Zero-place operations and functional completeness, and the definition of new connectives

Abstract

Tarski 1968 makes a move in the course of providing an account of ’definitionally equivalent’ classes of algebras with a businesslike lack of fanfare and commentary, the significance of which may accordingly be lost on the casual reader. In §1 we present this move as a response to a certain difficulty in the received account of what it is to define a function symbol (or ’operation symbol’). This difficulty, which presents itself as a minor technicality needing to be got around especially for the case of symbols for zero-place functions (for ’distinguished elements’), has repercussions—not widely recognised—for the account of functional completeness in sentential logic. A similarly stark comment in Church 1956 reveals an appreciation of this difficulty, though not every subsequent author on the topic has taken the point. We fill out this side of the picture in §2. The discussion of functional completeness in §2 is supplemented by some remarks on what is involved in defining a connective, which have been included in an Appendix. The emphasis throughout is on conceptual clarification rather than on proving theorems, and the main body of the paper may be regarded as an elaboration on the remarks just mentioned by Tarski and Church. The Appendix (§3) is intended to be similarly clarificatory, though this time with some corrective intent, of remarks made in and about Makinson 1973