Logical equality: Difference between revisions

'''Logical equality''' is a [[logical operator]] that correspondscompares totwo [[equalitytruth (mathematics)|equalityvalues]], inor [[Booleanmore algebragenerally, (logic)|Boolean algebra]] and to thetwo [[logicalWell-formed biconditionalformula|formulas]], insuch [[propositionalthat calculus]]. Itit gives the [[function (mathematics)|functional]] value ''[[Truth|trueTrue]]'' if both functional arguments have the same [[logicaltruth value]], and ''[[False (logic)|falseFalse]]'' if they are different. In the case where formulas have [[free variables]], we say two formulas are equal when their truth values are equal for all possible resolutions of free variables. It corresponds to [[equality (mathematics)|equality]] in [[Boolean algebra]] and to the [[logical biconditional]] in [[propositional calculus]].

Some logicians, however, draw a firm distinction between a ''functional form'', like those in the left column, which they interpret as an application of a function to a pair of arguments — and thus a mere indication that the value of the compound expression depends on the values of the component expressions — and an ''equational form'', like those in the right column, which they interpret as an assertion that the arguments have equal values, in other words, that the functional value of the compound expression is ''true''.^{{{Citation needed|date=July 2024}}}

In [[mathematics]], the plus sign "+" almost invariably indicates an operation that satisfies the axioms assigned to addition in the type of [[algebraic structure]] that is known as a ''[[field (mathematics)|field]]''. For booleanBoolean algebra, this means that the logical operation signified by "+" is not the same as the [[inclusive disjunction]] signified by "∨" but is actually equivalent to the logical inequality operator signified by "≠", or what amounts to the same thing, the [[exclusive disjunction]] signified by "XOR" or "⊕". Naturally, these variations in usage have caused some failures to communicate between mathematicians and switching engineers over the years. At any rate, one has the following array of corresponding forms for the symbols associated with logical inequality:

This explains why "EQ" is often called "[[XNOR gate|XNOR]]" in the [[combinational logic]] of circuit engineers, since it is the ''negation'' of the ''[[XOR]]'' operation; "NXOR" is a less commonly used alternative.<ref>{{citation|title=Using Java 2|first1=Brian|last1=Keeton|first2=Chuck|last2=Cavaness|first3=Geoff|last3=Friesen|publisher=Que Publishing|year=2001|isbn=9780789724687|page=112|url=https://books.google.com/books?id=yhFxiVyd1MgC&pg=PA112}}.</ref> Another rationalization of the admittedly circuitous name "XNOR" is that one begins with the "both false" operator NOR and then adds the eXception "or both true".