Contents

# 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 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 $=$.

## 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.

Last revised on April 18, 2019 at 08:30:52. See the history of this page for a list of all contributions to it.