10176

# 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.

### DETAILS

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.

### PERMANENT CITATION

 Share: Embed Interactive Demonstration New! Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details » Download Demonstration as CDF » Download Author Code »(preview ») Files require Wolfram CDF Player or Mathematica.

#### Related Topics

 RELATED RESOURCES
 The #1 tool for creating Demonstrations and anything technical. Explore anything with the first computational knowledge engine. The web's most extensive mathematics resource. An app for every course—right in the palm of your hand. Read our views on math,science, and technology. The format that makes Demonstrations (and any information) easy to share and interact with. Programs & resources for educators, schools & students. Join the initiative for modernizing math education. Walk through homework problems one step at a time, with hints to help along the way. Unlimited random practice problems and answers with built-in Step-by-step solutions. Practice online or make a printable study sheet. Knowledge-based programming for everyone.