Friday, 23 December 2011

Validity as a Primitive

Earlier this year, there was a bit of discussion on M-Phi about a nice forthcoming (in Journal of Philosophy) paper by Jc Beall and my MCMP colleague Julien Murzi, "Two Flavors of Curry's Paradox".

I've now written a shortish paper (11 pages) summarizing the main idea I had about it - i.e., to take validity as a primitive notion and add it to Peano arithmetic governed by a couple of reasonable principles:
if $\phi$ is valid, then $\phi$
if $\phi \rightarrow \theta$ is valid, then if $\phi$ is valid, then $\theta$ is valid
and a restricted necessitation rule, saying
if $\phi$ is logically derivable, then infer "$\phi$ is valid"
showing that one can get a consistent (indeed conservative) extension. I also add a truth predicate, along with "if $\phi$ is true, then $\phi$", "if $\phi \rightarrow \theta$ is true, then if $\phi$ is true, then $\theta$ is true", along with the principle that validities are true. This is also conservative over PA.

The basic idea of the conservativeness proof is quite simple: just replace "$\phi$ is valid" (and "$\phi$ is true") by "$\phi$ is a theorem of pure logic (in the relevant language)". Then everything comes out as a theorem of PA. Everything is classical.

The paper, "Validity as a Primitive", is here (at academia.edu) and will appear in Analysis.

4 comments:

  1. Sounds great, Jeff. I haven't had the chance to check out your paper yet, but two questions immediately spring to mind:

    1- Do you think one can draw 'deflationist' conclusions wrt the validity predicate, given the conservativeness results?

    2- Do you have the same kind of speed-up with the validity predicate as you have with the truth predicate? In other words, does the validity predicate offer the same kind of 'expressive boost' as the truth predicate, despite the conservativeness results?

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  2. Thanks!
    On (1), no, I don't think there are any implications, although surely it's compatible with deflationism.

    On (2), I suspect yes, because Val("A") -> A is quite a powerful principle: its proof inside PA requires full induction: and I suspect that if you add it to a finitely axiomatized theory, the result will be non-conservative. It's basically the (local) reflection principle for pure logic.
    So, while PA_V conservatively extends PA, I suspect PA_V may prove some arithmetic sentences faster than PA does. I'm not sure how to prove this though, except by giving a detailed analysis of V-logic derivations.

    Merry christmas!

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  3. I've just read your paper and the one from Beall and Murzi. Really interesting stuff.
    Some comments:
    1. V-T seems to me very strong. It is right when the theory has a compatibility between the external and the internal logic. But that is not always the case. For example, that doesn't seem to happen, for example, in a kripkean system (like Maudlin's), where some valid sentences are not true but indeterminate (i.e. validity means never-falsity).
    Something like Val(x) --> -T-(ucl(x)) may be better for those cases.
    2. I think it's a good idea to rule out NEC for Validity, because of the intromission of arithmetical and not logical theorems. Beall and Murzi's derivation is a strange one. But in the same line, there is more to say. NEC for Necessity is also strange when we are combining intensional predicates (although it's frequently used in Fitch-like paradoxes). For example, most of modal-epistemological theories can prove (by Epistemological Nec), that S knows that (p v -p), and applying the Modal Nec, can infer that Necessarily, S knows that (p v -p). But that is not so clearly truth preserving. After all, knowledge (unlike logic) could be thought as an empirical and not a conceptual/analytical matter.

    Happy New Year

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  4. Many thanks for the comment Deigo.
    Right, V-T is not a theorem of KF+Cons (though it's consistent with KF+Cons, I think: not sure).
    On necessitation, the restricted version I use seems well-motivated for the notion of validity--at least in the classical first-order case. Still, it opens a huge can of worms about what counts as a "logical notion" more generally (e.g., for extended or non-classical logics), something I conveniently sidestep in the paper! Your point about combining intensional predicates/operators (e.g., K and []) seems right.

    Feliz año nuevo

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