"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."
Predicate logic extends propositional logic to include variables, quantifiers, and predicates. It is used in mathematics and computer science to make precise statements.
Propositions and Predicates: The building blocks of predicate logic. Propositions are statements that are either true or false, while predicates are statements that contain variables.
First-Order Logic: A formal system for reasoning about propositions and predicates, including quantifiers like "for all" and "there exists".
Syntax and Semantics: The rules for constructing valid statements in predicate logic and the meanings of those statements.
Validity and Soundness: The concepts of validity and soundness in predicate logic, and how to determine if a statement is valid or sound.
Interpreting Predicate Logic: The process of interpreting statements in predicate logic, including the use of models and truth assignments.
Inference and Deduction: The techniques for making inferences and deductions in predicate logic, such as using the rules of inference and proof strategies.
Translation into Predicate Logic: The process of translating natural language statements into statements in predicate logic.
Predicate Logic Applications: The various ways in which predicate logic can be applied, such as in computer science, linguistics, and mathematics.
Second-Order Logic: An extension of first-order logic that allows for quantification over predicates.
Modal Logic: A system of logic that deals with modalities such as "necessarily", "possibly", and "impossibly".
Propositional logic: Deals with propositions or statements that are either true or false, and Boolean operators like AND, OR, NOT.
Modal logic: Deals with the concepts of necessity and possibility, and introduces operators like "must", "may", "can".
Quantificational logic: Deals with the quantification of variables, and introduces symbols like "for all" (∀) and "there exists" (∃).
Higher-order logic: Allows quantification over functions and predicates, in addition to individual variables.
Intuitionistic logic: A type of logic that rejects the law of excluded middle, and allows only for constructive proofs.
Linear logic: A type of logic that models resource allocation and consumption, and allows for the manipulation of resources without duplication or deletion.
Relevance logic: A type of logic that emphasizes the relevance of premises to conclusions, and allows for the rejection of irrelevant premises.
Fuzzy logic: A type of logic that allows for degrees of truth, and is suitable for dealing with imprecise or uncertain information.
Deontic logic: A type of modal logic that deals with concepts of obligation and permission, and introduces operators like "should", "must", "may not".
Epistemic logic: A type of modal logic that deals with concepts of knowledge and belief, and introduces operators like "knows", "believes", "hypothesizes".
Temporal logic: A type of modal logic that deals with concepts of time and causality, and introduces operators like "before", "after", "causes".
Dynamic logic: A type of logic that deals with the evolution of knowledge and beliefs over time, in response to new information or actions.
Game logic: A type of dynamic logic that models strategic interactions and decision-making between multiple agents.
Non-monotonic logic: A type of logic that allows for the revision of beliefs and assumptions in the light of new evidence, without necessarily discarding old beliefs.
Paraconsistent logic: A type of logic that allows for contradictions to coexist without leading to triviality or inconsistency.
"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, such as set theory, a theory for groups, or a formal theory of arithmetic, 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."
"The term 'theory' is sometimes understood in a more formal sense as just a set of sentences in first-order logic."
"The term '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, 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."
"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, i.e. 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)."