If-logic Requires Regular Play

for the conference: Logic, Games and Philosophy: Foundational Perspectives Hilbert and Ackermann defined the language of first order predicate logic in such a way that a variable never occurs both free and bound within a single formula, nor can a formula contain nested quantification over the same variable [3, p. 74]. We will call formulas that satisfy these properties regular. Although first order languages are now usually defined without these restrictions, imposing them does not affect the properties of the logic because by renaming of bound variables (‘alpha-conversion’) any irregular first order formula maybe transformed into a regular equivalent: e.g. the irregular formula ∀x∃y[R(x, y) ∧ ∃xP (x)] is classically equivalent to the regular one ∀x∃y[R(x, y) ∧ ∃zP (z)]. We show that the difference between regular and irregular formulas does have semantical consequences in the area of game theoretical semantics and IF-logic, whose language is an extension of the language of first order predicate logic. First, although it is never explicitly mentioned, a restriction to regular formulas is crucial in Hintikka’s claim that IF-logic is a conservative extension of classical first order logic. Note that in Hintikka’s approach of [4], the ‘slashing convention’ is incorporated: “moves connected with existential quantifiers are always independent of earlier moves with existential quantifiers” ([4, p. 63]). This makes the classically true first order formula ∀x∃y[x = y ∧ ∀x∃z[y = z]] (1) if read as an IF-sentence, equivalent to the formula in the more general language and semantics of [1]: ∀x∃y[x = y ∧ ∀x∃z/y [y = z]] (2) The latter formula can be seen to be true in one-element models only. This example shows that, due to the slashing convention, the conservativeextension-claim does not hold if we include irregular first order sentences.