Logic

[2] Abductive arguments are inferences to the best explanation, for example, when a doctor concludes that a patient has a certain disease which explains the symptoms they suffer.

[4] Logic is traditionally defined as the study of the laws of thought or correct reasoning,[5] and is usually understood in terms of inferences or arguments.

[12] For example, modus ponens is a rule of inference according to which all arguments of the form "(1) p, (2) if p then q, (3) therefore q" are valid, independent of what the terms p and q stand for.

[33] Another approach defines informal logic in a wide sense as the normative study of the standards, criteria, and procedures of argumentation.

Further approaches focus on the discussion of logical topics with or without formal devices and on the role of epistemology for the assessment of arguments.

But this approach comes with new problems of its own: sentences are often context-dependent and ambiguous, meaning an argument's validity would not only depend on its parts but also on its context and on how it is interpreted.

[65] According to an influential view by Alfred Tarski, deductive arguments have three essential features: (1) they are formal, i.e. they depend only on the form of the premises and the conclusion; (2) they are a priori, i.e. no sense experience is needed to determine whether they obtain; (3) they are modal, i.e. that they hold by logical necessity for the given propositions, independent of any other circumstances.

[70] The third feature can be expressed by stating that deductively valid inferences are truth-preserving: it is impossible for the premises to be true and the conclusion to be false.

This characteristic is closely related to non-monotonicity and defeasibility: it may be necessary to retract an earlier conclusion upon receiving new information or in light of new inferences drawn.

As a consequence, the line between correct and incorrect arguments is blurry in some cases, such as when the premises offer weak but non-negligible support.

[83] For example, given the premise that there is a plate with breadcrumbs in the kitchen in the early morning, one may infer the conclusion that one's house-mate had a midnight snack and was too tired to clean the table.

For example, the conclusion that a burglar broke into the house last night, got hungry on the job, and had a midnight snack, would also explain the state of the kitchen.

Some theorists, like John Stuart Mill, give a more restrictive definition of fallacies by additionally requiring that they appear to be correct.

The strategic rules, on the other hand, describe how the allowed moves may be used to win a game, for instance, by controlling the center and by defending one's king.

Rules in a proof system are defined in terms of the syntactic form of formulas independent of their specific content.

[114] The central aspect of Aristotelian logic involves classifying all possible syllogisms into valid and invalid arguments according to how the propositions are formed.

[117] These intuitions include the law of excluded middle, the double negation elimination, the principle of explosion, and the bivalence of truth.

Graham Priest is an influential contemporary proponent of this position and similar views have been ascribed to Georg Wilhelm Friedrich Hegel.

[141] The pragmatic or dialogical approach to informal logic sees arguments as speech acts and not merely as a set of premises together with a conclusion.

A winning move is a successful argument that takes the opponent's commitments as premises and shows how one's own conclusion follows from them.

[166] Set theory originated in the study of the infinite by Georg Cantor, and it has been the source of many of the most challenging and important issues in mathematical logic.

Early influential theorists in this field were Richard Montague and Barbara Partee, who focused their analysis on the English language.

This is usually combined with the claim that the laws of logic express universal regularities found in the structural features of the world.

For example, it has been argued that certain insights of quantum mechanics refute the principle of distributivity in classical logic, which states that the formula

The school of Mohism also acknowledged the importance of language for logic and tried to relate the ideas in these fields to the realm of ethics.

It was not treated as a separate academic discipline and discussions of its topics usually happened in the context of epistemology and theories of dialogue or argumentation.

[199] A similar emphasis on the relation to epistemology is also found in Buddhist and Jainist schools of logic, where inference is used to expand the knowledge gained through other sources.

[200] Some of the later theories of Nyaya, belonging to the Navya-Nyāya school, resemble modern forms of logic, such as Gottlob Frege's distinction between sense and reference and his definition of number.

Alfred North Whitehead and Bertrand Russell, in turn, condensed many of these insights in their work Principia Mathematica.

It also made Alfred Tarski's approach to model theory possible and provided the foundation of modern mathematical logic.

Depiction of inference using modus ponens
Logic studies valid forms of inference like modus ponens .
Visualization of how to translate an English sentence into first-order logic
Formal logic needs to translate natural language arguments into a formal language, like first-order logic, to assess whether they are valid. In this example, the letter "c" represents Carmen while the letters "M" and "T" stand for "Mexican" and "teacher". The symbol "∧" has the meaning of "and".
Diagram of argument terminology used in logic
Argument terminology used in logic
Poster from 1901
Young America's dilemma: Shall I be wise and great, or rich and powerful? (poster from 1901) This is an example of a false dilemma : an informal fallacy using a disjunctive premise that excludes viable alternatives.
Diagram of the square of opposition
The square of opposition is often used to visualize the relations between the four basic categorical propositions in Aristotelian logic. It shows, for example, that the propositions "All S are P" and "Some S are not P" are contradictory, meaning that one of them has to be true while the other is false.
Symbol introduced by Gottlob Frege for the universal quantifier
Gottlob Frege 's Begriffschrift introduced the notion of quantifier in a graphical notation, which here represents the judgment that is true.
Photograph of Bertrand Russell
Bertrand Russell made various contributions to mathematical logic. [ 163 ]
Diagram of an AND gate using transistors
Conjunction (AND) is one of the basic operations of Boolean logic. It can be electronically implemented in several ways, for example, by using two transistors .