On the connection of the first-order functional calculus with many-valued propositional calculi.
Juliusz Reichbach · Notre Dame Journal of Formal Logic · 1962
From the results presented in my paper [2] it follows that it is possible to approximate the first-order functional calculus by many valued propositional calculi; in this paper* we shall describe this approximation.We shall use the terminology of [2] and in particular:(2) individual variables: Xp x 2 , . . .[or simply x], (2) apparent individual variables: a^, a 2 , . . .[or simply a] 7 (3) finite number of functional variables: f^ , . . ., / c , (4) logical constants: ' (negation), + (alternative), Π (general quantifier), (5) atomic expressions: R, Rj, R 2 > expressions: E, F, G, Ep F^, G ι ,... 1 (6) ιυ(E) -the number of different individual [p(E)-apparent] variables occurring in the expression E, (7) U m \ -the sequence ί^, . . ., i m ; {^(fnl-*!! different indices of those and only those individual variables which occur in E, (8) n(E) = max \w(E) + p(E), max U w(E) \\, (9) n(E) = ?2(E),if E is an alternative of normal forms, n(E) = max {n(E) r n(F)\ 9 where F is the simplest alternative of normal forms equivalent to E, in the opposite case (we choose an arbitrary alternative), (10) Έ -maximum of arguments of /«, . ., / , (21) E(u/z) -the expression resulting from E by substitution of u for each occurrence of z in E (with usual conditions), (12) C(E) -the set of all significant parts of the formula E: H € C(E) . 2 = .H = E or there exist F, G, H t such that: (H = F) A (E = F') v {(H = F) v (H = G)] (E= F + G)v (Ξi) {H = H J[ (x/a)\ A (E = JlaHJ, (13) Skt -the set of all formulas of the form ΣΛ ; . . .Σtf χ .Π# t+ί . . .Ila^F,where F is a quantifierless expression containing no free variables, ΐla. is the sign of the universal quantifier binding the variable a. and