natural deduction metalanguage, practical foundations
type theory (dependent, intensional, observational type theory, homotopy type theory)
computational trinitarianism = propositions as types +programs as proofs +relation type theory/category theory
logic | category theory | type theory |
---|---|---|
true | terminal object/(-2)-truncated object | h-level 0-type/unit type |
proposition(-1)-truncated objecth-proposition, mere proposition
proofgeneralized elementprogram
cut rulecomposition of classifying morphisms / pullback of display mapssubstitution
cut elimination for implicationcounit for hom-tensor adjunctionbeta reduction
introduction rule for implicationunit for hom-tensor adjunctioneta conversion
logical conjunctionproductproduct type
disjunctioncoproduct ((-1)-truncation of)sum type (bracket type of)
implicationinternal homfunction type
negationinternal hom into initial objectfunction type into empty type
universal quantificationdependent productdependent product type
existential quantificationdependent sum ((-1)-truncation of)dependent sum type (bracket type of)
equivalencepath space objectidentity type
equivalence classquotientquotient type
inductioncolimitinductive type, W-type, M-type
higher inductionhigher colimithigher inductive type
completely presented setdiscrete object/0-truncated objecth-level 2-type/preset/h-set
setinternal 0-groupoidBishop set/setoid
universeobject classifiertype of types
modalityclosure operator, (idemponent) monadmodal type theory, monad (in computer science)
linear logic(symmetric, closed) monoidal categorylinear type theory/quantum computation
proof netstring diagramquantum circuit
(absence of) contraction rule(absence of) diagonalno-cloning theorem
synthetic mathematicsdomain specific embedded programming language
</table>
Disjunctive logic is the internal logic of lextensive categories.
…
The (coherent) theory of (geometric) fields is obtained from the axioms for the algebraic theory of commutative unital rings by adding the sequents
Since inverse elements in a commutative ring are unique when they exist the second sequent involves a legitimate existential quantification plus a legitimate disjunction (due to the nontriviality) whence the resulting theory is (finitary) disjunctive.
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The main reference on disjunctive logic is Johnstone (1979) which was inspired by a non-syntactic concept of Yves Diers?. The elephant has some additional cursory remarks. Freyd (2002) gives disjunctive logic a short treatment under the name ‘alternating logic’ and discusses the theory of real closed fields as an example. Barr and Wells (1985) discuss the corresponding class of sketches without mentioning the syntactic side.
Jiří Adámek, Jiří Rosický, Locally presentable and accessible categories, Cambridge UP 1994. (exercise 5f., p.238)
Michael Barr, Charles Wells, Toposes, Triples and Theories , Springer Heidelberg 1985. (Reprinted as TAC reprint no.12 (2005); section 8.2, pp.292f)
Peter Freyd, Cartesian Logic , Theor. Comp. Sci. 278 (2002) pp.3-21.
Peter Johnstone, A Syntactic Approach to Diers’ Localizable Categories , pp.466-478 in Springer LNM 753 Heidelberg 1979.
Peter Johnstone, Sketches of an Elephant II, Oxford UP 2002. (D1.3.6 p.834, D2.1.2(e) p.863, D2.2.6 p.872, D2.4.8 p.886)
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