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Equivalential Calculus

The equivalential calculus is a subsystem of propositional calculus with equivalence (≡) as the only connective. Two equivalential tautologies are and . In Polish notation, discovered by Łukasiewicz, these formulas are written as and . This Demonstration shows the derivation of 21 theorems of the equivalential calculus based on one axiom, , and the rules of substitution and modus ponens.
Here is an example of substitution. The substitution means that the propositional variable is replaced by and that is replaced by . So applying that substitution to the formula gives .
Modus ponens is a derivation of from and .
To be a theorem of the equivalential calculus, its formula must follow from already proven theorems. The first theorem is the axiom, but the rest of the numbered formulas need explanation.
1. Axiom
2.
The second line means that the formula follows by modus ponens from formula 1 (the axiom), to which the substitution is applied, and from the formula 1, which is the axiom.
So gives , which is .
The formula in step 2 follows from the last two by modus ponens.

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In 1929, Leśniewski recognized that the equivalential calculus could be axiomatized. He used two axioms and the rules of detachment and substitution. He also observed that a formula of propositional calculus in which equivalence was the only propositional function was a tautology if and only if each propositional letter occurs an even number of times. In 1932, Wajsberg showed that the calculus can be based on a single axiom. In 1933, Łukasiewicz found the shortest axiom and gave a simple completeness proof.
This axiomatization of the equivalential calculus was presented in [1]. These 21 theorems were derived on pages 98–99.
Reference
[1] J. Łukasiewicz, "The Equivalential Calculus," in Polish Logic 1920–1939 (S. McCall, ed.), Oxford: Clarendon Press, 1967 pp. 88–115.
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