nLab cartesian logic

Idea

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.

Definition

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

$$\phi(x),\phi(y) \vdash x = y$$

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 $=$.

Semantics in Categories with Finite Limits

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.

Related concepts

category	functor	internal logic	theory	hyperdoctrine	subobject poset	coverage	classifying topos
finitely complete category	cartesian functor	cartesian logic	essentially algebraic theory
lextensive category		disjunctive logic
regular category	regular functor	regular logic	regular theory	regular hyperdoctrine	infimum-semilattice	regular coverage	regular topos
coherent category	coherent functor	coherent logic	coherent theory	coherent hyperdoctrine	distributive lattice	coherent coverage	coherent topos
geometric category	geometric functor	geometric logic	geometric theory	geometric hyperdoctrine	frame	geometric coverage	Grothendieck topos
Heyting category	Heyting functor	intuitionistic first-order logic	intuitionistic first-order theory	first-order hyperdoctrine	Heyting algebra
De Morgan Heyting category		intuitionistic first-order logic with weak excluded middle			De Morgan Heyting algebra
Boolean category		classical first-order logic	classical first-order theory	Boolean hyperdoctrine	Boolean algebra
star-autonomous category		multiplicative classical linear logic
symmetric monoidal closed category		multiplicative intuitionistic linear logic
cartesian monoidal category		fragment $\{\&, \top\}$ of linear logic
cocartesian monoidal category		fragment $\{\oplus, 0\}$ of linear logic
cartesian closed category		simply typed lambda calculus

References

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.