nLab
logic

Idea

As a discipline, logic is the study of methods of reasoning. While in the past (and often today in philosophical circles), this discipline was prescriptive (describing how one should reason), it is increasingly (and usually in mathematical circles) descriptive (describing how one does reason).

As a (potentially) mathematical object, a logic is a specific method of reasoning. There are several ways to formalise this which for the moment we will not to get into, but maybe later.

Category-theoretic logic

It turns out that logic is naturally expressed in the language of category theory: given any category $C$, its objects $T$ may be understood as types of terms, and morphisms $\psi \stackrel{\Psi }{\to }T$ as terms (generalized elements of $T$). A proposition is then a subobject, $\varphi \hookrightarrow T$ – the subobject of all those variables for which the proposition is true.

In summary:

• type theory deals with understanding categories in terms of their slice categories, collected in their codomain fibration;

• logic deals with understanding subobjects, hence subterminal objects of these slice categories.

For instance,

• limits and colimits, exponentials, and object classifiers belong to the type theory,

• while images, dual images, intersections, unions, and subobject classifiers belong to the logic.

See also:

• internal language,
• syntactic category.

Entries on logic

• axiom of choice

• Boolean algebra

• Boolean algebra with operators

• boolean domain

• boolean function

• boolean-valued function

• classical logic

• constructive mathematics

• context

• equality

• evil

• excluded middle

• geometric logic

• Heyting algebra

• higher-order logic

• internal logic

• intuitionistic logic

• linear logic

• minimal logic

• modal logic

  • algebraic models for modal logics

    • closure algebra
    • monadic algebra
    • temporal algebra

  • arrow logic

  • epistemic logic

    • the logic K(m)
    • the logic T(m)
    • the logic S4(m)
    • the logic S5(m)

  • frame (modal logic)

  • geometric models for modal logics

  • temporal logic

• negation

• Peirce's law

• predicate

• predicate logic

• predicative mathematics

• propositional logic

• relation

• truth value

• type

• type theory

Related concepts

• type theory, logic

• 2-type theory, 2-logic

• (∞,1)-type theory, (∞,1)-logic