically equivalent to ‘grass is grass’ (it both entails it and is entailed by it, if ‘grass is grass’ counts as a logical truth). So, a theory containing this axiom is not really-really-on-topic. But you suggested we should concentrate on really-really-on-topic theories; so, although a theory containing (G) seems perfectly fine, your suggestion amounts to excluding it. Why should we do that?

(ii) You might presumably reply that a biconditional is not a conjunction, but only equivalent to one.

(iii) But then take the following theory (which is a variant of the theories that you discussed above):

T := {‘S is true iff p’, ‘~( ~(S* is true iff (p&q)) v ~(∀e(e≠e) v q)’}.

T is a minimal really-really-on-topic truth theory for L. The two axioms are not conjunctions, so they do not have conjuncts which could be logically equivalent to a sentence that is not on-topic. Nor are the axioms themselves equivalent to such a sentence. Yet, T is equivalent to the following theory:

T* := {‘S is true iff p’, ‘(S* is true iff (p&q)) & (∀e(e≠e) v q)’}.

And this theory (and therefore T) has

‘S* is true iff p’

as a theorem.

(iv) So, unless I am mistaken in one of the points, it seems that your proposal is too restrictive if one counts biconditionals as conjunctions, and too permissive if one does not.

7 11 2007
Alex

Two questions:
(a) Why is the truth theory offerred under (2) minimal? Eliminate ‘p & q’ from the set and it seems you have another truth theory, i.e. {‘S is true iff p’, ‘S* is true iff (p&q)}. If ‘p’ is true it even happens to be a true truth theory, if not (and ‘q’ is true) it is false – but I can’t see why this should be excluded by what you said. (a minor point: eliminate ‘S* is true iff (p&q)’ as well and you have a truth theory again; it would be an incomplete truth theory for a language that contains both S and S*, but again, this doesn’t seem to be excluded by what you said).
(b) Why is the truth theory offered under (4) on topic? Is it because expressions of L are in the domain of the universal quantifier? I suppose your point could also be made by substituting this axiom by the following: ‘(S is true & S is not true) v q’ in which case it is easier to see why such a theory is on topic as you defined it (btw, given the truth functional equivalence of conjunctions to e.g. sentences in which only negation and disjunction occur the ‘on topic’ idea seems to be a quite obvious non-starter).

8 11 2007
Miguel

Hi Iggy, hi Alex, thanks for your replies!

I’ll take Alex’s questions first.

(a) I guess this is a misunderstanding. For all (1) through (4) the language L is supposed to contain both S and S*. (I should have made that clear.) Hence, you cannot remove “p&q” in the case of (2). There is no such thing as an incomplete truth theory if “truth theory” is introduced in the way that I introduced it.

(b) The quantifier in (4) is supposed to range over expressions of L (hence the “e”). Again, I should have made that clear. But of course, your example would work just as fine. As to the “non starter”-thing: first, your target seems to be the really-on-topic-idea, not the on-topic thing, right? As for it being an “obvious non starter”: well, here standards may vary. But mine might be too low, I’ll give you that.

Iggy’s question:

Nice question, but in the end, I think, not a big problem. I was afraid that the oversimplifications I made might make it hard to deal with logical truths and your post shows that this is indeed the case. So here is what I would suggest:

(i) Committing oneself to the view that biconditionals are not conjunctions is too adhoc.

(ii) But then, as you said, a theory containing (G) as an axiom is excluded by what I say.

(iii) My response: that doesn’t matter (or at least it doesn’t matter a lot). By presenting the theories the way I did I may have encouraged taking theories seriously that have axioms for whole sentences. But of course, they shouldn’t be taken (too) seriously. Let me change your example from “grass is grass” to “Peter = Peter”. A theory for a language containing this sentence will have an axiom for “Peter” and one for “=”. While the T-equivalence for “Peter = Peter” is – and must be – equivalent to a conjunction that contains a conjunct that is equivalent to a sentence that is not on topic, this T-equivalence is not an axiom. On the other hand, none of the two axioms has to be equivalent to a conjunction that contains a conjunct that is equivalent to a sentence that is not on topic (of course they might, but in that case they are rightly excluded by what I say).
Note that your example exploits a feature of the oversimplified theories that my examples do not exploit. In the cases that I mentioned, we can take ‘axioms’ like “S is true iff p” as a mere shorthand for those axioms that are used to derive “S is true iff p”. In your case, it is essential that the whole-sentence- ‘axiom’ is really an axiom.

Is that convincing?

11 11 2007
Iggy

Hi Miguel.

So, you want to focus on meaning theories that tackle the meaning of the subsentential parts of the sentences of the object-language.
Fair enough. However, even if your proposal works for such theories, one may think the fact that it doesn’t work for theories that have axioms for whole sentences shows that it does not go to the heart of the problem, but may at best yield an extensionally correct construct. But perhaps this is all you are looking for anyway.

What might be helpful (for me) is to see how SCP looks like within theories that do not use axioms for sentences but only for subsentential expressions, and also how your proposal then avoids the problem.

12 11 2007
Miguel

Hi Iggy!

I am not sure I understand your first sentence. And I am also not sure I understand what you mean by “the heart of the problem”. I am sure, though, that much of what I said was confused and inadequate. So let me, very briefly, restate (and somewhat extend) the main ‘point’ (small as it may be) of my post. I hope this will help.

What is the Simple Coextensionality Problem (SCP)? A theory suffers from SCP if the (non-syntactical) information it provides is insufficient to distinguish between word-by-word coextensional sentences. If we use “ITS(x)” to ascribe to x the totality of features that a sentence S has according to a theory T, we can state it thus (the quantifier ranges over sentences of L):

A theory T for a language L has SCP ↔
∀S, S* (S is word-by-word coextensional with S* → ITS(S*)).

All truth theories suffer from SCP. Can SCP be avoided by considering a theory about truth theories, namely one which tells us, for every S of L meaning that p, that there is a truth theory that has a theorem that says that S is true iff p? Bear in mind: I am not asking whether such a theory would be a theory of meaning. In my view, it is pretty clear that it wouldn’t. I am simply asking whether such a theory would fail simply because it cannot avoid SCP, or whether there is a way of avoiding SCP (while still failing for other reasons). Avoiding SCP is necessary, though not sufficient, for being a theory of meaning.

Let us concentrate on finite truth theories for infinite languages. Let MT be a theory that tells us, for every S of (an infinite language) L meaning that p, that there is a (finite) truth theory that has a theorem that says that S is true iff p. Does MT avoid SCP? Without further constraints, the answer would have to be “No”. Proof: Let “Fa” and “Gb” be two atomic non-synonymous word-by-word coextensional sentences, the first meaning that a is F, the second that b is G, and let T be a truth theory for L containing an axiom A1 = “∀t ( ‘F’^t is true iff Val(t) is F)” which is necessary to derive “S is true iff p”. (The particular form of the axiom does not matter much; the case could be made for other ways of treating, say, predicates.) Now let T* be the theory that results from T by replacing A1 with “∀t ( ‘F’^t is true iff (Val(t) is F and b is G))” and by adding “a is F & b is G” as a further axiom. T* is a truth theory for L, but T* has “‘Fa’ is true iff b is G” as a theorem. Generalizing this, we can see that in its simple form, MT will not avoid SCP. So my question was: which F guarantees that a theory MT which tells us, for every S of L meaning that p, that there is an F-truth theory that has a theorem that says that S is true iff p, avoids SCP? In the case sketched, the on-topic constrain helps, since T* is not on-topic. But this constraint is not enough, since we can simply form an on-topic axiom from the non-on-topic axiom and an on-topic one using ‘&’. The constraint that I proposed in the post is both inadequate and confused. It would have been somewhat less confused (but still inadequate) had I given the following constraint: no axiom of T is equivalent to a conjunction that contains a conjunct that is equivalent to a sentence that is not on-topic. The inadequacy can be seen from the following example: Let T* be the theory that results from T above by replacing A1 with “∀t (‘F’^t is true iff (Val(t) is F and b is G))” and by adding “(a is F & b is G) or not-NLT” where NLT is an on-topic theorem of T that is not a logical truth. Hence, T* is a truth theory for L, and T* has the unwanted “‘Fa’ is true iff b is G” as a theorem, but the trouble-causing axiom “(a is F & b is G) or not-NLT” is not excluded by the constraint.

So here is my current (and last?) guess: the proper constraint is that every theorem of T has to be on-topic or a logical truth. Let us call such theories properly-on-topic. The intuitive motivation is that a truth theory should provide only information about the language that it is a truth theory for (plus information stated by logical truths, since no theory can avoid these anyway). The intuitive idea why this should work is the following: to cause trouble for MT we need a way of providing for arbitrary non-synonymous word-by-word coextensional sentences S and S* (meaning that p and that q respectively) a truth theory that has a theorem which states that S is true iff q. But this can only be achieved by including in the theory information to the effect that p iff q. Every such theory will be excluded by the above constraint, since every such theory will have a theorem to the effect that p iff q that is neither on-topic nor logically true. Hence, a theory MT which tells us, for every S of L meaning that p, that there is a properly-on-topic truth theory that has a theorem that says that S is true iff p, avoids SCP.

Let me emphasize (again) that this is of course not sufficient for MT to be theory of meaning. All I was trying to do was to get clear about the dialectical situation: which kinds of theories fail for which kinds of reasons. It seems to me that metatheoretical theories as sketched above and properly constrained, do not fail because of SCP. They still fail, though. (And now we may go on to consider canonical proof procedures and what not.)

Thanks again!




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