• Corpus ID: 2337874


  author={F. William Lawvere},
The unity of opposites in the title is essentially that between logic and geometry, and there are compelling reasons for maintaining that geometry is the leading aspect. At the same lime, in the present joint work with Myles Tierney there are important influences in the other direction: a Grothendieck " topology " appears most naturally as a modal operator, of the nature " it is locally the case that ", the usual logical operators such as V, 3, => have natural analogues which apply to families… 

Using the internal language of toposes in algebraic geometry

Any scheme has its associated little and big Zariski toposes. These toposes support an internal mathematical language which closely resembles the usual formal language of mathematics, but is “local

The History of Categorical Logic: 1963-1977

A categorical semantics for fuzzy predicate logic

Modes of Adjointness

It seems a worthy enterprise to trace the concept of adjunction back to the origins of the algebraic semantics of modal logic and to make explicit its ubiquity in this branch of mathematics.

Modes of Adjointness

The fact that many modal operators are part of an adjunction is probably folklore since the discovery of adjunctions. On the other hand, the natural idea of a minimal propositional calculus extended

CONSTRUCTIVE KRIPKE SEMANTICS AND REALIZABILITY Our construction is a straightforward adaptation of the Friedman –

What is the truth-value structure of realizability? How can realizability style models be integrated with forcing techniques from Kripke and Beth semantics, and conversely? These questions have

Topological Completeness of First-Order Modal Logic

As McKinsey and Tarski [20] showed, the Stone representation theorem for Boolean algebras extends to algebras with operators to give topological semantics for (classical) propositional modal logic,

Topological Completeness of First-Order Modal Logics

This paper proves the system of full first-order S4 modal logic to be deductively complete with respect to extended topological semantics, and is general enough to also apply to other modal systems.

Realizability Categories

This chapter introduces realizability and category theory and makes a small survey of their intersection and a selection of some later developments.

Generalized Spaces for Constructive Algebra

The purpose of this contribution is to give a coherent account of a particular narrative which links locales, geometric theories, sheaf semantics and constructive commutative algebra. We are hoping



— On contradiction. Where do correct ideas come from? Peking

  • — On contradiction. Where do correct ideas come from? Peking
  • 1966

Equality in Hyperdoctrines and the Comprehension Scheme as an Adjoint Functor

  • LAWVERE. — Adjointness in Foundations (Dialectica) Proc. of A.M. S. Symposium on Pure Math. XVII-Applications of Category Theory
  • 1969