VIDEOS 1 TO 50

A Crash Course in Formal Logic Pt 5a: Categorical Statements

Published: 2013/07/11

Channel: PhilHelper

Categorical Logic: All A are B

Published: 2013/02/04

Channel: Kevin deLaplante

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

Published: 2010/03/01

Channel: Mark Thorsby

A Crash Course in Formal Logic Pt 6a: Categorical Syllogisms, Terms, Mood and Figure

Published: 2013/07/15

Channel: PhilHelper

Categorical Logic Introduction

Published: 2017/07/23

Channel: xuan-gottfried YANG

Categorical Logic: Some A are B

Published: 2013/02/04

Channel: Kevin deLaplante

Categorical Logic: Only A are B

Published: 2013/02/05

Channel: Kevin deLaplante

Categorical Logic Basics: Part One

Published: 2010/07/19

Channel: Brad Morris

Categorical Syllogisms

Published: 2010/02/26

Channel: Mark Thorsby

categorical logic lecture 1

Published: 2015/10/25

Channel: ProfBoedeker

Categorical Logic: The Mechanics of Categorical Propositions and Immediate Inferences

Published: 2016/03/27

Channel: Symbolic Logic and Argumentation Skills (Critical Thinking)

Venn Diagrams and Testing Validity

Published: 2014/11/12

Channel: Mick Presnell

Major and Minor Premises (Categorical Logic)

Published: 2014/03/07

Channel: Carneades.org

Boole vs Aristotle (Categorical Logic)

Published: 2014/02/22

Channel: Carneades.org

A Crash Course in Formal Logic Pt 5d, Conversion, Obversion and Contraposition

Published: 2013/07/12

Channel: PhilHelper

Categorical Logic

Published: 2014/02/16

Channel: Carneades.org

Categorical Syllogism - Pt. 1 - Where to put the X?

Published: 2012/02/22

Channel: Vernon Martin

Propositional and Categorical Logic

Published: 2014/01/02

Channel: Carneades.org

Propositional Logic: Categorical vs Propositional Logic

Published: 2013/02/04

Channel: Kevin deLaplante

Categorical Logic: The Technology of Valid Argument

Published: 2011/03/06

Channel: Mark McIntire

4.1 The Components of Categorical Propositions

Published: 2012/07/31

Channel: Mark Thorsby

What is CATEGORICAL LOGIC? What does CATEGORICAL LOGIC mean? CATEGORICAL LOGIC meaning & explanation

Published: 2016/10/07

Channel: The Audiopedia

What is Distribution in Categorical Syllogisms?

Published: 2011/09/15

Channel: J Gainor

A Crash Course in Formal Logic Pt 6b: Venn Diagrams

Published: 2013/07/15

Channel: PhilHelper

Categorical Logic: Square of Opposition

Published: 2013/02/04

Channel: Kevin deLaplante

Translating into Standard Form for Categorical Propositions

Published: 2014/04/07

Channel: Symbolic Logic and Argumentation Skills (Critical Thinking)

Subcontrary (Categorical Logic)

Published: 2014/03/03

Channel: Carneades.org

Contraposition (Categorical Logic)

Published: 2014/02/26

Channel: Carneades.org

Obversion (Categorical Logic)

Published: 2014/02/27

Channel: Carneades.org

Venn Diagrams (Part 1) - Power of Logic

Published: 2013/02/21

Channel: Mark Womack

Subalternation (Categorical Logic)

Published: 2014/03/04

Channel: Carneades.org

Conversion (Categorical Logic)

Published: 2014/02/25

Channel: Carneades.org

Categorical Statements

Published: 2017/03/21

Channel: Adam Rosenfeld

Categorical Logic: Venn Diagramming Syllogisms, Part One

Published: 2010/07/19

Channel: Brad Morris

Distribution of Terms in Categorical Claims

Published: 2013/10/31

Channel: Mick Presnell

Categorical Logic - A-claim

Published: 2011/03/14

Channel: Vernon Martin

Categorical Logic - I-claim

Published: 2011/03/14

Channel: Vernon Martin

More on Categorical Logic

Published: 2016/04/17

Channel: Chase Fields

Categorical Logic Basic Symbolisation

Published: 2017/09/24

Channel: Alex Whalen

A Crash Course in Formal Logic Pt 6c: Rules for Categorical Syllogisms

Published: 2013/07/16

Channel: PhilHelper

Let's Practice Testing Categorical Syllogisms for Validity!

Published: 2016/03/27

Channel: Symbolic Logic and Argumentation Skills (Critical Thinking)

Venn Diagrams and Categorical Statements

Published: 2017/03/31

Channel: Michael Gavin

Illicit Conversion (Categorical Logic Fallacy)

Published: 2014/02/25

Channel: Carneades.org

Categorical Logic: Proving Complex Arguments

Published: 2007/08/01

Channel: Kevin Browne

Categorical Logic Validity Part 1

Published: 2017/09/24

Channel: Alex Whalen

Lesson 9 - Categorical Logic

Published: 2017/11/18

Channel: Coastline OLIT

1 Categorical Propositions

Published: 2015/06/15

Channel: Chico Jones

Categorical Propositions Pt 1

Published: 2013/10/03

Channel: Mick Presnell

Olivia Caramello - 2/4 ntroduction to categorical logic, classifying toposes...

Published: 2015/12/04

Channel: Institut des Hautes Études Scientifiques (IHÉS)

**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.

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.

- 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.

- Barr, M. and Wells, C. (1990),
*Category Theory for Computing Science*. Hemel Hempstead, UK. - Lambek, J. and Scott, P.J. (1986),
*Introduction to Higher Order Categorical Logic*. Cambridge University Press, Cambridge, UK. - Lawvere, F.W., and Rosebrugh, R. (2003),
*Sets for Mathematics*. Cambridge University Press, Cambridge, UK. - Lawvere, F.W. (2000), and Schanuel, S.H.,
*Conceptual Mathematics: A First Introduction to Categories*. Cambridge University Press, Cambridge, UK, 1997. Reprinted with corrections, 2000.

**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.

**^**Lawvere, Quantifiers and Sheaves

- 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].

- Categorical Logic lecture notes by Steve Awodey

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