Saturday, 4 August 2012

So, what's your theory of truth?

I have written a little bit on theories of truth, so I'm sometimes asked what my own theory of truth is. I'm supposed to say something like,
Given $\phi$ one may infer $T\ulcorner \phi \urcorner$
or
$\forall x \in Sent(L_T) (T neg(x) \leftrightarrow \neg Tx)$
etc.
But I don't, because I think of them as being about the formal behaviour of a truth predicate, and not about truth.
Instead, I'd prefer this more representationalist view:
1. There are these things, called strings: they are sequences (usually finite) drawn from an alphabet.
2. There is syntax, which specifies that some of these syntactical strings are sentences (of some language $\mathcal{L}$).
3. And there are $\mathcal{L}$-interpretations, $\mathcal{I}$.
4. Then there is a semantic relation, written
$\mathcal{I} \models \phi$
which means "the sentence $\phi$ is true in the interpretation $\mathcal{I}$".
That's my theory of truth!
(And for my non-causal theory of reference, replace "sentence $\phi$" by "term $t$" and replace "$\mathcal{I} \models \phi$" by "$t^{\mathcal{I}} = a$").

But there are several objections:
1. (Non-classicist)
By an "interpretation" you mean a classical one? What if we want to discuss $n$-valued logic or truth in some Kripke frames or something exotic?
My response is: yes, entirely agree. The interpretations can be anything you find in a logic (or formal semantics) textbook: $n$-valued, $2^{\aleph_0}$-valued, supervaluational, Kripke frames, etc.
2. (Sentential deflationist)
Surely truth is a unary predicate, $Tx$, applicable to sentences. The function of $T$ is logical, to express schematic generalizations, which must satisfy disquotation/transparency?
My response is: I don't agree. Truth for linguistic entities is a binary relation, relating a string and an interpretation $\mathcal{I}$ (or an interpreted language $\mathbf{L} = (\mathcal{L}, \mathcal{I})$). The function of any unary truth predicate one studies is to denote some class of truths in an interpreted language, or to denotes some class of true propositions. For example, given $\mathbb{N}$, then define
$E = Th_{L}(\mathbb{N})$,
i.e., true arithmetic (well, codes of). Then, if the symbol $T$ is interpreted as $E$, we get,
$(\mathbb{N}, E) \models T\ulcorner \phi \urcorner \leftrightarrow \phi$,
for each $\phi \in L$. So, each T-sentence (so long as $\phi \in L$) comes out true in $(\mathbb{N}, E)$.
3. (Minimalist)
Yes, ok: truth for sentences is binary, not unary. Sentential deflationists are mistaken. But surely the primary notion of truth is for propositions; then it really is expressed by a unary predicate, $Tx$. The function of $T$ as applied to propositions is logical, to express generalizations, which must satisfy disquotation/transparency?
My response is: I'm not sure. I do tend to think that propositional truth is more basic somehow. But then no one has a good theory of propositions (for example, it's unclear how diagonalization and self-reference works for propositions).

[I am being a bit sneaky: my favourite formal theory of truth is KF, the Kripke-Feferman theory of truth.]