Non-classical logic | Logic in computer science | Constructivism (mathematics) | Systems of formal logic | Intuitionism
Intuitionistic logic, sometimes more generally called constructive logic, refers to systems of symbolic logic that differ from the systems used for classical logic by more closely mirroring the notion of constructive proof. In particular, systems of intuitionistic logic do not assume the law of the excluded middle and double negation elimination, which are fundamental inference rules in classical logic. Formalized intuitionistic logic was originally developed by Arend Heyting to provide a formal basis for L. E. J. Brouwer's programme of intuitionism. From a proof-theoretic perspective, Heyting’s calculus is a restriction of classical logic in which the law of excluded middle and double negation elimination have been removed. Excluded middle and double negation elimination can still be proved for some propositions on a case by case basis, however, but do not hold universally as they do with classical logic. The standard explanation of intuitionistic logic is the BHK interpretation. Several systems of semantics for intuitionistic logic have been studied. One of these semantics mirrors classical Boolean-valued semantics but uses Heyting algebras in place of Boolean algebras. Another semantics uses Kripke models. These, however, are technical means for studying Heyting’s deductive system rather than formalizations of Brouwer’s original informal semantic intuitions. Semantical systems claiming to capture such intuitions, due to offering meaningful concepts of “constructive truth” (rather than merely validity or provability), are Kurt Gödel’s dialectica interpretation, Stephen Cole Kleene’s realizability, Yurii Medvedev’s logic of finite problems, or Giorgi Japaridze’s computability logic. Yet such semantics persistently induce logics properly stronger than Heyting’s logic. Some authors have argued that this might be an indication of inadequacy of Heyting’s calculus itself, deeming the latter incomplete as a constructive logic. (Wikipedia).
Logic: The Structure of Reason
As a tool for characterizing rational thought, logic cuts across many philosophical disciplines and lies at the core of mathematics and computer science. Drawing on Aristotle’s Organon, Russell’s Principia Mathematica, and other central works, this program tracks the evolution of logic, be
From playlist Logic & Philosophy of Mathematics
LambdaConf 2015 - Introduction to Intuitionistic Type Theory Vlad Patryshev
Traditionally, in Computer Science, sets are assumed to be the basis of a type theory, together with Boolean logic. In this version of type theory, we do not need sets or Boolean logic; intuitionism is enough ("no principle of excluded middle required"). The underlying math is Topos Theory
From playlist LambdaConf 2015
Proof synthesis and differential linear logic
Linear logic is a refinement of intuitionistic logic which, viewed as a functional programming language in the sense of the Curry-Howard correspondence, has an explicit mechanism for copying and discarding information. It turns out that, due to these mechanisms, linear logic is naturally r
From playlist Talks
Introduction to Predicate Logic
This video introduces predicate logic. mathispower4u.com
From playlist Symbolic Logic and Proofs (Discrete Math)
Set Theory (Part 18): The Rational Numbers are Countably Infinite
Please feel free to leave comments/questions on the video and practice problems below! In this video, we will show that the rational numbers are equinumerous to the the natural numbers and integers. First, we will go over the standard argument listing out the rational numbers in a table a
From playlist Set Theory by Mathoma
Data structure intuition is something that develops naturally for most software developers. In all languages, we rely heavily on standard containers and collections. Need fast insertion/lookup? Hashmap. Need a sorted data structure that stores unique values? Set. Duplicate values? Multiset
From playlist Software Development
The Ultimate Guide to Propositional Logic for Discrete Mathematics
This is the ultimate guide to propositional logic in discrete mathematics. We cover propositions, truth tables, connectives, syntax, semantics, logical equivalence, translating english to logic, and even logic inferences and logical deductions. 00:00 Propositions 02:47 Connectives 05:13 W
From playlist Discrete Math 1
Arithmetical expressions as natural numbers | Data structures in Mathematics Math Foundations 194
Primitive natural numbers and Hindu Arabic numerals can be pinned down very concretely and precisely. But what about numbers expressed via more elaborate arithmetical expressions, perhaps involving towers of exponents, or hyperoperations? Is there a consistent and logical proper way of set
From playlist Math Foundations
This is a follow up to https://youtu.be/lDhKE2SKF08. In this video we zoom in on Negation and also discuss models such as the 3-valued one for intuitionistic propositional logic. The script I'm using you can find here: https://gist.github.com/Nikolaj-K/1478e66ccc9b7ac2ea565e743c904555
From playlist Logic
Squashing theories into Heyting algebras
This is the first of two videos on Heyting algebra, Tarski-Lindenbaum and negation: https://gist.github.com/Nikolaj-K/1478e66ccc9b7ac2ea565e743c904555 Followup video: https://youtu.be/ws6vCT7ExTY
From playlist Logic
Dale Miller: Focused proof systems
HYBRID EVENT Recorded during the meeting Linear Logic Winter School" the January 28, 2022 by the Centre International de Rencontres Mathématiques (Marseille, France) Filmmaker: Guillaume Hennenfent Find this video and other talks given by worldwide mathematicians on CIRM's Audiovisual
From playlist Logic and Foundations
Damiano Mazza: Heterodox exponential modalities in linear logic
HYBRID EVENT Recorded during the meeting Linear Logic Winter School" the January 28, 2022 by the Centre International de Rencontres Mathématiques (Marseille, France) Filmmaker: Guillaume Hennenfent Find this video and other talks given by worldwide mathematicians on CIRM's Audiovisual
From playlist Logic and Foundations
Discussion - Panel: What is the role of Topos in Information and Communication Technologies?
Mérouane Debbah (Huawei Technologies) Olivia Caramello (Università degli Studi dell'Insubria in Como) Daniel Bennequin (Univ. Paris-Diderot) Thierry Coquand (University of Göteborg) Jean-Claude Belfiore (Huawei France)
From playlist 4th Huawei-IHES Workshop on Mathematical Theories for Information and Communication Technologies
Will Troiani - Introduction to proof nets (Part 1)
In the first of several talks on linear logic and proof nets, building towards the proof of the sequentialisation theorem, Will introduces the sequent calculus of multiplicative linear logic, proof structures and the translation between them. Lecture notes - https://cglseminar.github.io/n
From playlist Computation, Geometry, Logic seminar
Model Theory - part 06 - Quantifiers as Adjoints
In this video we start to talk about how one can view quantifiers as adjoints of certain functors.
From playlist Model Theory
Klaus Mainzer: Constructivity and Computability. Perspectives for Mathematics [...]
Title: Klaus Mainzer: Constructivity and Computability. Perspectives for Mathematics, Computer Science, and Philosophy The lecture was held within the framework of the Hausdorff Trimester Program: Constructive Mathematics. Abstract: Since antiquity, mathematical proofs were realized by
From playlist Workshop: "Constructive Mathematics"
Hugo Herbelin: A constructive proof of dependent choice, compatible with classical logic
The lecture was held within the framework of the Hausdorff Trimester Program: Constructive Mathematics. Abstract: Martin-Löf's type theory has strong existential elimination (dependent sum type) what allows to prove the full axiom of choice. However the theory is intuitionistic. We give
From playlist Workshop: "Constructive Mathematics"
Logic for Programmers: Propositional Logic
Logic is the foundation of all computer programming. In this video you will learn about propositional logic. 🔗Homework: http://www.codingcommanders.com/logic.php 🎥Logic for Programmers Playlist: https://www.youtube.com/playlist?list=PLWKjhJtqVAbmqk3-E3MPFVoWMufdbR4qW 🔗Check out the Cod
From playlist Logic for Programmers