Editing Logical connective (section)

===Redundancy===

Such a logical connective as [[converse implication]] "<math>\leftarrow</math>" is actually the same as [[material conditional]] with swapped arguments; thus, the symbol for converse implication is redundant. In some logical calculi (notably, in [[classical logic]]), certain essentially different compound statements are [[logical equivalence|logically equivalent]]. A less [[Triviality (mathematics)|trivial]] example of a redundancy is the classical equivalence between <math>\neg p\vee q</math> and <math>p\to q</math>. Therefore, a classical-based logical system does not need the conditional operator "<math>\to</math>" if "<math>\neg</math>" (not) and "<math>\vee</math>" (or) are already in use, or may use the "<math>\to</math>" only as a [[syntactic sugar]] for a compound having one negation and one disjunction.

There are sixteen [[Boolean function]]s associating the input [[truth value]]s <math>p</math> and <math>q</math> with four-digit [[binary numeral system|binary]] outputs.<ref>[[Józef Maria Bocheński|Bocheński]] (1959), ''A Précis of Mathematical Logic'', passim.</ref> These correspond to possible choices of binary logical connectives for [[classical logic]]. Different implementations of classical logic can choose different [[Functional completeness|functionally complete]] subsets of connectives.

One approach is to choose a ''minimal'' set, and define other connectives by some logical form, as in the example with the material conditional above.
The following are the [[Functional completeness#Minimal functionally complete operator sets|minimal functionally complete sets of operators]] in classical logic whose arities do not exceed 2:
;One element: <math>\{\uparrow\}</math>, <math>\{\downarrow\}</math>.
;Two elements: <math>\{\vee, \neg\}</math>, <math>\{\wedge, \neg\}</math>, <math>\{\to, \neg\}</math>, <math>\{\gets, \neg\}</math>, <math>\{\to, \bot\}</math>, <math>\{\gets, \bot\}</math>, <math>\{\to, \nleftrightarrow\}</math>, <math>\{\gets, \nleftrightarrow\}</math>, <math>\{\to, \nrightarrow\}</math>, <math>\{\to, \nleftarrow\}</math>, <math>\{\gets, \nrightarrow\}</math>, <math>\{\gets, \nleftarrow\}</math>, <math>\{\nrightarrow, \neg\}</math>, <math>\{\nleftarrow, \neg\}</math>, <math>\{\nrightarrow, \top\}</math>, <math>\{\nleftarrow, \top\}</math>, <math>\{\nrightarrow, \leftrightarrow\}</math>, <math>\{\nleftarrow, \leftrightarrow\}</math>.
;Three elements: <math>\{\lor, \leftrightarrow, \bot\}</math>, <math>\{\lor, \leftrightarrow, \nleftrightarrow\}</math>, <math>\{\lor, \nleftrightarrow, \top\}</math>, <math>\{\land, \leftrightarrow, \bot\}</math>, <math>\{\land, \leftrightarrow, \nleftrightarrow\}</math>, <math>\{\land, \nleftrightarrow, \top\}</math>.

Another approach is to use with equal rights connectives of a certain convenient and functionally complete, but ''not minimal'' set. This approach requires more propositional [[axiom]]s, and each equivalence between logical forms must be either an [[axiom]] or provable as a theorem.

The situation, however, is more complicated in [[intuitionistic logic]]. Of its five connectives, {∧, ∨, →, ¬, ⊥}, only negation "¬" can be reduced to other connectives (see {{Section link|False (logic)|False, negation and contradiction}} for more). Neither conjunction, disjunction, nor material conditional has an equivalent form constructed from the other four logical connectives.