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 first-order theory in which the only connectives used are $=$, $\top$, $\wedge$ and existential quantifiers over provably subterminal formulae, that is, $\exists x:A. \phi$ can only be used if
is provable. Unlike other first-order theories, this side-condition makes the order of presentation of axioms matter, as an axiom might require another in order to prove that a formula involved is subterminal.
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 can be given models interpreting formulae as subobjects in a category with finite limits. See internal logic for more details. The most interesting component of this is the existential quantifiers. As with regular logic, the existential quantifier is interpreted as an image factorization, but the subterminality side condition ensures the image exists even in an arbitrary category with finite limits. A formula $\phi(x)$ with $x:A$ in a context $\Gamma$ is interpreted as a subobject $\phi \to \Gamma \times A$, and the subterminality side condition is satisfied in the internal logic if and only if the first projection $\Gamma \times A \to \Gamma$ is mono, in which case the composition $\phi \to \Gamma \times A \to \Gamma$ is already the inclusion into its image.
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 July 3, 2023 at 19:02:33. See the history of this page for a list of all contributions to it.