equivalences: full and partial

A full equivalence between two statements (technically ‘propositions’) P and Q can be expressed as “P if, and only if, Q”. The ‘partial equivalence’ “P only if Q” is called an ‘implication’, and the ‘partial equivalence’ “P if Q” is called an ‘inference’. (And it is a common mistake to confuse ‘imply’ and ‘infer’.) Keeping track of full and partial equivalences is done by means of what are called ‘truth tables’ and ‘symbolic logic’.
(A famous book along this line is ‘Symbolic Logic & the Real Number System’ by A. H. Lightstone.)
keywords: Mathematics, proofs, deductions, deductive chains