Predicate logic | Model theory | Systems of formal logic
First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantified variables over non-logical objects, and allows the use of sentences that contain variables, so that rather than propositions such as "Socrates is a man", one can have expressions in the form "there exists x such that x is Socrates and x is a man", where "there exists" is a quantifier, while x is a variable. This distinguishes it from propositional logic, which does not use quantifiers or relations; in this sense, propositional logic is the foundation of first-order logic. A theory about a topic is usually a first-order logic together with a specified domain of discourse (over which the quantified variables range), finitely many functions from that domain to itself, finitely many predicates defined on that domain, and a set of axioms believed to hold about them. Sometimes, "theory" is understood in a more formal sense as just a set of sentences in first-order logic. The adjective "first-order" distinguishes first-order logic from higher-order logic, in which there are predicates having predicates or functions as arguments, or in which quantification over predicates or functions, or both, are permitted. In first-order theories, predicates are often associated with sets. In interpreted higher-order theories, predicates may be interpreted as sets of sets. There are many deductive systems for first-order logic which are both sound (i.e., all provable statements are true in all models) and complete (i.e. all statements which are true in all models are provable). Although the logical consequence relation is only semidecidable, much progress has been made in automated theorem proving in first-order logic. First-order logic also satisfies several metalogical theorems that make it amenable to analysis in proof theory, such as the Löwenheim–Skolem theorem and the compactness theorem. First-order logic is the standard for the formalization of mathematics into axioms, and is studied in the foundations of mathematics.Peano arithmetic and Zermelo–Fraenkel set theory are axiomatizations of number theory and set theory, respectively, into first-order logic.No first-order theory, however, has the strength to uniquely describe a structure with an infinite domain, such as the natural numbers or the real line. Axiom systems that do fully describe these two structures (that is, categorical axiom systems) can be obtained in stronger logics such as second-order logic. The foundations of first-order logic were developed independently by Gottlob Frege and Charles Sanders Peirce. For a history of first-order logic and how it came to dominate formal logic, see José Ferreirós (2001). (Wikipedia).
Differential Equations: Linear First Order DEs Introduction
The second of the three analytic methods for solving first order differential equations is only valid if the differential equation is linear. In this video, we look at what it means for a differential equation to be linear and how it can then be solved.
From playlist Differential Equations
Lars Kristiansen: First order concatenation theory vs first order number theory
The lecture was held within the framework of the Hausdorff Trimester Program: Types, Sets and Constructions. Abstract: First-order concatenation theory can be compared to first-order number theory, e.g., Peano Arithmetic or Robinson Arithmetic. The universe of a standard structure for fir
From playlist Workshop: "Proofs and Computation"
Differential Equations: First Order Linear Example 2
We present a solution to a first order linear differential equation.
From playlist First Order Linear Differential Equations
Differential Equations: First Order Linear Example 1
We present a solution to a first order linear differential equation.
From playlist First Order Linear Differential Equations
Differential Equations: First Order Linear Example 3
We present a solution to a first order linear differential equation.
From playlist First Order Linear Differential Equations
Differential Equations | First Order Linear System of DEs.
We solve a nonhomogeneous system of first order linear differential equations using a strategy inspired from solving a single first order linear differential equation. http://www.michael-penn.net http://www.randolphcollege.edu/mathematics/
From playlist Systems of Differential Equations
Find the Interval That a Linear First Order Differential Equation Has a Unique Solution
This video explains how to determine the interval that a first order differential equation initial value problem would have a unique solution. Library: http://mathispower4u.com Search: http://mathispower4u.wordpress.com
From playlist Introduction to Differential Equations
Nicole Schweikardt: Databases and descriptive complexity – lecture 2
Recording during the meeting "Spring school on Theoretical Computer Science (EPIT) - Databases, Logic and Automata " the April 11, 2019 at the Centre International de Rencontres Mathématiques (Marseille, France) Filmmaker: Guillaume Hennenfent Find this video and other talks given by wor
From playlist Numerical Analysis and Scientific Computing
Logic 7 - First Order Logic | Stanford CS221: AI (Autumn 2021)
For more information about Stanford's Artificial Intelligence professional and graduate programs visit: https://stanford.io/ai Associate Professor Percy Liang Associate Professor of Computer Science and Statistics (courtesy) https://profiles.stanford.edu/percy-liang Assistant Professor
From playlist Stanford CS221: Artificial Intelligence: Principles and Techniques | Autumn 2021
Logic 2 - First-order Logic | Stanford CS221: AI (Autumn 2019)
For more information about Stanford’s Artificial Intelligence professional and graduate programs, visit: https://stanford.io/3bg9F0C Topics: First-order Logic Percy Liang, Associate Professor & Dorsa Sadigh, Assistant Professor - Stanford University http://onlinehub.stanford.edu/ Associa
From playlist Stanford CS221: Artificial Intelligence: Principles and Techniques | Autumn 2019
Diego Figueira: Semistructured data, Logic, and Automata – lecture 2
Semistructured data is an umbrella term encompassing data models which are not logically organized in tables (i.e., the relational data model) but rather in hierarchical structures using markers such as tags to separate semantic elements and data fields in a ‘self-describing’ way. In this
From playlist Logic and Foundations
Logic 1 - Overview: Logic Based Models | Stanford CS221: AI (Autumn 2021)
For more information about Stanford's Artificial Intelligence professional and graduate programs visit: https://stanford.io/ai This lecture covers logic-based models: propositional logic, first order logic Applications: theorem proving, verification, reasoning, think in terms of logical f
From playlist Stanford CS221: Artificial Intelligence: Principles and Techniques | Autumn 2021
Gödel's Incompleteness Theorems: An Informal Introduction to Formal Logic #SoME2
My entry into SoME2. Also, my first ever video. I hope you enjoy. The Book List: Logic by Paul Tomassi A very good first textbook. Quite slow at first and its treatment of first-order logic leaves a little to be desired in my opinion, but very good on context, i.e. why formal logic is im
From playlist Summer of Math Exposition 2 videos
algebraic geometry 30 The Ax Grothendieck theorem
This lecture is part of an online algebraic geometry course, based on chapter I of "Algebraic geometry" by Hartshorne. It covers the Ax-Grothendieck theorem, which states that an injective regular map between varieties is surjective. The proof uses a strange technique: first prove the resu
From playlist Algebraic geometry I: Varieties
Thomas Colcombet : Algebra vs Logic over (generalised) words
CONFERENCE Recording during the thematic meeting : « Discrete mathematics and logic: between mathematics and the computer science » the January 17, 2023 at the Centre International de Rencontres Mathématiques (Marseille, France) Filmmaker: Jean Petit Find this video and other talks give
From playlist Logic and Foundations
Differential Equations: First Order Linear
We derive the solution to an arbitrary first order linear differential equation.
From playlist First Order Linear Differential Equations