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 linguistics, a mass noun, uncountable noun, or non-count noun is a noun with the syntactic property that any quantity of it is treated as an undifferentiated unit, rather than as something with discrete elements. Non-count nouns are distinguished from count nouns.
In mathematical logic, a universal quantification is a type of quantifier, a logical constant which is interpreted as "given any" or "for all". 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 formal logic and related branches of mathematics, a functional predicate, or function symbol, is a logical symbol that may be applied to an object term to produce another object term. Functional predicates are also sometimes called mappings, but that term has additional meanings in mathematics. In a model, a function symbol will be modelled by a function.
Understood in a narrow sense, philosophical logic is the area of philosophy that studies the application of logical methods to philosophical problems, often in the form of extended logical systems like modal logic. Some theorists conceive philosophical logic in a wider sense as the study of the scope and nature of logic in general. In this sense, philosophical logic can be seen as identical to the philosophy of logic, which includes additional topics like how to define logic or a discussion of the fundamental concepts of logic. The current article treats philosophical logic in the narrow sense, in which it forms one field of inquiry within the philosophy of logic.
In logic and mathematics, second-order logic is an extension of first-order logic, which itself is an extension of propositional logic. Second-order logic is in turn extended by higher-order logic and type theory.
In logic, philosophy and related fields, mereology is the study of parts and the wholes they form. Whereas set theory is founded on the membership relation between a set and its elements, mereology emphasizes the meronomic relation between entities, which—from a set-theoretic perspective—is closer to the concept of inclusion between sets.
In mathematics and logic, plural quantification is the theory that an individual variable x may take on plural, as well as singular, values. As well as substituting individual objects such as Alice, the number 1, the tallest building in London etc. for x, we may substitute both Alice and Bob, or all the numbers between 0 and 10, or all the buildings in London over 20 stories.
In formal semantics and philosophy of language, a definite description is a denoting phrase in the form of "the X" where X is a noun-phrase or a singular common noun. The definite description is proper if X applies to a unique individual or object. For example: "the first person in space" and "the 42nd President of the United States of America", are proper. The definite descriptions "the person in space" and "the Senator from Ohio" are improper because the noun phrase X applies to more than one thing, and the definite descriptions "the first man on Mars" and "the Senator from some Country" are improper because X applies to nothing. Improper descriptions raise some difficult questions about the law of excluded middle, denotation, modality, and mental content.
The identity of indiscernibles is an ontological principle that states that there cannot be separate objects or entities that have all their properties in common. That is, entities x and y are identical if every predicate possessed by x is also possessed by y and vice versa. It states that no two distinct things can be exactly alike, but this is intended as a metaphysical principle rather than one of natural science. A related principle is the indiscernibility of identicals, discussed below.
Predicate transformer semantics were introduced by Edsger Dijkstra in his seminal paper "Guarded commands, nondeterminacy and formal derivation of programs". They define the semantics of an imperative programming paradigm by assigning to each statement in this language a corresponding predicate transformer: a total function between two predicates on the state space of the statement. In this sense, predicate transformer semantics are a kind of denotational semantics. Actually, in guarded commands, Dijkstra uses only one kind of predicate transformer: the well-known weakest preconditions.
In formal ontology, a branch of metaphysics, and in ontological computer science, mereotopology is a first-order theory, embodying mereological and topological concepts, of the relations among wholes, parts, parts of parts, and the boundaries between parts.
In linguistic semantics, an expression X is said to have cumulative reference if and only if the following holds: If X is true of both of a and b, then it is also true of the combination of a and b. Example: If two separate entities can be said to be "water", then combining them into one entity will yield more "water". If two separate entities can be said to be "a house", their combination cannot be said to be "a house". Hence, "water" has cumulative reference, while the expression "a house" does not. The plural form "houses", however, does have cumulative reference. If two entities are both "houses", then their combination will still be "houses".
In formal semantics, truth-value semantics is an alternative to Tarskian semantics. It has been primarily championed by Ruth Barcan Marcus, H. Leblanc, and M. Dunn and N. Belnap. It is also called the substitution interpretation or substitutional quantification.
In linguistics, telicity is the property of a verb or verb phrase that presents an action or event as having a specific endpoint. A verb or verb phrase with this property is said to be telic, while a verb or verb phrase is said to be atelic if the situation it describes is not heading for any particular endpoint.
In mathematics, point-free geometry is a geometry whose primitive ontological notion is region rather than point. Two axiomatic systems are set out below, one grounded in mereology, the other in mereotopology and known as connection theory. A point can mark a space or objects.
Donkey sentences are sentences that contain a pronoun with clear meaning but whose syntactical role in the sentence poses challenges to grammarians. Such sentences defy straightforward attempts to generate their formal language equivalents. The difficulty is with understanding how English speakers parse such sentences.
According to Rudolf Carnap, in logic, an interpretation is a descriptive interpretation if at least one of the undefined symbols of its formal system becomes, in the interpretation, a descriptive sign. In his Introduction to Semantics he makes a distinction between formal interpretations which are logical interpretations and descriptive interpretations: a formal interpretation is a descriptive interpretation if it is not a logical interpretation.
In logic, a quantifier is an operator that specifies how many individuals in the domain of discourse satisfy an open formula. For instance, the universal quantifier in the first order formula expresses that everything in the domain satisfies the property denoted by . On the other hand, the existential quantifier in the formula expresses that there is something in the domain which satisfies that property. A formula where a quantifier takes widest scope is called a quantified formula. A quantified formula must contain a bound variable and a subformula specifying a property of the referent of that variable.
