# nLab disjunctive logic

## Idea

Disjunctive logic is the internal logic of lextensive categories.

## Example

The (coherent) theory of (geometric) fields is obtained from the axioms for the algebraic theory of commutative unital rings by adding the sequents

$(0=1)\vdash \bot\,\text{ (nontriviality) and }\,\top \vdash_x ((x=0) \vee ((\exists y)(xy=1)))\quad.$

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.

## Properties

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.