Hypersequent Calculi for Contingent Existence

5 August 2015

Congress of Logic, Methodology and Philosophy of Science, Helsinki

It is well known that the most straightforward way of adding rules for the quantifiers to any adequate sequent calculus for the modal logic S5 allows for the derivation of the Barcan formula—$\diamondsuit\exists x\varphi \to \exists x \diamondsuit\varphi$. This result is philosophically undesirable as, contra the Barcan Formula, common-sense metaphysics would have it that things can possibly exist without actually existing. In this talk I give an account of what is suspicious about such derivations by making use of a modal object language with primitive scope indicators. Using this modal object language we see derivations of the Barcan formula as requiring a certain kind of scope elision between “possibly, $a$ is $F$” (written in this language as “$\diamondsuit\ a\ Fa$“) and “concerning $a$, possibly it is $F$” (written as “$a\ \diamondsuit Fa$“), where we are only able to infer “$\exists x \diamondsuit Fx$” from the second of these formulas.

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