A simplified formalization of predicate logic with identity

On Tarski's formalization of predicate logic with identity

“…) If q0 is any finite sequence, then OC (q0) is the set of objects qn for n < I q0 [; thus ff q0 is a formula of a language L, then OC (~) is the set of symbols occurring in q0. We say that a occurs bound at the n t~ place in q~, in symbols OB (a, n, ~v), if a e VR, and, for some language L, q~ ~ FML, n < I qo 1, qn = a, and there exist ~v and m such that (i) ~v e FMz and m < n < m ]~v ]< [ ~ ], (ii) ~v 0 = q and ~v~ = a, (iii) ~v~= ~m+~ for every k< l~v ];and that a occurs [ree at the nt~ ~/ace i~ q, or OF (a, n, q~ ), if a ~ YR, ~v is a formula of some language, n < [ 9 [, 9n = a, and a does not occur bound at the n tn place in 9-BV (9), or the set of bound variables ofg, is the set of a such that OB (a, n, 9) for some n; and FV (9), or the set of/tee variables of 9, is the set of a such that OF (a, n, 9) for some n. As was mentioned in [10], the notion 'FV (9)' can be introduced by a simple reeursion, without use of the complicated notions 'OF (a, n, 9)' and 'OB (a, n, 9)'; the same remark applies to 'BV (9)'. STz, or the set of sentences of a language L, is the set of 9 e FML such that FV (9) = 0.…”

“…We shall adopt much of the terminology and symbolism of [10], for example, all of the set-theoretical notions and symbols given there, the notations 'VR' for the set of variables (whose members are v o, v 1 .... ) and 'LC' for the set of logical constants (whose members are i, n, q, and e). Like Tarski we assume that the objects i, n, q, e, v0, v 1 .... are pairwise distinct.…”

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In [10] (the preceding paper) Tarski haS shown the completeness of a system of predicate logic with identity (but without operation symbols and individual constants) whose axioms consist of all formulas of the following eight kinds : We shall show that axiom (B5) can be omitted, that the remaining axiom system is independent, and that Tarski's approach can be extended to predicate logic with operation symbols and individual constants; 1 we shall be * Eingegangen am 30. 10.

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Singulary cylindric and polyadic equality algebras

“…(') See [14]. In particular, the equivalence written before Theorem 3 can be proved using only three variables.…”

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“…(') See [14]. dicate symbol it of & let g(it)= {x eU: there exist yu---,ya-1 eU suchthat (x,yu •••,y"-i >e/(«p0/Br)}.…”