Logic

Logic (from Classical Greek λόγος (logos), originally meaning the word, or what is spoken, but coming to mean thought or reason) is most often said to be the study of arguments, although the exact definition of logic is a matter of controversy amongst philosophers (see below). However the subject is grounded, the task of the logician is the same: to advance an account of valid and fallacious inference to allow one to distinguish good from bad arguments.

Topics in logic

Throughout history, there has been interest in distinguishing good from bad arguments, and so logic has been studied in some more or less familiar form.

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Aristotelian logic has principally been concerned with teaching good argument, and is still taught with that end today, while in mathematical logic and analytical philosophy much greater emphasis is placed on logic as an object of study in its own right, and so logic is studied at a more abstract level.

Related Topics:
Aristotelian logic - Mathematical logic - Analytical philosophy

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Consideration of the different types of logic explains that logic is not studied in a vacuum. While logic often seems to provide its own motivations, the subject develops most healthily when the reason for our interest is made clear.

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Syllogistic

The Organon was Aristotle's body of work on logic, with the Prior Analytics constituting the first explicit work in formal logic, introducing the syllogistic. The parts of syllogistic, also known by the name term logic, were the analysis of the judgements into propositions consisting of two terms that are related by one of a fixed number of relations, and the expression of inferences by means of syllogisms that consisted of two propositions sharing a common term as premise, and a conclusion which was a proposition involving the two unrelated terms from the premises.

Related Topics:
Organon - Aristotle - Prior Analytics - Formal logic - Syllogistic - Term logic - Syllogism

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Aristotle's work was regarded in classical times and from medieval times in Europe and the Middle East as the very picture of a fully worked out system. It was not alone: the Stoics proposed a system of propositional logic that was studied by medieval logicians; nor was the perfection of Aristotle's system undisputed; for example the problem of multiple generality was recognised in medieval times. Nonetheless, problems with syllogistic were not seen as being in need of revolutionary solutions.

Related Topics:
Stoics - Propositional logic - Problem of multiple generality

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Today, Aristotle's system is mostly seen as of historical value (though there is some current interest in extending term logics), regarded as made obsolete by the advent of the predicate calculus.

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Predicate logic

Logic as it is studied today is a very different subject to that studied before, and the principal difference is the innovation of predicate logic. Whereas Aristotelian syllogistic specified the forms that the relevant parts of the involved judgements took, predicate logic allows sentences to be analysed into subject and argument in several different ways, thus allowing predicate logic to solve the problem of multiple generality that had perplexed medieval logicians. With predicate logic, for the first time, logicians were able to give an account of quantifiers general enough to express all arguments occurring in natural language.

Related Topics:
Problem of multiple generality - Quantifiers

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The discovery of predicate logic is usually attributed to Gottlob Frege, who is also credited as one of the founders of analytical philosophy, but the formulation of predicate logic most often used today is the first-order logic presented in Principles of Theoretical Logic by David Hilbert and Wilhelm Ackermann in 1928. The analytical generality of the predicate logic allowed the formalisation of mathematics, and drove the investigation of set theory, allowed the development of Alfred Tarski's approach to model theory; it is no exaggeration to say that it is the foundation of modern mathematical logic.

Related Topics:
Gottlob Frege - Analytical philosophy - First-order logic - Principles of Theoretical Logic - David Hilbert - Wilhelm Ackermann - 1928 - Set theory - Alfred Tarski - Model theory - Mathematical logic

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Modal logic

In language, modality deals with the phenomenon that subparts of a sentence may be have their semantics modified perhaps by special verbs or modal particles. For example, "We go to the games" can be modified to give "We should go to the games", and "We can go to the games"" and perhaps "We will go to the games". More abstractly, we might say that modality affects the circumstances in which we take an assertion to be satisfied.

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The logical study of modality dates back to Aristotle, who was concerned with the alethic modalities of necessity and possibility, which he observed to be dual in the sense of De Morgan duality. While the study of necessity and possibility remained important to philosophers, little logical innovation happened until the landmark investigations of Clarence Irving Lewis in 1918, who formulated a family of rival axiomatisations of the alethic modalities. His work unleashed a torrent of new work on the topic, expanding the kinds of modality treated to include deontic logic, epistemic logic. The seminal work of Arthur Prior applied the same formal language to treat temporal logic and paved the way for the marriage of the two subjects. Saul Kripke discovered contemporaneously with rivals his theory of frame semantics which revolutionised the formal technology available to modal logicians and gave a new graph-theoretic way of looking at modality that has driven many applications in computational linguistics and computer science, such as dynamic logic.

Related Topics:
Alethic modalities - De Morgan duality - Clarence Irving Lewis - 1918 - Deontic logic - Epistemic logic - Arthur Prior - Temporal logic - Saul Kripke - Frame semantics - Graph-theoretic - Computational linguistics - Computer science - Dynamic logic

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Deduction and reasoning

The motivation for the study of logic in ancient times was clear, as we have described: it is so that we may learn to distinguish good from bad arguments, and so become more effective in argument and oratory, and perhaps also, to become a better person.

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This motivation is still alive, although it no longer takes centre stage in the picture of logic; typically dialectical logic will form the heart of a course in critical thinking, a compulsory course at many universities, especially those that follow the American model.

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Mathematical logic

Mathematical logic really refers to two distinct areas of research: the first is the application of the techniques of formal logic to mathematics and mathematical reasoning, and the second, in the other direction, the application of mathematical techniques to the representation and analysis of formal logic.

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The boldest attempt to apply logic to mathematics was undoubtedly the logicism pioneered by philosopher-logicians such as Gottlob Frege and Bertrand Russell: the idea was that mathematical theories were logical tautologies, and the programme was to show this by means to a reduction of mathematics to logic. The various attempts to carry this out met with a series of failures, from the crippling of Frege's project in his Grundgesetze by Russell's paradox, to the defeat of Hilbert's Program by Gödel's incompleteness theorems.

Related Topics:
Logicism - Gottlob Frege - Bertrand Russell - Russell's paradox - Hilbert's Program - Gödel's incompleteness theorem

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Both the statement of Hilbert's Program and its refutation by Gödel depended upon their work establishing the second area of mathematical logic, the application of mathematics to logic in the form of proof theory. Despite the negative nature of the incompleteness theorems, Gödel's completeness theorem, a result in model theory and another application of mathematics to logic, can be understood as showing how close logicism came to being true: every rigorously defined mathematical theory can be exactly captured by a first-order logical theory; Frege's proof calculus is enough to describe the whole of mathematics, though not equivalent to it. Thus we see how complementary the two areas of mathematical logic have been.

Related Topics:
Proof theory - Gödel's completeness theorem - Model theory - Proof calculus

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If proof theory and model theory have been the foundation of mathematical logic, they have been but two of the four pillars of the subject. 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, from Cantor's theorem, through the status of the Axiom of Choice and the question of the independence of the continuum hypothesis, to the modern debate on large cardinal axioms.

Related Topics:
Set theory - Georg Cantor - Cantor's theorem - Axiom of Choice - Continuum hypothesis - Large cardinal

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Recursion theory captures the idea of computation in logical and arithmetic terms; its most classical achievements are the undecidability of the Entscheidungsproblem by Alan Turing, and his presentation of the Church-Turing thesis. Today recursion theory is mostly concerned with the more refined problem of complexity classes -- when is a problem efficiently solvable? -- and the classification of degrees of unsolvability.

Related Topics:
Recursion theory - Arithmetic - Entscheidungsproblem - Alan Turing - Church-Turing thesis - Complexity class - Degrees of unsolvability

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Philosophical logic

Philosophical logic deals with formal descriptions of natural language. Most philosophers assume that the bulk of "normal" proper reasoning can be captured by logic, if one can find the right method for translating ordinary language into that logic. Philosophical logic is essentially a continuation of the traditional discipline that was called "Logic" before it was supplanted by the invention of mathematical logic. Philosophical logic has a much greater concern with the connection between natural language and logic. As a result, philosophical logicians have contributed a great deal to the development of non-standard logics (e.g., free logics, tense logics) as well as various extensions of classical logic (e.g., modal logics), and non-standard semantics for such logics (e.g., supervaluation semantics).

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Logic and computation

Logic is extensively applied in the fields of artificial intelligence, and computer science, and these fields provide a rich source of problems in formal logic.

Related Topics:
Artificial intelligence - Computer science

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In the 1950s and 1960s, researchers predicted that when human knowledge could be expressed using logic with mathematical notation, it would be possible to create a machine that reasons, or artificial intelligence. This turned out to be more difficult than expected because of the complexity of human reasoning. In logic programming, a program consists of a set of axioms and rules. Logic programming systems such as Prolog compute the consequences of the axioms and rules in order to answer a query.

Related Topics:
Mathematical notation - Logic programming - Prolog

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In symbolic logic and mathematical logic, proofs by humans can be computer-assisted. Using automated theorem proving the machines can find and check proofs, as well as work with proofs too lengthy to be written out by hand.

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In computer science, Boolean algebra is the basis of hardware design, as well as much software design.

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There are also various systems for reasoning about computer programs. Hoare logic is one of the earliest of such systems. Other systems are CSP, CCS, pi-calculus for reasoning about concurrent processes or mobile processes. There is interest in the idea of finding a logical calculus that naturally captures computability; there are several efforts to provide computational logics for this purpose.

Related Topics:
Computer program - Hoare logic - CSP - CCS - Pi-calculus - Logical calculus

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