A propositional calculus: Based on the Łukasiewicz's system
Ford, Dayna Lee
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In Symbolic Logic, 5th Ed., by Irving M. Copi (Copi, 1979), there is a problem asking for the proof of the independence of each axiom from a set of axioms for a propositional calculus. This problem reads: 4. The Łukasiewicz's system (L.S.) has as primitive operators ∼ and →, with P v Q defined as ∼P→Q. Its Rule is: from P and P□Q to infer Q. Its three axioms are: Axiom 1. P→(Q→P); Axiom 2. [P→(Q→R)]→[(P→Q)→(P→R)]; Axiom 3. (∼P→∼Q)→(Q→P). This thesis extends this problem into the complete development of the Łukasiewicz's system. The study follows Copi's outline in developing the L. S. and involves discussing Object Language and Metalanguage, primitive symbols and well-formed-formulas, axioms and demonstrations, independence of the axioms, and the development of the calculus.