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