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...lent ways in [[constructive mathematics]] to capture the [[classical logic|classical]] notion of a [[field (mathematics)|field]]. It is essentially a field with ===Relation to classical logic=== ...

In [[abstract algebra|algebra]] and [[logic]], a '''modal algebra''' is a structure <math>\langle A,\land,\lor,-,0,1,\B ...its subvarieties is dually [[isomorphic]] to the lattice of [[normal modal logic]]s. ...

...</sub>]]-theorems of various [[first-order theories]] of [[classical logic|classical mathematics]] are also theorems of intuitionistic mathematics. It is named ...|decidable]], allowing the unquantified theorems of the intuitionistic and classical theories to coincide. ...

...in [[classical logic]], this means that for every [[Sentence (mathematical logic)|sentence]], the theory contains either the sentence or its negation but no A theory <math>T</math> with [[signature (logic)|signature]] ''σ'' is [[κ-categorical|<math>\kappa</math>-categorical]] for ...

...ut|the Hegelian and Marxist method|the formalisation of dialectic in logic|Logic and dialectic}} ...c]]. The precise nature of the relation between [[dialectic]]al and formal logic was hotly debated within the Soviet Union and China. ...

{{Short description|Principle of classical logic}} ...port-export''' is a name given to the propositional form of [[Exportation (logic)|Exportation]]: ...

...se they are not [[validity (logic)|valid]] in classical systems of [[modal logic]]. If they were valid, then the semantics of natural language would validat ...s as [[scalar implicature]]s which arise on the basis of [[classical logic|classical]] lexical entries for disjunction and modality.<ref name = "alonisep" /><re ...

...= On disjunctions and existential statements in intuitionistic systems of logic | journal = [[Mathematische Annalen]] | volume = 132| issue = 4 | pages = 3 ...junction]] and [[existential quantification]] (except in the [[antecedent (logic)|antecedent]] of implication), [[Constructivism (mathematics)|non-construct ...

...ref name="Chierchia2013">{{cite book| first=Gennaro| last=Chierchia| title=Logic in Grammar: Polarity, Free Choice, and Intervention| publisher=Oxford Unive [[Category:Non-classical logic]] ...

}}</ref> is a proof interpretation of intuitionistic logic ([[Heyting arithmetic]]) into a finite type extension of [[primitive recurs Via the [[Gödel–Gentzen negative translation]], the consistency of classical [[Peano arithmetic]] had already been reduced to the consistency of intuiti ...

The main application of the principle is in the study of [[intuitionistic logic]], where the principle is not generally valid. Its crucial equivalent speci The principle is valid in [[classical logic]]. ...

...uit measurement-commute.svg|thumb|Two equivalent [[quantum circuit|quantum logic circuits]]. One where measurement happens first, and one where an operation ...sical bits control if the 1-qubit [[quantum logic gate#X|X]] and [[quantum logic gate#Z|Z]] gates are executed, allowing teleportation.<ref>{{Cite book|titl ...

In classical mathematics, its axioms constitute a formulation of a [[strict total order] ...lternative axioms of a strict total order, which are often employed in the classical context. ...

...[[Hilbert system|Hilbert-style]] [[deductive system]]s for [[propositional logic]]s. ==Classical propositional calculus systems== ...

{{Short description|Axiom used in logic and philosophy}} ...[[Axiom#Mathematics|axiom]] in his first axiomatisation of [[propositional logic]]. It can be thought of as the [[law of excluded middle]] written in a for ...

{{Short description|Rule of replacement in propositional logic}} ...onal statement]]s having [[Logical conjunction|conjunctive]] [[antecedent (logic)|antecedent]]s to be replaced by statements having conditional [[consequent ...

{{Short description|Concept in modal logic}} ...s W.|chapter=Quantification in modal logic|title=Handbook of philosophical logic|volume=3|publisher=Springer Netherlands|year=2001|pages=267–323|doi=10.1007 ...

{{Short description|A less restrictive form of modal logic}} ...[[modal logic]] that deviates from the basic principles of [[normal modal logic]]s. ...

{{Short description|Framework in logic and natural language semantics}} ...y of Amsterdam.}}</ref><ref>{{cite journal | title=Generalized inquisitive logic: completeness via intuitionistic Kripke models| ...

...te. The difference to the [[axiom of choice]] is that in the [[Antecedent (logic)|antecedent]], the existence of <math>y</math> is already granted to be uni ...ive Set Theory |last1=Myhill |first1=John |journal=The Journal of Symbolic Logic |date=1975 |volume=40 |issue=3 |pages=347–382 }}</ref> or some [[Second-ord ...