Internalizing Case-Relative Truth in CIFOL$$+$$
Nuel Belnap · Outstanding contributions to logic · 2014
CIFOL is defined in Belnap and Müller 2013 (J Phil Logic 2013) as the first-order fragment of Aldo Bressan’s higher-order modal typed calculus $$MC^ u $$ . Bressan based his calculus on Carnap’s “method of extension and intension”: In CIFOL, truth is relative to “cases,” where cases play the formal role of “worlds” (but with less pretension). CIFOL $$+$$ results by following Bressan in adding term-constants t for the true and f for the false, and a single predicate constant, $$P_0$$ , which together with a couple of simple axioms enable the representation of “sentence $$\Phi $$ is true in case $$x$$ ” by means of a defined expression, $$T(\Phi , x)$$ , where $$\Phi $$ is the sentence of CIFOL $$+$$ in question and where $$x$$ ranges over a defined family of “elementary cases.” (Whereas being a case is defined in the semantic metalanguage, elementary cases are squarely in the (first order) domain of CIFOL $$+$$ .) A suitable suite of axioms guarantees that one can prove (in CIFOL $$+$$ ) that there is exactly one elementary case, $$x$$ , such that $$x$$ happens (i.e., such that $$x=\mathbf t $$ ), a fact that underlies the equivalence of $$\square (x=\mathbf t \rightarrow \Phi )$$ and $$\lozenge (x=\mathbf t \,\,\wedge \,\,\Phi )$$ . (Proofs are surprisingly intricate for first order modal logic.) One can then go on to show that $$T(\Phi , x)$$ is well-behaved in terms of its relation to the connectives of CIFOL $$+$$ , a result required for ensuring that $$T(\Phi , x)$$ is properly read as “that $$\Phi $$ is true in elementary case $$x$$ .”