# Equivalential Calculus

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.

p≡p

(p≡q)≡(q≡p)

Epp

EEpqEqp

EEpqEErqEpr

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 .

{pEpq,qEEpqs}

p

Epq

q

EEpqs

EpEpq

EEpqEEpqEEpqs

Modus ponens is a derivation of from and .

y

x

Exy

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

EEpqEErqEpr

2.

EEsEErqEprEEpqs

1{pEpq,qEErqEpr,rs},1

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.

EEsEErqEprEEpqs

S={pEpq,qEErqErp,rs}

So gives , which is .

1S

EE(Epq)(EErqEpr)EEs(EErqEpr)E(Epq)s

EEpqEErqEpr≡EEsEErqEprEEpqs

The formula in step 2 follows from the last two by modus ponens.

EEsEErqEprEEpqs