Sentential constants in R and \(R^{\neg}\)
This paper is a study of sentential constants in the relevant logic R, its intensional fragment \(R_ i\) and its Boolean extension \(R^{\neg}\). The basic constants are the Church constants T and F introduced in A. Church [``The weak theory of implication'' in Kontrolliertes Denken. Untersuchungen zum Logikkalkül und der Logik der Einzelwissenschaften, ed. Menne, Wilhelmy and Angsil, Komissions-Verlag Karl Alber, Munich, 1951, pp. 22-37] and the Ackermann constants t and f of \textit{W. Ackermann} [J. Symb. Logic 21, 113-128 (1956; Zbl 0072.001)]. The author makes interesting use of matrix and general algebraic techniques to show (among other things): 1. There are exactly 6 non-equivalent formulae in \(R_ i\) (and in R, and in \(R^{\neg})\) that can be built out of the Ackermann constants and the intensional connectives; 2. There are exactly 14 non-equivalent formulae in \(R_ i\) (and in R) built out of the sentential constants and the intensional connectives; 3. There are exactly 8 formulae in \(R^{\neg}\) built up from the sentential constants and all of the connectives; 4. Defining (in \(R^{\neg})\) necessity, N, as \(NA=T\to A\), possibility, P, as \(PA=\neg N\neg A\) and strict implication, \(\Rightarrow\), as \(A\Rightarrow B=N(\neg A\vee B)\) and letting the class of modal formulae be the class of formulae built up from propositional variables, T, F, and Boolean and modal connectives: a modal formula is a theorem of \(R^{\neg}\) iff it is a theorem of S5; and \(A\to B\) is a theorem of \(R^{\neg}\) iff \(A\Rightarrow B\) is. The author refers the reader to \textit{J. K. Slaney} [J. Symb. Logic 50, 487-501 (1985; Zbl 0576.03015)] for the solution to the problem of the number of non-equivalent Ackermann constants in R. (The answer is 3088.)
scientific article; zbMATH DE number 4012574