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.

Controversies in logic

Just as we have seen there is disagreement over what logic is about, so there is disagreement about what logical truths there are.

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Bivalence and the law of the excluded middle

The logics discussed above are all "bivalent" or "two-valued"; that is, they are most naturally understood as dividing propositions into the true and the false propositions. Systems which reject bivalence are known as non-classical logics.

Related Topics:
Bivalent - Non-classical logic

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In the early 20th century Jan Łukasiewicz investigated the extension of the traditional true/false values to include a third value, "possible", so inventing ternary logic, the first multi-valued logic.

Related Topics:
20th century - Jan Łukasiewicz - Ternary logic - Multi-valued logic

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Intuitionistic logic was proposed by L. E. J. Brouwer as the correct logic for reasoning about mathematics, based upon his rejection of the law of the excluded middle as part of his intuitionism. Brouwer rejected formalisation in mathematics, but his student Arend Heyting studied intuitionistic logic formally, as did Gerhard Gentzen. Intuitionistic logic has come to be of great interest to computer scientists, as it is a constructive logic, and is hence a logic of what computers can do.

Related Topics:
Intuitionistic logic - L. E. J. Brouwer - Law of the excluded middle - Intuitionism - Arend Heyting - Gerhard Gentzen - Constructive logic

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Modal logic is not truth conditional, and so it has often been proposed as a non-classical logic. However modal logic is normally formalised with the principle of the excluded middle, and its relational semantics is bivalent, so this inclusion is disputable. However, modal logic can be used to encode non-classical logics, such as intuitionistic logic.

Related Topics:
Modal logic - Relational semantics

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Logics such as fuzzy logic have since been devised with an infinite number of "degrees of truth", represented by a real number between 0 and 1. Bayesian probability can be interpreted as a system of logic where probability is the subjective truth value.

Related Topics:
Fuzzy logic - Real number - Bayesian probability

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Implication: strict or material?

It is easy to observe that the notion of implication formalised in classical logic does not comfortably translate into natural language by means of "if... then...", due to a number of

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problems called the paradoxes of material implication.

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The first class of paradoxes are those that involve counterfactuals, such as "If the moon is made of green cheese, then 2+2=4", puzzling because natural language does not support the principle of explosion. Eliminating these classes of paradox led to David Lewis's formulation of strict implication, and to a more radically revisionist logics such as relevance logic and dialetheism.

Related Topics:
Principle of explosion - David Lewis - Strict implication - Relevance logic - Dialetheism

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The second class of paradox are those that involve redundant premises, falsely suggesting that we know the succedent because of the antecedent: thus "if that man gets elected, granny will die" is materially true if granny happens to be in the last stages of a terminal illness, regardless of the man's election prospects. Such sentences violate the Gricean maxim of relevance, and can be modeled by logics that reject the principle of monotonicity of entailment, such as relevance logic.

Related Topics:
Gricean maxim - Monotonicity of entailment

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Tolerating the impossible

Closely related to questions arising from the paradoxes of implication comes the radical suggestion that logic ought to tolerate inconsistency. Again, relevance logic and dialetheism are the most important approaches here, though the concerns are different: the key issue that classical logic and some of its rivals, such as intuitionistic logic have is that they respect the principle of explosion, which means that the logic collapses if it is capable of deriving a contradiction. Graham Priest, the proponent of dialetheism, has argued for paraconsistency on the striking grounds that there are in fact, true contradictions (Priest 2004).

Related Topics:
Inconsistency - Relevance logic - Dialetheism - Classical logic - Intuitionistic logic - Principle of explosion - Graham Priest

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Is logic empirical?

What is the epistemological status of the laws of logic? What sort of arguments are appropriate for criticising purported principles of logic? In an influential paper entitled Is logic empirical? Hilary Putnam, building on a suggestion of W.V.O. Quine, argued that in general that the facts of propositional logic have a similar epistemological status as facts about the physical universe, for example as the laws of mechanics or of general relativity, and in particular that what physicists have learned about quantum mechanics provides a compelling case for abandoning certain familiar principles of classical logic: if we want to be realists about the physical phenomena described by quantum theory, then we should abandon the principle of distributivity, substituting for classical logic the quantum logic proposed by Garrett Birkhoff and John von Neumann.

Related Topics:
Laws of logic - Hilary Putnam - W.V.O. Quine - General relativity - Realists - Principle of distributivity - Quantum logic - Garrett Birkhoff - John von Neumann

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Another paper by the same name by Sir Michael Dummett argues that Putnam's desire for realism mandates the law of distributivity: distributivity of logic is essential for the realist's understanding of how propositions are true of the world, in just the same way as he has argued the principle of bivalence is. In this way, the question Is logic empirical? can be seen to lead naturally into the fundamental controversy in metaphysics on realism versus anti-realism.

Related Topics:
Sir Michael Dummett - Metaphysics - Realism versus anti-realism

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