In so many possible worlds.
Kit Fine · Notre Dame Journal of Formal Logic · 1972
Ordinary modal logic deals with the notion of a proposition being true in at least one possible world. This makes it natural to consider the notion of a proposition being true in n possible worlds for any nonnegative integer n. Such a notion would stand to Tarski’s numerical quantifiers as ordinary possibility stands to the existential quantifier. In this paper1 I present several logics for numerical possibility. First I give the syntax and semantics for a minimal such logic (sections 1 and 2); then I prove its completeness (sections 3 and 4); and finally I show how to extend this result to other logics (section 5). 1 The logic GrK The logic GrK is defined as follows. Formation rules: Formulas are constructed in the usual way from a set V of propositional variables p1, p2,..., the binary operator ∨ (or), the unary operators ¬ (not), (necessity) and ♦>n, n = 2, 3,..., and parentheses ( and). Throughout the paper I observe some familiar conventions: A,B,C, with or without subscripts, range over formulas; →, ↔, ♦ (possibility) etc. are given standard definitions; all expressions are used autonomously; and parentheses are omitted from formulas in an obvious way. ♦>0A abbreviates A → A, ♦>1A abbreviates ♦A and ♦=nA abbreviates ♦>nA ∧ ¬♦>n+1A, n = 0, 1,.... ♦>nA is taken to mean “A is true in at least n possible worlds”; so ♦=nA means “A is true in exactly n possible worlds ” (see section 2). ` A means “A is a theorem of GrK”. Transformation rules: Axiom schemes (where n,m = 1, 2,...)