## Truth as an operator and as a predicate

November 5, 2009

Suppose we add to the propositional calculus a new unary operator, T, whose truth table is just the trivial one that leaves the truth value of its operand untouched. By adding

• $(Tp \leftrightarrow p)$

to a standard axiomatization of the propositional calculus we completely fix the meaning of T. Moreover this is a consistent classical account of truth that gives us a kind of unrestricted “T-schema” for the truth operator.

On the face of it, then, it seems that if we treat truth as an operator operating on sentences rather than a predicate applying to names of sentences we somehow avoid the semantic paradoxes. But this seems almost like magic: both ways of talking about truth supposed to be expressing the same property – how could a grammatical difference in their formulation be the true source of the paradox?

My gut feeling is that there isn’t anything particularly deep about the consistency of the operator theory of truth: it just boils down to an accidental grammatical fact about the kinds of languages we usually speak. The grammatical fact is this. One can have syntactically simple expressions of type e but not of type t. Without the type theory jargon this just means we can have names that can be the argument of a predicate but not “names” that can be the argument of an operator. Call these latter kind of expressions “name*s”. If $p$ is a name* then $\neg p$ is grammatically well formed and is evaluated as the same as $\neg \phi$ where $\phi$ is whatever sentence p refers* to. If pick $p$ so that it refers* to “$\neg p$” then we are in just the same predicament we were in the case where we were considering names and treating truth like a predicate. One could simply pick a constant and stipulate that it refers to the sentence “~Tr(c)”.

We could make this a little more precise. By restricting our attention to languages without name*s we’re remaining silent about propositions that we could have expressed if we removed the restriction. Indeed, there is a natural translation between operator talk (in the propositional language with truth described at the beginning) and predicate talk. So, on the looks of it, it seems we could make exactly the same move in the predicate case: accept only sentences that are translations of sentences we accept. The natural translation I’m referring to is this:

• $p^* \mapsto p$
• $(\phi \wedge \psi)^* \mapsto (\phi^*\wedge\psi^*)$
• $(\neg \phi)^* \mapsto \neg \phi^*$
• $(T\phi)^* \mapsto Tr(\ulcorner\phi^*\urcorner)$

Here’s a neat little fact which is quite easy to prove. Let M be a model of the propositional calculus (a truth value assignment.)

Theorem. $\phi$ is the translation a true formula in M if and only if $\phi$ appears in Kripke’s minimal fixedpoint construction using the weak Kleene valuation with ground model M.

Note that, because we don’t have quantifiers, the construction tapers out at $\omega$ so we can prove the right-left direction by induction over the finite initial stages of the construction. Left-right is an induction over formula complexity.

If the rule is to simply reject all sentences which aren’t translations of an operator sentence then it appears that the neat classical operator view is really just the well known non-classical view based on the weak Kleene valuation scheme. It is well known that the latter only appears to be classical when we restrict attention to grounded formulae; it seems the appearance is just as shallow for the former view.

Incidentally, note that there’s no natural way to extend this result to languages with quantifiers. This is because there’s no “natural” translation between the propositional calculus with propositional quantifiers and a quantified language with the truth predicate capable of talking about its own syntax.

April 23, 2009

I’ve been casually reading Field’s “Saving Truth from Paradox” for some time now. I think it’s a fantastic book, and I highly recommend it to anyone interested in the philosophy of logic, truth or vagueness.

I’ve just read Ch. 21 where he discusses a paradox presented in Restall 2006. The discussion was very enlightening for me, since I had often thought this paradox to be fatal to non-classical solutions to the liar. But although Fields discussion convinced me Restall’s argument wasn’t as watertight as I thought it was, I was still left a bit uneasy. (I think there is something wrong with Restall’s argument that Field doesn’t consider, but I’ll come to that.)

Before I continue, I should state the paradox. The problem is that if one has a strong negation in the language, $\neg$, one can generate a paradoxical liar sentence which says of itself that it’s strongly not true. Strong negation has the following properties which ensures that that last sentence is inconsistent:

1. $p, \neg p \models \bot$
2. If $\Gamma , p \models \bot$ then $\Gamma \models \neg p$

Roughly, the strong negation of p is the weakest proposition inconsistent with p – the first condition guarantees that it’s inconsistent with p, the second that it’s the weakest such proposition. It’s not too hard to see why having such a connective will cause havoc.

Restall’s insight (which was originally made to motivate a “strong” conditional, but it amounts to the same thing) was that one can get such a proposition by brute force: the weakest proposition inconsistent with p is equivalent to the disjunction of all propositions inconsistent with p. Thus, introducing infinitary disjunction into the language, we may just “define” $\neg p$ to be $\bigvee \{q \mid p \wedge q \models \bot \}$. Each disjunct is inconsistent with p so the whole disjunction must be inconsistent with p, giving us the first condition. If q is inconsistent with p, then q is one of the disjuncts in $\neg p$ so q entails $\neg p$, giving us (more or less) the second condition.

An initial problem Field points out is that this definition is horribly impredicative – $\neg p$ is inconsistent with p, so $\neg p$ must be one of it’s own disjuncts. Field complains that such non-well founded sentences give rise to paradoxes even without the truth predicate, for example, the sentence that is it’s own negation. (I personally don’t find these kinds of languages too bad, but maybe that’s best left for another post.) This problem is overcome since you can run a variant of the argument by only disjoining atomic formulae so long as you have a truth predicate.

The second point, Field’s supposed rebuttal of the argument, is that to specify a disjunction by a condition, F say, on the disjuncts, you must first show F isn’t vague or indeterminate, or else you’ll end up with sentences such that it is vague/indeterminate what their components are. Allowing such sentences means they can enter into vague/indeterminate relations of validity – for example, it is vague whether a sentence such that it is vague whether it has “snow is white” as a conjunct entails “snow is white”. But the property F, in this case, is the property of entailing a contradiction if conjoined with p. Thus to assess whether F is vague/indeterminate or not, we must ask if entailment can ever be vague. But to do this we must determine whether there are sentences in the language such that it is indeterminate what their components are. Since the language contains the disjunction of the F’s, this requires us to determine whether F is vague – so we have gone in a circle.

Clearly something weird is going on. That said, I don’t quite see how this observation refutes the argument. It’s perfectly consistent with what’s been said above that entailment for the expanded language with infinitary disjunction is precise, that there is a precise disjunction of the things inconsistent with p, and that Restall’s argument goes through unproblematically. It’s also consistent that there *are* vague cases of entailment – but that the two conditions for strong negation above determinately obtain (there are some subtle issues that must be decided here, e.g., is “p and q” determinately distinct from the sentence that has p as its first conjunct, but only has q as its second conjunct indeterminately.)

Even so, I think there are a couple of problems with Restall’s argument. The first is a minor problem. To define the relevant disjunction, we must talk about the property of “entailing a contradiction if conjoined with p”. But to do this we are treating “entails” like it was a connective in the language. However, one of Fields crucial insights is that “A entails B” is not an assertion of some kind of implication holding between A and B, but rather the conditional assertion of A on B. “entails” cannot be thought of like a connective. For one thing, connectives are embeddable, whereas it doesn’t make much sense to talk of embedded conditional assertions. Secondly, a point which I don’t think Field makes explicit, is that it is crucial that “entails” doesn’t work like an embeddable connective, otherwise one could run a form of Curry’s paradox using entailment instead of the conditional.

This not supposed to be a knockdown problem. After all, so what if you can’t *define* strong negation, there is, nonetheless, this disjunction whose disjuncts are just those propositions inconistent with p. We may not be able to define it or refer to it, but God knows which one it is all the same.

The real problem, I think, is the following. How are we construing $\neg p$? Is it a new connective in the language, stipulated to mean the same as “the disjunction of those things inconsistent with p”? If it is, how do we know it is a logical connective? (If $\neg$ weren’t logical neither (1) nor (2) would hold, since there would be no logical principles governing it.) Field objects to a similar argument from Wright, because “inconsistent with p” is not logical. Inconsistency is not logical: for a start it can only be had by sentences, so it is not topic neutral.

The way of construing $\neg p$ that makes it different from Wright’s argument, and allegedy problematic, is to construe $\neg p$ as schematic for a large disjunction. The symbol $\neg$ does not actually belong to the language at all – writing $\neg p$ is just a metalinguistic shorthand for a very long disjunction, a disjunction that will change, depending in each case, on p. Treating it as such guarantees that (1) and (2) hold, since when they are expanded out, are just truths about the logic of disjunction and don’t contain $\neg$ at all.

But treating $\neg p$ as schematic for a disjunction means it doesn’t behave like an ordinary connective. For one you can’t quantify into it’s scope. What sentence would $\exists x\neg Fx$ be schematic for? What we want it to mean is that there is some object, a, such that the disjunction of things inconsistent with Fa holds. But there’s no single sentence involved here.

Another crucial shortcoming is that it’s not clear that we can “put a dot” under $\neg$. That is, define a function which takes the Gödel number of p, to the Gödel number of the disjunction of things inconsistent with p. Firstly there might not be enough Gödel numbers to do this (since we have an uncountable language now!) But secondly, how do we know we can code “inconsistent with p” in arithmetic? Fields logic isn’t recursively axiomatizable (Welch, forthcoming) so it seems like we’re not going to be able to code “inconsistent with p” or the strong negation of p – and thus it seems we’re not going to be able to run the Gödel diagonalisation argument. (I was always asleep in Gödel class so maybe someone can check I’m not missing something here.)

So you can’t get a strongly negated liar sentence through Gödel diagonalisation, but what about indexical self reference? “This sentence is strongly not true” is schematic for a sentence not including “strongly not”, but with a large disjunction instead. However, which disjunction is it? We’re in the same pickle we were in when we tried to quantify into the scope of $\neg$. In both cases, the disjunction needed to vary depending on the value of the variable “x” or in this case, the indexical “this”.

I can’t say I’ve gotten to the bottom of this, but it’s no longer clear to me how problematic Restall’s argument is for the non classical logician.