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.
The proof of Gödel's completeness theorem given by Kurt Gödel in his doctoral dissertation of 1929 is not easy to read today; it uses concepts and formalisms that are no longer used and terminology that is often obscure. The version given below attempts to represent all the steps in the proof and all the important ideas faithfully, while restating the proof in the modern language of mathematical logic. This outline should not be considered a rigorous proof of the theorem.
In mathematical logic, a universal quantification is a type of quantifier, a logical constant which is interpreted as "given any", "for all", or "for any". It expresses that a predicate can be satisfied by every member of a domain of discourse. In other words, it is the predication of a property or relation to every member of the domain. It asserts that a predicate within the scope of a universal quantifier is true of every value of a predicate variable.
In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such as Russell's paradox. Today, Zermelo–Fraenkel set theory, with the historically controversial axiom of choice (AC) included, is the standard form of axiomatic set theory and as such is the most common foundation of mathematics. Zermelo–Fraenkel set theory with the axiom of choice included is abbreviated ZFC, where C stands for "choice", and ZF refers to the axioms of Zermelo–Fraenkel set theory with the axiom of choice excluded.
In logic, negation, also called the logical not or logical complement, is an operation that takes a proposition to another proposition "not ", standing for " is not true", written , or . It is interpreted intuitively as being true when is false, and false when is true. Negation is thus a unary logical connective. It may be applied as an operation on notions, propositions, truth values, or semantic values more generally. In classical logic, negation is normally identified with the truth function that takes truth to falsity. In intuitionistic logic, according to the Brouwer–Heyting–Kolmogorov interpretation, the negation of a proposition is the proposition whose proofs are the refutations of .
A formula of the predicate calculus is in prenex normal form (PNF) if it is written as a string of quantifiers and bound variables, called the prefix, followed by a quantifier-free part, called the matrix. Together with the normal forms in propositional logic, it provides a canonical normal form useful in automated theorem proving.
The theory of descriptions is the philosopher Bertrand Russell's most significant contribution to the philosophy of language. It is also known as Russell's theory of descriptions. In short, Russell argued that the syntactic form of descriptions is misleading, as it does not correlate their logical and/or semantic architecture. While descriptions may seem like fairly uncontroversial phrases, Russell argued that providing a satisfactory analysis of the linguistic and logical properties of a description is vital to clarity in important philosophical debates, particularly in semantic arguments, epistemology and metaphysical elements.
In linguistics, focus is a grammatical category that conveys which part of the sentence contributes new, non-derivable, or contrastive information. In the English sentence "Mary only insulted BILL", focus is expressed prosodically by a pitch accent on "Bill" which identifies him as the only person Mary insulted. By contrast, in the sentence "Mary only INSULTED Bill", the verb "insult" is focused and thus expresses that Mary performed no other actions towards Bill. Focus is a cross-linguistic phenomenon and a major topic in linguistics. Research on focus spans numerous subfields including phonetics, syntax, semantics, pragmatics, and sociolinguistics.
A free logic is a logic with fewer existential presuppositions than classical logic. Free logics may allow for terms that do not denote any object. Free logics may also allow models that have an empty domain. A free logic with the latter property is an inclusive logic.
Tarski's axioms are an axiom system for Euclidean geometry, specifically for that portion of Euclidean geometry that is formulable in first-order logic with identity. As such, it does not require an underlying set theory. The only primitive objects of the system are "points" and the only primitive predicates are "betweenness" and "congruence". The system contains infinitely many axioms.
In the philosophy of language, the descriptivist theory of proper names is the view that the meaning or semantic content of a proper name is identical to the descriptions associated with it by speakers, while their referents are determined to be the objects that satisfy these descriptions. Bertrand Russell and Gottlob Frege have both been associated with the descriptivist theory, which is sometimes called the mediated reference theory or Frege–Russell view.
"On Denoting" is an essay by Bertrand Russell. It was published in the philosophy journal Mind in 1905. In it, Russell introduces and advocates his theory of denoting phrases, according to which definite descriptions and other "denoting phrases ... never have any meaning in themselves, but every proposition in whose verbal expression they occur has a meaning." This theory later became the basis for Russell's descriptivism with regard to proper names, and his view that proper names are "disguised" or "abbreviated" definite descriptions.
In the study of formal theories in mathematical logic, bounded quantifiers are often included in a formal language in addition to the standard quantifiers "∀" and "∃". Bounded quantifiers differ from "∀" and "∃" in that bounded quantifiers restrict the range of the quantified variable. The study of bounded quantifiers is motivated by the fact that determining whether a sentence with only bounded quantifiers is true is often not as difficult as determining whether an arbitrary sentence is true.
Markov's principle, named after Andrey Markov Jr, is a conditional existence statement for which there are many equivalent formulations, as discussed below.
In semantics, donkey sentences are sentences that contain a pronoun with clear meaning but whose syntactic role in the sentence poses challenges to linguists. Such sentences defy straightforward attempts to generate their formal language equivalents. The difficulty is with understanding how English speakers parse such sentences.
Karel Lambert is an American philosopher and logician at the University of California, Irvine and the University of Salzburg. He has written extensively on the subject of free logic, a term which he coined.
Dynamic semantics is a framework in logic and natural language semantics that treats the meaning of a sentence as its potential to update a context. In static semantics, knowing the meaning of a sentence amounts to knowing when it is true; in dynamic semantics, knowing the meaning of a sentence means knowing "the change it brings about in the information state of anyone who accepts the news conveyed by it." In dynamic semantics, sentences are mapped to functions called context change potentials, which take an input context and return an output context. Dynamic semantics was originally developed by Irene Heim and Hans Kamp in 1981 to model anaphora, but has since been applied widely to phenomena including presupposition, plurals, questions, discourse relations, and modality.
Computable topology is a discipline in mathematics that studies the topological and algebraic structure of computation. Computable topology is not to be confused with algorithmic or computational topology, which studies the application of computation to topology.
In mathematical logic, the spectrum of a sentence is the set of natural numbers occurring as the size of a finite model in which a given sentence is true. By a result in descriptive complexity, a set of natural numbers is a spectrum if and only if it can be recognized in non-deterministic exponential time.
