ISSN:
1572-8730
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
,
Philosophy
Notes:
Abstract The propositional fragment L 1 of Leśniewski's ontology is the smallest class (of formulas) containing besides all the instances of tautology the formulas of the forms: ɛ(a, b) ⊃ ɛ(a, a), ɛ(a, b) ∧ ɛ(b,).⊃ ɛ(a, c) and ɛ(a, b) ∧ ɛ(b, c). ⊃ ɛ(b, a) being closed under detachment. The purpose of this paper is to furnish another more constructive proof than that given earlier by one of us for: Theorem A is provable in L 1 iff TA is a thesis of first-order predicate logic with equality, where T is a translation of the formulas of L 1 into those of first-order predicate logic with equality such that Tɛ(a, b) = FblxFax (Russeltian-type definite description), TA ∨ B = TA ∨ TB, T ∼ A = ∼TA, etc. For the proof of this theorem use is made of a tableau method based upon the following reduction rules: $$\begin{gathered} \frac{{G\left[ {A \vee B} \right]}}{{G\left[ {A \vee B{\text{\_}}} \right] \vee \sim A|G|[A \vee B\_] \vee \sim B,}}{\text{ }}\frac{{G[\varepsilon (a,b)\_]}}{{G[\varepsilon (a,b)\_] \vee \sim \varepsilon (a,a),}} \hfill \\ \frac{{G[\varepsilon (a,b)\_,\varepsilon (b,c)\_]}}{{G[\varepsilon (a,b)\_,\varepsilon (b,c)\_], \vee \sim \varepsilon (a,c),}}{\text{ }}\frac{{G[\varepsilon (a,b)\_,\varepsilon (b,c)\_]}}{{G[\varepsilon (a,b)\_,\varepsilon (b,c)\_] \vee \sim \varepsilon (b,a),}} \hfill \\ \end{gathered} $$ where F[A +] (G[A−]) means that A occurs in F[A +] (G[A−]) as its positive (negative) part in accordance with the definition given by Schütte.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF00370344
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