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
Cartesian logic or finite limit logic is the internal logic of finitely complete categories (which the Elephant calls cartesian categories).
An important property is that every cartesian theory has an initial model. It follows that model reduct functors between cartesian theories have left adjoints: broadly, the free algebra constructions supplied by universal algebra for algebraic theories are also available for cartesian theories.
Various definitions and names for the logic can be found in the references. The Elephant definition amounts to saying that a cartesian theory is a geometric theory in which there is no occurrence of $\vee$, and every use of $\exists$ requires a proof that the existence in unique.
Palmgren and Vickers showed that in a geometric logic of partial terms (with a non-reflexive equality), the cartesian theories are precisely those that can be presented using only $\wedge$, $\top$ and $=$.
Cartesian logic was introduced in the early seventies by John Isbell, Peter Freyd and Michel Coste (cf. Johnstone 1979). A standard source is Johnstone (2002). Palmgren and Vickers gave a new proof of the initial model theorem, valid in weak foundational settings.
Jiří Adámek, Jiří Rosický, Locally presentable and accessible categories , Cambridge UP 1994.
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. (Around D1.3.4 p.833)
Erik Palmgren, Steve Vickers, Partial Horn logic and cartesian categories, Ann Pure Appl Logic 145 (3) (2007) pp.314-353.
Last revised on November 16, 2022 at 06:07:02. See the history of this page for a list of all contributions to it.