A binary Sheffer operator which does the work of quantifiers and sentential connectives.
Robert B. Brandom · Notre Dame Journal of Formal Logic · 1979
In recent years, the range of propositional systems for which binary Sheffer operators have been discovered has broadened to include various systems with multiple truth-values, modalities, and multigrade connectives (see [1] for a review).In this paper*, I present an indigenously definable binary Sheffer operator for the first order predicate calculus, and show how the technique employed there to combine quantifiers and sentential connectives in a single operator can be used to extend the previously discovered binary Sheffer operators to capture quantified modal systems.We consider a stroke language containing a countable number of individual variables X\,x 2 y•> and for each positive integer n, a countable number of w-ary predicates.If P is an w-ary predicate letter, the result of concatenating P to the left of n variable letters is a well-formed formula.If A and B are wffs, A/B is a wff.For any positive integer k and any wff A, we introduce the notation A k (the name of a formula) as follows:We interpret the stroke language in terms of a standard predicate language with the usual sentential connectives and quantifiers as follows.For any wffs A and J5, letwhere ^ indicates semantic equivalence, and v is an individual variable such that:is alphabetically the first variable which does not occur in A or B iff forallfc,B^4*.