Logic

logicianlogicallogicsformal logicillogicallogicallyConundrumillogicalogicalanalytical
Logic (from the ), originally meaning "the word" or "what is spoken", but coming to mean "thought" or "reason", is a subject concerned with the most general laws of truth, and is now generally held to consist of the systematic study of the form of valid inference.wikipedia
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Validity (logic)

validityvalidinvalid
Logic (from the ), originally meaning "the word" or "what is spoken", but coming to mean "thought" or "reason", is a subject concerned with the most general laws of truth, and is now generally held to consist of the systematic study of the form of valid inference.
In logic, an argument is valid if and only if it takes a form that makes it impossible for the premises to be true and the conclusion nevertheless to be false.

Mathematics

mathematicalmathmathematician
Historically, logic has been studied in philosophy (since ancient times) and mathematics (since the mid-19th century), and recently logic has been studied in computer science, linguistics, psychology, and other fields.
Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, and the systematic study of the shapes and motions of physical objects.

Informal logic

informal reasoninginformal
Informal logic is the study of natural language arguments. The study of fallacies is an important branch of informal logic. Since much informal argument is not strictly speaking deductive, on some conceptions of logic, informal logic is not logic at all. See 'Rival conceptions', below.
Informal logic, intuitively, refers to the principles of logic and logical thought outside of a formal setting.

Universal logic

However, agreement on what logic is has remained elusive, and although the field of universal logic has studied the common structure of logics, in 2007 Mossakowski et al. commented that "it is embarrassing that there is no widely acceptable formal definition of 'a logic'".
Universal logic is the field of logic that studies the common features of all logical systems, aiming to be to logic what universal algebra is to algebra.

Argument

argumentslogical argumentproof
Informal logic is the study of natural language arguments. The study of fallacies is an important branch of informal logic. Since much informal argument is not strictly speaking deductive, on some conceptions of logic, informal logic is not logic at all. See 'Rival conceptions', below.
In logic and philosophy, an argument is a series of statements (in a natural language), called the premises or premisses (both spellings are acceptable), intended to determine the degree of truth of another statement, the conclusion.

Philosophy

philosophicalphilosopherhistory of philosophy
Historically, logic has been studied in philosophy (since ancient times) and mathematics (since the mid-19th century), and recently logic has been studied in computer science, linguistics, psychology, and other fields.
Major sub-fields of academic philosophy include metaphysics ("concerned with the fundamental nature of reality and being"), epistemology (about the "nature and grounds of knowledge [and]...its limits and validity" ), ethics, aesthetics, political philosophy, logic and philosophy of science.

Propositional calculus

propositional logicpropositionalsentential logic
Symbolic logic is the study of symbolic abstractions that capture the formal features of logical inference. Symbolic logic is often divided into two main branches: propositional logic and predicate logic.
Propositional calculus is a branch of logic.

Aristotle

AristotelianAristotelianismAristote
Formal logic is the study of inference with purely formal content. An inference possesses a purely formal content if it can be expressed as a particular application of a wholly abstract rule, that is, a rule that is not about any particular thing or property. The works of Aristotle contain the earliest known formal study of logic. Modern formal logic follows and expands on Aristotle. In many definitions of logic, logical inference and inference with purely formal content are the same. This does not render the notion of informal logic vacuous, because no formal logic captures all of the nuances of natural language.
His writings cover many subjects – including physics, biology, zoology, metaphysics, logic, ethics, aesthetics, poetry, theatre, music, rhetoric, psychology, linguistics, economics, politics and government – and constitute the first comprehensive system of Western philosophy.

Logical consequence

entailsentailmentfollows from
Formal logic is the study of inference with purely formal content. An inference possesses a purely formal content if it can be expressed as a particular application of a wholly abstract rule, that is, a rule that is not about any particular thing or property. The works of Aristotle contain the earliest known formal study of logic. Modern formal logic follows and expands on Aristotle. In many definitions of logic, logical inference and inference with purely formal content are the same. This does not render the notion of informal logic vacuous, because no formal logic captures all of the nuances of natural language.
Logical consequence (also entailment) 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.

Logical conjunction

conjunctionANDlogical AND
It requires, first, ignoring those grammatical features irrelevant to logic (such as gender and declension, if the argument is in Latin), replacing conjunctions irrelevant to logic (such as "but") with logical conjunctions like "and" and replacing ambiguous, or alternative logical expressions ("any", "every", etc.) with expressions of a standard type (such as "all", or the universal quantifier ∀).
In logic, mathematics and linguistics, And is the truth-functional operator of logical conjunction; the and of a set of operands is true if and only if all of its operands are true.

Jan Łukasiewicz

ŁukasiewiczJ. LukasiewiczJan '''Ł'''ukasiewicz
Aristotle uses variable letters to represent valid inferences in Prior Analytics, leading Jan Łukasiewicz to say that the introduction of variables was "one of Aristotle's greatest inventions".
Jan Łukasiewicz (21 December 1878 – 13 February 1956) was a Polish logician and philosopher born in Lemberg, a city in the Galician kingdom of Austria-Hungary (now Lviv, Ukraine).

Inference

inferredinferlogical inference
Logic (from the ), originally meaning "the word" or "what is spoken", but coming to mean "thought" or "reason", is a subject concerned with the most general laws of truth, and is now generally held to consist of the systematic study of the form of valid inference. Formal logic is the study of inference with purely formal content. An inference possesses a purely formal content if it can be expressed as a particular application of a wholly abstract rule, that is, a rule that is not about any particular thing or property. The works of Aristotle contain the earliest known formal study of logic. Modern formal logic follows and expands on Aristotle. In many definitions of logic, logical inference and inference with purely formal content are the same. This does not render the notion of informal logic vacuous, because no formal logic captures all of the nuances of natural language. There is no universal agreement as to the exact scope and subject matter of logic (see, below), but it has traditionally included the classification of arguments, the systematic exposition of the 'logical form' common to all valid arguments, the study of inference, including fallacies, and the study of semantics, including paradoxes.
Deduction is inference deriving logical conclusions from premises known or assumed to be true, with the laws of valid inference being studied in logic.

Organon

Aristotelian logicAristotelianLogic
Aristotle's Organon, especially On Interpretation, gives a cursory outline of semantics which the scholastic logicians, particularly in the thirteenth and fourteenth century, developed into a complex and sophisticated theory, called Supposition Theory.
The Organon (Greek: Ὄργανον, meaning "instrument, tool, organ") is the standard collection of Aristotle's six works on logic.

Model theory

modelmodelsmodel-theoretic
Mathematical logic is an extension of symbolic logic into other areas, in particular to the study of model theory, proof theory, set theory, and recursion theory.
:universal algebra + logic = model theory.

Sum of Logic

Summa LogicaeSumma Totius Logicaetextbooks
For example, in part II of his Summa Logicae, William of Ockham presents a comprehensive account of the necessary and sufficient conditions for the truth of simple sentences, in order to show which arguments are valid and which are not.
The Summa Logicae ("Sum of Logic") is a textbook on logic by William of Ockham.

Alfred Tarski

TarskiTarski, AlfredTarskian
The main modern approach is model-theoretic semantics, based on Alfred Tarski's semantic theory of truth.
Alfred Tarski (January 14, 1901 – October 26, 1983), born Alfred Teitelbaum, was a Polish-American logician and mathematician of Polish-Jewish descent.

William of Ockham

OckhamOccamWilliam of Occam
For example, in part II of his Summa Logicae, William of Ockham presents a comprehensive account of the necessary and sufficient conditions for the truth of simple sentences, in order to show which arguments are valid and which are not. The theory of inference (or 'consequences') was systematically developed in medieval times by logicians such as William of Ockham and Walter Burley.
He is commonly known for Occam's razor, the methodological principle that bears his name, and also produced significant works on logic, physics, and theology.

Structural proof theory

display logicstructural proof theories
Modern semantics also admits rival approaches, such as the proof-theoretic semantics that associates the meaning of propositions with the roles that they can play in inferences, an approach that ultimately derives from the work of Gerhard Gentzen on structural proof theory and is heavily influenced by Ludwig Wittgenstein's later philosophy, especially his aphorism "meaning is use".
In logic, structural proof theory is the subdiscipline of proof theory that studies proof calculi that support a notion of analytic proof.

Natural language

linguisticnaturalnatural languages
Informal logic is the study of natural language arguments. The study of fallacies is an important branch of informal logic. Since much informal argument is not strictly speaking deductive, on some conceptions of logic, informal logic is not logic at all. See 'Rival conceptions', below.
They are distinguished from constructed and formal languages such as those used to program computers or to study logic.

Truth

truetheory of truthtruth theory
Logic (from the ), originally meaning "the word" or "what is spoken", but coming to mean "thought" or "reason", is a subject concerned with the most general laws of truth, and is now generally held to consist of the systematic study of the form of valid inference.
Logic is concerned with the patterns in reason that can help tell us if a proposition is true or not.

Gerhard Gentzen

GentzenDr Gerhard GentzenGentzen, Gerhard
Modern semantics also admits rival approaches, such as the proof-theoretic semantics that associates the meaning of propositions with the roles that they can play in inferences, an approach that ultimately derives from the work of Gerhard Gentzen on structural proof theory and is heavily influenced by Ludwig Wittgenstein's later philosophy, especially his aphorism "meaning is use".
Gerhard Karl Erich Gentzen (November 24, 1909 – August 4, 1945) was a German mathematician and logician.

Paradox

paradoxesparadoxicallogical paradox
There is no universal agreement as to the exact scope and subject matter of logic (see, below), but it has traditionally included the classification of arguments, the systematic exposition of the 'logical form' common to all valid arguments, the study of inference, including fallacies, and the study of semantics, including paradoxes.
Paradoxes that arise from apparently intelligible uses of language are often of interest to logicians and philosophers.

Walter Burley

BurleyWalter BurleighWalter of Burley
The theory of inference (or 'consequences') was systematically developed in medieval times by logicians such as William of Ockham and Walter Burley.
Walter Burley (or Burleigh) (c. 1275–1344/5) was a medieval English scholastic philosopher and logician with at least 50 works attributed to him.

Axiom

axiomspostulateaxiomatic
As an example, Kurt Gödel's incompleteness theorems show that sufficiently complex formal systems of arithmetic cannot be consistent and complete; however, first-order predicate logics not extended by specific axioms to be arithmetic formal systems with equality can be complete and consistent.
As used in modern logic, an axiom is a premise or starting point for reasoning.

Semantic theory of truth

Convention Tsemantic theoryTarski's theory of truth
The main modern approach is model-theoretic semantics, based on Alfred Tarski's semantic theory of truth.
The semantic conception of truth, which is related in different ways to both the correspondence and deflationary conceptions, is due to work published by Polish logician Alfred Tarski in the 1930s.