logical connective

A logical connective is a word that glues propositions together, or modifies one, to build a bigger proposition — exactly the way “and”, “or”, “not”, “if … then” work in ordinary speech, but pinned down so that the truth value of the whole is fixed by the truth values of the parts. The five standard connectives are negation (not), conjunction (and), disjunction (or), implication (if … then), and biconditional (if and only if).

Each connective is defined by a truth table. “P and Q” is true exactly when both P and Q are true; “P or Q” is the inclusive or, true when at least one holds (so it is true even when both hold); “not P” flips the truth value. A subtlety that trips up newcomers: in mathematics “or” is always inclusive unless explicitly stated otherwise, unlike the often-exclusive “or” of everyday English.

These connectives obey an algebra. De Morgan's laws say “not (P and Q)” is equivalent to “(not P) or (not Q)”, and “not (P or Q)” is equivalent to “(not P) and (not Q)”. Such equivalences are the workhorses of proof: they let you rewrite a goal into a more attackable shape, for instance turning the negation of a conjunction into a disjunction you can split into cases.