|logic||category theory||type theory|
|true||terminal object/(-2)-truncated object||h-level 0-type/unit type|
|false||initial object||empty type|
|proposition||(-1)-truncated object||h-proposition, mere proposition|
|cut rule||composition of classifying morphisms / pullback of display maps||substitution|
|cut elimination for implication||counit for hom-tensor adjunction||beta reduction|
|introduction rule for implication||unit for hom-tensor adjunction||eta conversion|
|logical conjunction||product||product type|
|disjunction||coproduct ((-1)-truncation of)||sum type (bracket type of)|
|implication||internal hom||function type|
|negation||internal hom into initial object||function type into empty type|
|universal quantification||dependent product||dependent product type|
|existential quantification||dependent sum ((-1)-truncation of)||dependent sum type (bracket type of)|
|equivalence||path space object||identity type|
|equivalence class||quotient||quotient type|
|induction||colimit||inductive type, W-type, M-type|
|higher induction||higher colimit||higher inductive type|
|completely presented set||discrete object/0-truncated object||h-level 2-type/preset/h-set|
|set||internal 0-groupoid||Bishop set/setoid|
|universe||object classifier||type of types|
|modality||closure operator, (idemponent) monad||modal type theory, monad (in computer science)|
|linear logic||(symmetric, closed) monoidal category||linear type theory/quantum computation|
|proof net||string diagram||quantum circuit|
|(absence of) contraction rule||(absence of) diagonal||no-cloning theorem|
|synthetic mathematics||domain specific embedded programming language|
In (classical) logic, the negation of a statement is a statement which is true if and only if is false. Hence, viewed algebraically, the negation corresponds to the complement operator of the corresponding Boolean algebra which satisfies as well as .
More generally, as different logics correspond to different types of lattices, one calls negation antitone, or polarity reversing, lattice operators that mimic or approximate the algebraic and proof-theoretic behavior of .
In intuitionistic logic, we only have
while in paraconsistent logic, we instead have
You can interpret intuitionistic negation as ‘denial’ and paraconsistent negation as ‘doubt’. So when one says that one doesn't deny , that's weaker than actually asserting ; while when one says that one doesn't doubt , that's stronger than merely asserting . Paraconsistent logic has even been applied to the theory of law: if is a judgment that normally requires only the preponderance of evidence, then is a judgment of beyond reasonable doubt.
Linear logic features (at least) three different forms of negation, one for each of the above. (The default meaning of the term ‘negation’ in linear logic, , is the one that satisfies the classical double-negation law.)
Y. Gauthier, A Theory of Local Negation: The Model and some Applications , Arch. Math. Logik 25 (1985) pp.127-143. (gdz)
H. Wansing, Negation , pp.415-436 in Goble (ed.), The Blackwell Guide to Philosophical Logic , Blackwell Oxford 2001.