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A Crash Course in Formal Logic Pt  5a: Categorical Statements
A Crash Course in Formal Logic Pt 5a: Categorical Statements
Published: 2013/07/11
Channel: PhilHelper
Categorical Logic: All A are B
Categorical Logic: All A are B
Published: 2013/02/04
Channel: Kevin deLaplante
Olivia Caramello - 1/4 Introduction to categorical logic, classifying toposes...
Olivia Caramello - 1/4 Introduction to categorical logic, classifying toposes...
Published: 2015/12/03
Channel: Institut des Hautes Études Scientifiques (IHÉS)
5.2 Venn Diagrams and Categorical Syllogisms
5.2 Venn Diagrams and Categorical Syllogisms
Published: 2010/03/01
Channel: Mark Thorsby
Categorical Logic Basics:  Part One
Categorical Logic Basics: Part One
Published: 2010/07/19
Channel: Brad Morris
A Crash Course in Formal Logic Pt  6a: Categorical Syllogisms, Terms, Mood and Figure
A Crash Course in Formal Logic Pt 6a: Categorical Syllogisms, Terms, Mood and Figure
Published: 2013/07/15
Channel: PhilHelper
Boole vs Aristotle (Categorical Logic)
Boole vs Aristotle (Categorical Logic)
Published: 2014/02/22
Channel: Carneades.org
Categorical Syllogisms
Categorical Syllogisms
Published: 2010/02/26
Channel: Mark Thorsby
Venn Diagrams and Testing Validity
Venn Diagrams and Testing Validity
Published: 2014/11/12
Channel: Mick Presnell
Categorical Logic Introduction
Categorical Logic Introduction
Published: 2016/06/14
Channel: Lernvideos und Vorträge
Major and Minor Premises (Categorical Logic)
Major and Minor Premises (Categorical Logic)
Published: 2014/03/07
Channel: Carneades.org
4.1 The Components of Categorical Propositions
4.1 The Components of Categorical Propositions
Published: 2012/07/31
Channel: Mark Thorsby
Propositional Logic: Categorical vs Propositional Logic
Propositional Logic: Categorical vs Propositional Logic
Published: 2013/02/04
Channel: Kevin deLaplante
Obversion (Categorical Logic)
Obversion (Categorical Logic)
Published: 2014/02/27
Channel: Carneades.org
categorical logic lecture 1
categorical logic lecture 1
Published: 2015/10/25
Channel: ProfBoedeker
Propositional and Categorical Logic
Propositional and Categorical Logic
Published: 2014/01/02
Channel: Carneades.org
Categorical Logic: The Mechanics of Categorical Propositions and Immediate Inferences
Categorical Logic: The Mechanics of Categorical Propositions and Immediate Inferences
Published: 2016/03/27
Channel: Symbolic Logic and Argumentation Skills (Critical Thinking)
A Crash Course in Formal Logic Pt  5d, Conversion, Obversion and Contraposition
A Crash Course in Formal Logic Pt 5d, Conversion, Obversion and Contraposition
Published: 2013/07/12
Channel: PhilHelper
Translating into Standard Form for Categorical Propositions
Translating into Standard Form for Categorical Propositions
Published: 2014/04/07
Channel: Symbolic Logic and Argumentation Skills (Critical Thinking)
Categorical Logic
Categorical Logic
Published: 2014/02/16
Channel: Carneades.org
Subalternation (Categorical Logic)
Subalternation (Categorical Logic)
Published: 2014/03/04
Channel: Carneades.org
Conversion (Categorical Logic)
Conversion (Categorical Logic)
Published: 2014/02/25
Channel: Carneades.org
Venn Diagrams (Part 1) - Power of Logic
Venn Diagrams (Part 1) - Power of Logic
Published: 2013/02/21
Channel: Mark Womack
Subcontrary (Categorical Logic)
Subcontrary (Categorical Logic)
Published: 2014/03/03
Channel: Carneades.org
Categorical Logic: Square of Opposition
Categorical Logic: Square of Opposition
Published: 2013/02/04
Channel: Kevin deLaplante
Categorical Propositions Pt 1
Categorical Propositions Pt 1
Published: 2013/10/03
Channel: Mick Presnell
Categorical Logic: Only A are B
Categorical Logic: Only A are B
Published: 2013/02/05
Channel: Kevin deLaplante
What is Distribution in Categorical Syllogisms?
What is Distribution in Categorical Syllogisms?
Published: 2011/09/15
Channel: J Gainor
Contrary (Categorical Logic)
Contrary (Categorical Logic)
Published: 2014/03/02
Channel: Carneades.org
Categorical Syllogism - Pt. 1 - Where to put the X?
Categorical Syllogism - Pt. 1 - Where to put the X?
Published: 2012/02/22
Channel: Vernon Martin
Categorical Logic: Immediate Inferences on the Traditional and Modern Interpretations
Categorical Logic: Immediate Inferences on the Traditional and Modern Interpretations
Published: 2016/03/26
Channel: Symbolic Logic and Argumentation Skills (Critical Thinking)
4.2 Quantity, Quality, & Distribution
4.2 Quantity, Quality, & Distribution
Published: 2012/07/31
Channel: Mark Thorsby
Categorical Logic:  Venn Diagramming Syllogisms, Part One
Categorical Logic: Venn Diagramming Syllogisms, Part One
Published: 2010/07/19
Channel: Brad Morris
Categorical Logic: Some A are B
Categorical Logic: Some A are B
Published: 2013/02/04
Channel: Kevin deLaplante
Categorical Logic: Immediate Inferences, cont.
Categorical Logic: Immediate Inferences, cont.'
Published: 2016/03/27
Channel: Symbolic Logic and Argumentation Skills (Critical Thinking)
What is CATEGORICAL LOGIC? What does CATEGORICAL LOGIC mean? CATEGORICAL LOGIC meaning & explanation
What is CATEGORICAL LOGIC? What does CATEGORICAL LOGIC mean? CATEGORICAL LOGIC meaning & explanation
Published: 2016/10/07
Channel: The Audiopedia
Illicit Contraposition (Categorical Logic Fallacy)
Illicit Contraposition (Categorical Logic Fallacy)
Published: 2014/02/26
Channel: Carneades.org
Olivia Caramello - 2/4 ntroduction to categorical logic, classifying toposes...
Olivia Caramello - 2/4 ntroduction to categorical logic, classifying toposes...
Published: 2015/12/04
Channel: Institut des Hautes Études Scientifiques (IHÉS)
Illicit Conversion (Categorical Logic Fallacy)
Illicit Conversion (Categorical Logic Fallacy)
Published: 2014/02/25
Channel: Carneades.org
Mood and Figure - Power of Logic
Mood and Figure - Power of Logic
Published: 2013/02/21
Channel: Mark Womack
Categorical Logic - A-claim
Categorical Logic - A-claim
Published: 2011/03/14
Channel: Vernon Martin
Categorical Logic Basics:  Part Two
Categorical Logic Basics: Part Two
Published: 2010/07/19
Channel: Brad Morris
Edip Yuksel (E) Logic - Categorical Syllogism
Edip Yuksel (E) Logic - Categorical Syllogism
Published: 2017/04/05
Channel: Edip Yuksel
Categorical Logic: Proving Complex Arguments
Categorical Logic: Proving Complex Arguments
Published: 2007/08/01
Channel: Kevin Browne
A Crash Course in Formal Logic Pt  6c: Rules for Categorical Syllogisms
A Crash Course in Formal Logic Pt 6c: Rules for Categorical Syllogisms
Published: 2013/07/16
Channel: PhilHelper
Categorical Logic: The Technology of Valid Argument
Categorical Logic: The Technology of Valid Argument
Published: 2011/03/06
Channel: Mark McIntire
Edip Yuksel (E) Logic - Categorical Syllogism
Edip Yuksel (E) Logic - Categorical Syllogism
Published: 2013/04/16
Channel: Edip Yuksel
Categorical Propositions Pt. 2
Categorical Propositions Pt. 2
Published: 2013/10/04
Channel: Mick Presnell
categorical logic lecture 2
categorical logic lecture 2
Published: 2015/10/25
Channel: ProfBoedeker
Olivia Caramello - 4/4 ntroduction to categorical logic, classifying toposes...
Olivia Caramello - 4/4 ntroduction to categorical logic, classifying toposes...
Published: 2015/12/08
Channel: Institut des Hautes Études Scientifiques (IHÉS)
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WIKIPEDIA ARTICLE

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Categorical logic is the branch of mathematics in which tools and concepts from category theory are applied to the study of mathematical logic. It is also notable for its connections to theoretical computer science. In broad terms, categorical logic represents both syntax and semantics by a category, and an interpretation by a functor. The categorical framework provides a rich conceptual background for logical and type-theoretic constructions. The subject has been recognisable in these terms since around 1970.

Overview[edit]

There are three important themes in the categorical approach to logic:

Categorical semantics
Categorical logic introduces the notion of structure valued in a category C with the classical model theoretic notion of a structure appearing in the particular case where C is the category of sets and functions. This notion has proven useful when the set-theoretic notion of a model lacks generality and/or is inconvenient. R.A.G. Seely's modeling of various impredicative theories, such as system F is an example of the usefulness of categorical semantics.
It was found that the connectives of pre-categorical logic were more clearly understood using the concept of adjoint functor, and that the quantifiers were also best understood using adjoint functors.[1]
Internal languages
This can be seen as a formalization and generalization of proof by diagram chasing. One defines a suitable internal language naming relevant constituents of a category, and then applies categorical semantics to turn assertions in a logic over the internal language into corresponding categorical statements. This has been most successful in the theory of toposes, where the internal language of a topos together with the semantics of intuitionistic higher-order logic in a topos enables one to reason about the objects and morphisms of a topos "as if they were sets and functions".[citation needed] This has been successful in dealing with toposes that have "sets" with properties incompatible with classical logic. A prime example is Dana Scott's model of untyped lambda calculus in terms of objects that retract onto their own function space. Another is the Moggi–Hyland model of system F by an internal full subcategory of the effective topos of Martin Hyland.
Term-model constructions
In many cases, the categorical semantics of a logic provide a basis for establishing a correspondence between theories in the logic and instances of an appropriate kind of category. A classic example is the correspondence between theories of βη-equational logic over simply typed lambda calculus and Cartesian closed categories. Categories arising from theories via term-model constructions can usually be characterized up to equivalence by a suitable universal property. This has enabled proofs of meta-theoretical properties of some logics by means of an appropriate categorical algebra. For instance, Freyd gave a proof of the existence and disjunction properties of intuitionistic logic this way.

See also[edit]

References[edit]

Books
  • Abramsky, Samson; Gabbay, Dov (2001). Handbook of Logic in Computer Science: Logic and algebraic methods. Oxford: Oxford University Press. ISBN 0-19-853781-6. 
  • Gabbay, Dov (2012). Handbook of the History of Logic: Sets and extensions in the twentieth century. Oxford: Elsevier. ISBN 978-0-444-51621-3. 
  • Kent, Allen; Williams, James G. (1990). Encyclopedia of Computer Science and Technology. New York: Marcel Dekker Inc. ISBN 0-8247-2272-8. 

Seminal papers

  • Lawvere, F.W., Functorial Semantics of Algebraic Theories. In Proceedings of the National Academy of Sciences 50, No. 5 (November 1963), 869-872.
  • Lawvere, F.W., Elementary Theory of the Category of Sets. In Proceedings of the National Academy of Sciences 52, No. 6 (December 1964), 1506-1511.
  • Lawvere, F.W., Quantifiers and Sheaves. In Proceedings of the International Congress on Mathematics (Nice 1970), Gauthier-Villars (1971) 329-334.

Notes[edit]

  1. ^ Lawvere, Quantifiers and Sheaves

Further reading[edit]

  • Michael Makkai and Gonzalo E. Reyes, 1977, First order categorical logic, Springer-Verlag.
  • Lambek, J. and Scott, P. J., 1986. Introduction to Higher Order Categorical Logic. Fairly accessible introduction, but somewhat dated. The categorical approach to higher-order logics over polymorphic and dependent types was developed largely after this book was published.
  • Jacobs, Bart (1999). Categorical Logic and Type Theory. Studies in Logic and the Foundations of Mathematics 141. North Holland, Elsevier. ISBN 0-444-50170-3.  A comprehensive monograph written by a computer scientist; it covers both first-order and higher-order logics, and also polymorphic and dependent types. The focus is on fibred category as universal tool in categorical logic, which is necessary in dealing with polymorphic and dependent types.
  • John Lane Bell (2005) The Development of Categorical Logic. Handbook of Philosophical Logic, Volume 12. Springer. Version available online at John Bell's homepage.
  • Jean-Pierre Marquis and Gonzalo E. Reyes (2012). The History of Categorical Logic 1963–1977. Handbook of the History of Logic: Sets and Extensions in the Twentieth Century, Volume 6, D. M. Gabbay, A. Kanamori & J. Woods, eds., North-Holland, pp. 689–800. A preliminary version is available at [1].

External links[edit]

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