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In logic, a logical connective is a logical constant. Connectives can be used to connect logical formulas. For instance in the syntax of propositional logic, the binary connective can be used to join the two atomic formulas and , rendering the complex formula .
Propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. It deals with propositions and relations between propositions, including the construction of arguments based on them. Compound propositions are formed by connecting propositions by logical connectives. Propositions that contain no logical connectives are called atomic propositions.
In logic and proof theory, natural deduction is a kind of proof calculus in which logical reasoning is expressed by inference rules closely related to the "natural" way of reasoning. This contrasts with Hilbert-style systems, which instead use axioms as much as possible to express the logical laws of deductive reasoning.
Deductive reasoning is the mental process of drawing deductive inferences. An inference is deductively valid if its conclusion follows logically from its premises, i.e. it is impossible for the premises to be true and the conclusion to be false.
In mathematical logic, sequent calculus is a style of formal logical argumentation in which every line of a proof is a conditional tautology instead of an unconditional tautology. Each conditional tautology is inferred from other conditional tautologies on earlier lines in a formal argument according to rules and procedures of inference, giving a better approximation to the natural style of deduction used by mathematicians than to David Hilbert's earlier style of formal logic, in which every line was an unconditional tautology. More subtle distinctions may exist; for example, propositions may implicitly depend upon non-logical axioms. In that case, sequents signify conditional theorems in a first-order language rather than conditional tautologies.
In mathematical logic, a sequent is a very general kind of conditional assertion.
Understood in a narrow sense, philosophical logic is the area of logic 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.
Metalogic is the study of the metatheory of logic. Whereas logic studies how logical systems can be used to construct valid and sound arguments, metalogic studies the properties of logical systems. Logic concerns the truths that may be derived using a logical system; metalogic concerns the truths that may be derived about the languages and systems that are used to express truths.
Robert Boyce Brandom is an American philosopher who teaches at the University of Pittsburgh. He works primarily in philosophy of language, philosophy of mind and philosophical logic, and his academic output manifests both systematic and historical interests in these topics. His work has presented "arguably the first fully systematic and technically rigorous attempt to explain the meaning of linguistic items in terms of their socially norm-governed use, thereby also giving a non-representationalist account of the intentionality of thought and the rationality of action as well."
In proof theory, ludics is an analysis of the principles governing inference rules of mathematical logic. Key features of ludics include notion of compound connectives, using a technique known as focusing or focalisation, and its use of locations or loci over a base instead of propositions.
In logic, the semantics of logic or formal semantics is the study of the semantics, or interpretations, of formal languages and natural languages usually trying to capture the pre-theoretic notion of logical consequence.
Inferential role semantics is an approach to the theory of meaning that identifies the meaning of an expression with its relationship to other expressions, in contradistinction to denotationalism, according to which denotations are the primary sort of meaning.
The cut-elimination theorem is the central result establishing the significance of the sequent calculus. It was originally proved by Gerhard Gentzen in his landmark 1934 paper "Investigations in Logical Deduction" for the systems LJ and LK formalising intuitionistic and classical logic respectively. The cut-elimination theorem states that any judgement that possesses a proof in the sequent calculus making use of the cut rule also possesses a cut-free proof, that is, a proof that does not make use of the cut rule.
Proof-theoretic semantics is an approach to the semantics of logic that attempts to locate the meaning of propositions and logical connectives not in terms of interpretations, as in Tarskian approaches to semantics, but in the role that the proposition or logical connective plays within a system of inference.
Semantic holism is a theory in the philosophy of language to the effect that a certain part of language, be it a term or a complete sentence, can only be understood through its relations to a larger segment of language. There is substantial controversy, however, as to exactly what the larger segment of language in question consists of. In recent years, the debate surrounding semantic holism, which is one among the many forms of holism that are debated and discussed in contemporary philosophy, has tended to centre on the view that the "whole" in question consists of an entire language.
In philosophy—more specifically, in its sub-fields semantics, semiotics, philosophy of language, metaphysics, and metasemantics—meaning "is a relationship between two sorts of things: signs and the kinds of things they intend, express, or signify".
Logic is the formal science of using reason and is considered a branch of both philosophy and mathematics and to a lesser extent computer science. Logic investigates and classifies the structure of statements and arguments, both through the study of formal systems of inference and the study of arguments in natural language. The scope of logic can therefore be very large, ranging from core topics such as the study of fallacies and paradoxes, to specialized analyses of reasoning such as probability, correct reasoning, and arguments involving causality. One of the aims of logic is to identify the correct and incorrect inferences. Logicians study the criteria for the evaluation of arguments.
Logical consequence is a fundamental concept in logic which describes the relationship between statements that hold true when one statement logically follows from one or more statements. A valid logical argument is one in which the conclusion is entailed by the premises, because the conclusion is the consequence of the premises. The philosophical analysis of logical consequence involves the questions: In what sense does a conclusion follow from its premises? and What does it mean for a conclusion to be a consequence of premises? All of philosophical logic is meant to provide accounts of the nature of logical consequence and the nature of logical truth.
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or logical truths. It studies how conclusions follow from premises due to the structure of arguments alone, independent of their topic and content. Informal logic is associated with informal fallacies, critical thinking, and argumentation theory. It examines arguments expressed in natural language while formal logic uses formal language. When used as a countable noun, the term "a logic" refers to a logical formal system that articulates a proof system. Logic plays a central role in many fields, such as philosophy, mathematics, computer science, and linguistics.
References
- Arthur Prior, "The runabout inference ticket." Analysis , 21, pp. 38–39, 1960–61.
- Nuel D. Belnap Jr., "Tonk, Plonk, and Plink", Analysis, 22, pp. 130–134, 1961–62.
- Michael Dummett, The Logical Basis of Metaphysics (Harvard University Press, 1991)