Generalized Quantification in an Axiomatic Truth Theory

ABSTRACT

Bruno Whittle (2019) has recently extended Kripke’s semantical theory of truth to languages containing generalized quantifiers. There are reasons for axiomatizing semantical theories, and for regarding Halbach and Horsten’s PKF as a good axiomatization of Kripke’s. PKF is a theory in Partial Logic. The present paper complements Whittle’s by showing how Partial Logic, and then PKF, may be extended to cover binary quantifiers meaning ‘every’, ‘some’, and ‘most’.

KEYWORDS:

• Axiomatic truth-theories
• generalized quantifiers
• partial logic
• PKF
• Bruno Whittle

Acknowledgements

For their comments on drafts, I thank Stephen Blamey, Hartry Field, Volker Halbach, Christopher Scambler, Daniel Waxman, Bruno Whittle, and two referees for this journal. I acknowledge an old debt to the late Saul Kripke, whose seminar on truth, given at Princeton in the Spring Semester of 1987, I had the good fortune to attend. Requiescat.

Disclosure Statement

No potential conflict of interest was reported by the author(s).

Notes

1 Recall that ‘$X\iff Y$’ means ‘$X\Rightarrow Y$ and $Y\Rightarrow X$’. When I write that a double-headed sequent follows ‘by Axiom 7’ (say), I mean that one component sequent follows by 7a and its converse follows by 7b.

2 PKF may not satisfy all of Feferman’s desiderata on axiomatizations of truth. In earlier work, he complained that systems, such as Partial Logic, which lack the Deduction Theorem impede the ‘sustained reasoning’ that is needed for mathematics (Feferman 1984: 95). There is, though, reason to doubt that a single axiomatic theory meets all of Feferman’s desiderata. See Rumfitt 2023.

3 $\mu$ simplifies an example of Whittle’s (2019: 343), which in turn sharpens a paradoxical case in Maudlin 2004: 63–4.

4 (Most) will not do when the domain of quantification is infinite. We want ‘Most natural numbers are natural numbers’ to come out true, yet ${\mathrm{\aleph }}_0=1/2{\mathrm{\aleph }}_0$. Whittle takes the natural semantic clause for ‘most’ to be $\mathrm{Mx}\left(\mathrm{\phi x},\mathrm{\psi x}\right)$ is true iff $\mathrm{card}\left({\left|\phi \right|}_{+}\cap {\left|\psi \right|}_{+}\right)>\mathrm{card}\left({\left|\phi \right|}_{+}\setminus {\left|\psi \right|}_{+}\right)$, but these truth conditions better fit ‘More $\phi$s are $\psi$ than are not $\psi$’, in which the predicates ‘$\psi$’ and ‘not$\psi$’ are inserted into the second and third places in the ternary form ‘More $\phi$s are $\psi$ than are $\chi$’. In the absence of Excluded Middle, the two truth conditions may diverge.

5 Whittle’s example is $\mathrm{Mx}\left(T\left(x\right)\wedge W\left(x\right),\mathrm{\neg }\left(x=⌜\rho ⌝\right)\right),$ in which ‘$x$’ ranges freely but $W\left(x\right)$ is a predicate meaning ‘$x$ is written on the wall’ (Whittle 2019: 344). It is given that the only sentences written on the wall are ‘$0=0$’ and $\rho$ itself.

6 To derive (∀1), remark that an instance of (∀B1) is $\mathrm{\forall x}\left(x=x,\phi \right)\Rightarrow \mathrm{\neg }\left(t=t\right),\phi \left(t/x\right)$. (¬3) and (¬4) then yield $t=t,\mathrm{\neg \phi }\left(t/x\right)\Rightarrow \mathrm{\neg \forall x}\left(x=x,\phi \right)$. By (=1), $\Rightarrow t=t$, so Cut yields $\mathrm{\neg \phi }\left(t/x\right)\Rightarrow \mathrm{\neg \forall x}\left(x=x,\phi \right)$, whence $\mathrm{\forall x}\left(x=x,\phi \right)\Rightarrow \phi \left(t/x\right)$, by (¬3) and (¬4) again. Under our definitions, this last sequent is precisely (∀1).

To derive (∀2), suppose $X\Rightarrow \phi \left(t/x\right),Y$. By Dilution, we have $X\Rightarrow \mathrm{\neg }\left(t=t\right),\phi \left(t/x\right),Y$. We may then apply (∀B2) to obtain $X\Rightarrow \mathrm{\forall x}\left(x=x,\phi \right),Y$ when $t$ does not occur in $X\cup Y.$ Given $X\Rightarrow \phi \left(t/x\right),Y$, then, we may infer $X\Rightarrow \mathrm{\forall x\phi },Y$, when $t$ does not occur in $X\cup Y$, which is precisely what (∀2) says.

7 No substantial logic is needed in deriving homophonic truth conditions from the axioms of PKF or PKF+. The derivations simply take the form of substituting formulae already shown to be equivalent. For the operative principle of substitution, see Blamey 2002: 327.

8 The definition which follows is my own, but it was suggested by Blamey’s remark that ‘we have no use for the notion of the consistency of a set of formulae’ (2002: 340) and by his use of pairs of sets of formulae at 2002: 344.