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# Contents

## 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.

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.