Editing Propositional calculus (section)

==== Connective definition methods ====
Some of these connectives may be defined in terms of others: for instance, implication, <math>p \rightarrow q</math>, may be defined in terms of disjunction and negation, as <math>\neg p \lor q</math>;<ref name="ms30"/> and disjunction may be defined in terms of negation and conjunction, as <math>\neg(\neg p \land \neg q</math>.<ref name=":29" /> In fact, a ''[[Functional completeness|truth-functionally complete]]'' system,{{refn|group=lower-alpha|A truth-functionally complete set of connectives<ref name=":2" /> is also called simply ''[[Functional completeness|functionally complete]]'', or ''adequate for truth-functional logic'',<ref name=":13" /> or ''expressively adequate'',<ref name="Smith2003"/> or simply ''adequate''.<ref name=":13" /><ref name="Smith2003" />}} in the sense that all and only the classical propositional tautologies are theorems, may be derived using only disjunction and negation (as [[Bertrand Russell|Russell]], [[Alfred North Whitehead|Whitehead]], and [[David Hilbert|Hilbert]] did), or using only implication and negation (as [[Gottlob Frege|Frege]] did), or using only conjunction and negation, or even using only a single connective for "not and" (the [[Sheffer stroke]]),<ref name=":18" /> as [[Jean Nicod]] did.<ref name=":2" /> A ''joint denial'' connective ([[logical NOR]]) will also suffice, by itself, to define all other connectives. Besides NOR and NAND, no other connectives have this property.<ref name=":29" />{{efn|[[Truth_table#Overview_table|See a table]] of all 16 bivalent truth functions.}}

Some authors, namely [[Colin Howson|Howson]]<ref name=":13" /> and Cunningham,<ref name="ms31"/> distinguish equivalence from the biconditional. (As to equivalence, Howson calls it "truth-functional equivalence", while Cunningham calls it "logical equivalence".) Equivalence is symbolized with ⇔ and is a metalanguage symbol, while a biconditional is symbolized with ↔ and is a logical connective in the object language <math>\mathcal{L}</math>. Regardless, an equivalence or biconditional is true if, and only if, the formulas connected by it are assigned the same semantic value under every interpretation. Other authors often do not make this distinction, and may use the word "equivalence",<ref name=":3" /> and/or the symbol ⇔,<ref name="ms32"/> to denote their object language's biconditional connective.