eolas/Logic/Logical_equivalence.md

tags
Logic propositional-logic

Two sentences, P and Q, are truth-functionally equivalent if and only if there is no truth assignment in which P is true and Q is false

Informal expression

P: If it is raining then the pavement will be wet.
Q: The pavement is not wet unless it is raining.

Formal expression

P \supset Q \equiv \sim P \lor Q

Truth-tables

P	Q				P	⊃	Q	
T	T					T		
T	F					F		
F	T					T		
F	F					T

P	Q				~	P	∨	Q	
T	T						T		
T	F						F		
F	T						T		
F	F						T

Derivation

Propositions P and Q are equivalent in a system of derivation for propositional logic if Q is derivable from P and P is derivable from Q.

Note that the property of equivalence stated in terms of derivablity above is identical to the derivation rule for the material biconditional:

//TODO: Add demonstration of this by deriving two equivalents from one of DeMorgan's Laws