Tharp’s theorems of metaphysics and the notion of necessary truth

Abstract

Leslie Tharp proves three theorems concerning epistemic and metaphysical modality for conventional modal predicate logic: every truth is a priori equivalent to a necessary truth, every truth is necessarily equivalent to an a priori truth, and every truth is a priori equivalent to a contingent truth. Lloyd Humberstone has shown that these theorems also hold in the modal system Actuality Modal Logic (AML), the logic that results from the addition of the actuality operator to conventional modal logic. We show that Tharp’s theorems fail for the expressively equivalent Subjunctive Modal Logic (SML), the logic that was developed by Kai Wehmeier as an alternative to AML. We then argue that the existence of Tharp’s theorems for AML is due to a faulty interpretation of the notion of necessary truth, a feature that is not shared by SML. The paper concludes with an argument for the thesis that the existence of the distinction between truth at all worlds w and truth at all worlds w from the point of view of w as the actual world is an artifact owing to the interaction of the necessity and actuality operator.

Technical Appendix

[Kripke Structure]: A Kripke structure is a tuple consisting of a set of worlds W, a distinguished world @ of the model we call the actual world, a domain D, and for each world w and n-ary predicate G, I(G, w) is the interpretation of G at w.

Lemma 1

For all Kripke Structures K, world sequences ${\upomega }$, and variable assignments ${\upsigma }$,

$$\begin{aligned} K, {\upomega } \models {\upvarphi }^{s}_{i}[{\upsigma }] \hbox { iff } K', {\upomega } \models {\upvarphi } [{\upsigma }], \end{aligned}$$

where $K'$ is the variant of K with the i-th world of ${\upomega }$ as the actual world.

Proof

By induction on the complexity of ${\upvarphi }$:

• Base Case: ${\upvarphi }$ is of the form $R(t_{1},{\ldots }, t_{n})$

• $K, {\upomega } \models R^{s}_{i}(t_{1},{\ldots }, t_{n}) [{\upsigma }]$ iff

• iff

• $K', {\upomega } \models R(t_{1},{\ldots }, t_{n}) [{\upsigma }]$

• ${\upvarphi }$ is of the form $\lnot \Psi$

• $K, {\upomega } \models \lnot \Psi ^{s}_{i} [{\upsigma }]$ iff

• $K, {\upomega }$ $\nvDash \Psi ^{s}_{i} [{\upsigma }]$ iff

• $K',{\upomega }$ $\nvDash \Psi [{\upsigma }]$ (by IH) iff

• $K', {\upomega } \models \lnot \Psi [{\upsigma }]$

• ${\upvarphi }$ is of the form ($\Psi$ & $\Phi$)

• $K, {\upomega } \models (\Psi ^{s}_{i }$& $\Phi ^{s}_{i}) [{\upsigma }]$ iff

• $K, {\upomega } \models \Psi ^{s}_{i } [{\upsigma }]$ and $K, {\upomega } \models \Phi ^{s}_{i }[{\upsigma }]$ iff

• $K', {\upomega } \models \Psi [{\upsigma }]$ (by IH) and $K'$, ${\upomega } \models \Phi [{\upsigma }]$ (by IH) iff

• $K', {\upomega } \models$ ($\Psi$ & $\Phi$) [${\upsigma }$]

• ${\upvarphi }$ is of the form $\exists x\Psi$

• $K, {\upomega } \models \exists x\Psi ^{s}_{i} [{\upsigma }]$ iff

• For some x-variant assignment ${\upsigma }'$ of ${\upsigma }: K, {\upomega } \models \Psi ^{s}_{i} [{\upsigma }']$ iff

• For some x-variant assignment ${\upsigma }'$ of ${\upsigma }: K', {\upomega } \models \Psi [{\upsigma }']$ (by IH) iff

• $K', {\upomega }\models \exists x\Psi [{\upsigma }]$

• ${\upvarphi }$ is of the form $\Box _{j}\Psi$

• $K, {\upomega } \models \Box _{j}\Psi ^{s}_{i} [{\upsigma }]$ iff

• For all j-variants ${\upomega }_{j}$ of ${\upomega }: K, {\upomega }_{j} \models \Psi ^{s}_{i} [{\upsigma }]$ iff

• For all j-variants ${\upomega }_{j}$ of ${\upomega }: K', {\upomega }_{j}\models \Psi [{\upsigma }]$ (by IH) iff

• $K', {\upomega } \models \Box _{j}\Psi [{\upsigma }]$ $\square$

Theorem 4

For all Kripke Structures K, world sequences ${\upomega }$, and variable assignments ${\upsigma }$,

$$\begin{aligned} K, {\upomega } \models \Box _{i}{\upvarphi }^{s}_{i }[{\upsigma }] \,\mathrm{iff }\, K, {\upomega } \models \mathrm{F}({\upvarphi }) [{\upsigma }], \end{aligned}$$

where ${\upvarphi }^{s}_{i}$ results from ${\upvarphi }$ by subjunctivizing every indicative predicate in ${\upvarphi }$ with an index that does not already occur in ${\upvarphi }$:

Proof

• $K, {\upomega } \models \Box _{i}{\upvarphi }^{s}_{i} [{\upsigma }]$ iff

• For all i-variants ${\upomega }_{i}$ of ${\upomega }: K, {\upomega }_{i} \models {\upvarphi }^{s}_{i} [{\upsigma }]$ iff

• For all i-variants ${\upomega }_{i}$ of ${\upomega }$ and all variants $K'$ of K with the actual world of $K'$ the i-th world of ${\upomega }_{i}: K', {\upomega }_{i} \models {\upvarphi } [{\upsigma }]$ (by Lemma 1) iff

• For all variants $K'$ of $K: K', {\upomega }\models {\upvarphi } [{\upsigma }]$

• $K, {\upomega } \models \hbox {F}({\upvarphi }) [{\upsigma }]$. $\square$