Abstract
Harold Hodes in [1] introduces an extension of first-order modal logic featuring a backtracking operator, and provides a possible worlds semantics, according to which the operator is a kind of device for ‘world travel’; he does not provide a proof theory. In this paper, I provide a natural deduction system for modal logic featuring this operator, and argue that the system can be motivated in terms of a reading of the backtracking operator whereby it serves to indicate modal scope. I prove soundness and completeness theorems with respect to Hodes’ semantics, as well as semantics with fewer restrictions on the accessibility relation (Hodes restricts his attentionto S5).
Highlights
Harold Hodes in [1] introduces an extension of first-order modal logic featuring a backtracking operator ‘↓’
The aim of this paper is to provide a proof system for modal logics featuring the operator, which I will claim does justice to this reading; the proof theory gives rules for ‘looking inside’ the scope of a modal operator, and when the ↓ operator is encountered, pulling the appended formula out of that scope
I hope that a proof theory of the sort presented in this paper will be useful to those who would like to understand backtracking operators in a way which does not essentially involve possible worlds
Summary
Harold Hodes in [1] introduces an extension of first-order modal logic featuring a backtracking operator ‘↓’ The purpose of this operator is similar to that of an actuality operator. Hodes gives a semantics (which I give an overview of in Section 2), but does not supply a proof theory.. Hodes gives a semantics (which I give an overview of in Section 2), but does not supply a proof theory.1 This semantics reflects a reading of the operator as one which allows more flexible ‘travel’ through worlds in evaluating the truth value of a formula.
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