Leibnitz Style Arithmetization of Monadic Predicate Calculus with Equality

Vladimir Sotirov

Leibnitz’s idea to translate syllogistic into arithmetic was realized in [1]. In this paper, Leibnitz style translation is extended on the whole monadic predicate calculus including equality. To build an arithmetical model, let N > 1 be an integer without multiple factors. Any predicate P_{i} is interpreted by arbitrary divisor of N (possibly 1 or N) denoted with d(P_{i}), and any individual variable x_{i} is evaluated by d(x_{i}), a prime divisor of N. Following the construction of a formula F, its arithmetical statement AR[F] corresponding to given evaluation will be obtained. Namely, for atomic formulas, AR[P_{i}(x_{j})] is “d(x_{j}) divides d(P_{i})” and AR[x_{i} = x_{j}] is “d(x_{i}) = d(x_{j})”; for a subformula G, AR[(∀x)G] is “for any prime divisor d, AR_{d}^{x}[G]” where AR_{d}^{x} differs from AR attaching d to x; finally, all propositional connectives are replaced with their non-formal analogues. If AR[F] is a true arithmetical sentence for any N under arbitrary evaluation, F is named arithmetically true.

Theorem 1 Any formula of the monadic predicate calculus with equality is a tautology iff it is arithmetically true.

First of all, a predicate formula is a tautology iff its universal closure is a tautology. Because of the decidability of the monadic predicate calculus with equality, the domains may be supposed to be finite [2, Theorem 25.1]. Any such model can be transformed in an isomorphic arithmetical model: if all elements of the domain D are a_{1}, …, a_{n}, take an integer N = p_{1} · . . . · p_{n} where p_{1},…, p_{n} are different prime numbers. If the interpretation of P_{i} is {a_{i_{1}}, …, a_{i_{k}}} ≠ ∅, let d(P_{i}) be p_{i_{1}} · . . . · p_{i_{k}}; if the subset is empty, d(P_{i}) is 1. The evaluation of individual variables is obvious.

[1] V. Sotirov, “Arithmetizations of syllogistic à la Leibnitz”, J. Appl. Non-Class. Logics, 9 (1999), n. 2–3, pp. 387–405.

[2] G. Boolos, R. Jeffrey, “Computability and Logic”, Cambridge Univ. Press, 3rd ed. 1989.

Address: Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, 8, Acad. G.Bonchev Str., 1113 Sofia, Bulgaria

E-mail: vlsot@math.bas.bg; sotirov@fmi.uni-sofia.bg