# Logic

logicianlogicallogicsformal logiclogicallyillogicalillogicalogicalanalyticalcoherence
Logic (from the ) is the systematic study of the form of valid inference, and the most general laws of truth.wikipedia
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### Validity (logic)

validityvalidinvalid
Logic (from the ) is the systematic study of the form of valid inference, and the most general laws of truth.
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 cognitive science (encompasses computer science, linguistics, philosophy and psychology).
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, intuitively, refers to the principles of logic and logical thought outside of a formal setting.

### Argument

logical argumentargumentsproof
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.

### Universal logic

However, agreement on what logic actually is has remained elusive, although the field of universal logic has studied the common structure of logics.
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.

### 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 cognitive science (encompasses computer science, linguistics, philosophy and psychology).
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
However, it was not alone: the Stoics proposed a system of propositional logic that was studied by medieval logicians.
Propositional calculus is a branch of logic.

### Aristotle

AristotelianAristotelesAristote
In the Western World, logic was first developed by Aristotle, who called the subject 'analytics'.
His writings cover many subjects – including physics, biology, zoology, metaphysics, logic, ethics, aesthetics, poetry, theatre, music, rhetoric, psychology, linguistics, economics, politics and government.

### Logical consequence

entailmententailsfollows from
Deductive reasoning concerns the logical consequence of given premises and is the form of reasoning most closely connected to logic.
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.

### Syntax

syntacticsyntacticalsyntactically
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 proof and inference, including paradoxes and fallacies, and the study of syntax and semantics.
It became apparent that there was no such thing as the most natural way to express a thought, and therefore logic could no longer be relied upon as a basis for studying the structure of language.

### 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.

### Inference

inferredinferlogical inference
Logic (from the ) is the systematic study of the form of valid inference, and the most general laws of truth. 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 proof and inference, including paradoxes and fallacies, and the study of syntax and semantics.
Deduction is inference deriving logical conclusions from premises known or assumed to be true, with the laws of valid inference being studied in logic.

### Jan Łukasiewicz

ŁukasiewiczJan LukasiewiczJ. Lukasiewicz
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).

### Organon

Aristotelian logicAristotle's logicLogic
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.

### Sum of Logic

Summa LogicaeSumma Totius LogicaeSuma Logicae
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.

### Model theory

modelmodelsmodel-theoretic
Model-theoretic semantics is one of the fundamental concepts of model theory.
:universal algebra + logic = model theory.

### 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

OckhamWilliam OckhamOccam
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.

### Natural language

linguisticnaturalnatural languages
Philosophical logic deals with formal descriptions of ordinary, non-specialist ("natural") language, that is strictly only about the arguments within philosophy's other branches.
They are distinguished from constructed and formal languages such as those used to program computers or to study logic.

### 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.

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 proof and inference, including paradoxes and fallacies, and the study of syntax and semantics.
Paradoxes that arise from apparently intelligible uses of language are often of interest to logicians and philosophers.

### Truth

trueTruth theorytheory of truth
Logic (from the ) is the systematic study of the form of valid inference, and the most general laws of truth.
Logic is concerned with the patterns in reason that can help tell us if a proposition is true or not.

### Walter Burley

BurleyGualterus de BurleyWalter Burleigh
The theory of inference (or 'consequences') was systematically developed in medieval times by logicians such as William of Ockham and Walter Burley.
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

Tarski's theory of truthConvention Tsemantic theory
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.