First-order logic | Wikipedia audio article
Channel: wikipedia tts
Date: 2019-10-05

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/First-order_logic


00:03:17 1 Introduction
00:07:07 2 Syntax
00:07:31 2.1 Alphabet
00:09:03 2.1.1 Logical symbols
00:11:16 2.1.2 Non-logical symbols
00:14:47 2.2 Formation rules
00:15:26 2.2.1 Terms
00:16:16 2.2.2 Formulas
00:20:23 2.2.3 Notational conventions
00:23:51 2.3 Free and bound variables
00:27:42 2.4 Example: ordered abelian groups
00:28:25 3 Semantics
00:30:46 3.1 First-order structures
00:34:06 3.2 Evaluation of truth values
00:35:51 3.3 Validity, satisfiability, and logical consequence
00:45:33 3.4 Algebraizations
00:47:06 3.5 First-order theories, models, and elementary classes
00:48:30 3.6 Empty domains
00:50:30 4 Deductive systems
00:52:59 4.1 Rules of inference
00:55:16 4.2 Hilbert-style systems and natural deduction
00:58:35 4.3 Sequent calculus
00:59:51 4.4 Tableaux method
01:01:26 4.5 Resolution
01:02:35 4.6 Provable identities
01:04:21 5 Equality and its axioms
01:08:34 5.1 First-order logic without equality
01:09:32 5.2 Defining equality within a theory
01:10:37 6 Metalogical properties
01:11:06 6.1 Completeness and undecidability
01:11:20 6.2 The Löwenheim–Skolem theorem
01:11:31 6.3 The compactness theorem
01:11:35 6.4 Lindström's theorem
01:13:25 7 Limitations
01:17:08 7.1 Expressiveness
01:17:45 7.2 Formalizing natural languages
01:20:17 8 Restrictions, extensions, and variations
01:21:43 8.1 Restricted languages
01:24:03 8.2 Many-sorted logic
01:25:06 8.3 Additional quantifiers
01:26:17 8.4 Infinitary logics
01:27:19 8.5 Non-classical and modal logics
01:28:02 8.6 Fixpoint logic
01:28:43 8.7 Higher-order logics
01:35:32 9 Automated theorem proving and formal methods
01:38:35 10 See also
01:39:54 11 Notes
01:42:31 12 References
01:44:14 13 External links
01:44:32 Higher-order logics
01:47:05 Automated theorem proving and formal methods
01:49:57 See also



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SUMMARY
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First-order logic—also known as predicate 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" and 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 for those things. Sometimes "theory" is understood in a more formal sense, which is 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 one or both of predicate quantifiers or function quantifiers 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 (all provable statements are true in all models) and complete (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 p ...