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Quantification theory and empty individual-domains

  • Theodore Hailperin (a1)

In a recent paper by Mostowski [1] we find an investigation of those formulas of quantification theory which are valid in all domains of individuals, including the empty domain. Mostowski gives a complete set of axioms for such a first order functional calculus (the system is called “”) and a comparison is made with a form of the usual calculus, Church's in [2]. It is pointed out that is much less elegant; in particular, the distributivity laws for quantifiers (e.g., (x){A . B) .{x)A . (x)B) do not hold in general, and likewise the rule of modus ponens does not preserve validity in all cases.

In this paper we show that a not inelegant system is obtained if one modifies Mostowski's approach in two respects; and once this is done a somewhat neater proof of completeness can be given.

The first respect in which we diverge from Mostowski is in the treatment of vacuous quantifiers. For him if p has no free x, then (x)p and (∃x)p are both to have the same value (interpretation) as p1. But this is not the only way to assign values to vacuous quantifications. For when universal quantification is viewed as a generalized conjunction, the formula (x)Fx has the significance of Fa . Fb…. for as many conjunctands as there are individuals in the domain, and if Fx should have the “constant” value p, then (x)p is to mean the conjunction of p with itself for as many times as there are individuals in the domain (compare the arithmetical ).

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[1]Mostowski A., On the rules of proof in the pure functional calculus of the first order, this Journal, vol. 16 (1951), pp. 107111.
[2]Church A., Introduction to mathematical logic, Part I, Annals of Mathematics Studies, Princeton University Press, 1944.
[3]Hilbert D., and Bernays P., Grundlagen der Mathematik, vol. 1, Berlin, 1934.
[4]Quine W. V., Mathematical logic, First edition, New York, 1940.
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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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