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
Disjunctive logic is the internal logic of lextensive categories.
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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)