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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.[61] Another paper of the same name by Michael Dummett argues that Putnam's desire for realism mandates the law of distributivity.[62] 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.
Implication: Strict or material Main article: Paradoxes of material implication The notion of implication formalized in classical logic does not comfortably translate into natural language by means of "if ... then ...", due to a number of problems called the paradoxes of material implication. The first class of paradoxes involves counterfactuals, such as If the moon is made of green cheese, then 2+2=5, which are puzzling because natural language does not support the principle of explosion. Eliminating this class of paradoxes was the reason for C. I. Lewis's formulation of strict implication, which eventually led to more radically revisionist logics such as relevance logic.

The second class of paradoxes involves 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 since granny is mortal, regardless of the man's election prospects. Such sentences violate the Gricean maxim of relevance, and can be modelled by logics that reject the principle of monotonicity of entailment, such as relevance logic. Tolerating the impossible Main article: Paraconsistent logic Georg Wilhelm Friedrich Hegel was deeply critical of any simplified notion of the law of non-contradiction. It was based on Gottfried Wilhelm Leibniz's idea that this law of logic also requires a sufficient ground to specify from what point of view (or time) one says that something cannot contradict itself. A building, for example, both moves and does not move; the ground for the first is our solar system and for the second the earth. In Hegelian dialectic, the law of non-contradiction, of identity, itself relies upon difference and so is not independently assertable. Closely related to questions arising from the paradoxes of implication comes the suggestion that logic ought to tolerate inconsistency. Relevance logic and paraconsistent logic are the most important approaches here, though the concerns are different: a key consequence of classical logic and some of its rivals, such as intuitionistic logic, 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 main proponent of dialetheism, has argued for paraconsistency on the grounds that there are in fact, true contradictions.[63][clarification needed] Rejection of logical truth The philosophical vein of various kinds of skepticism contains many kinds of doubt and rejection of the various bases on which logic rests, such as the idea of logical form, correct inference, or meaning, typically leading to the conclusion that there are no logical truths. This is in contrast with the usual views in philosophical skepticism, where logic directs skeptical enquiry to doubt received wisdoms, as in the work of Sextus Empiricus. Friedrich Nietzsche provides a strong example of the rejection of the usual basis of logic: his radical rejection of idealization led him to reject truth as a "... mobile army of metaphors, metonyms, and anthropomorphisms—in short ... metaphors which are worn out and without sensuous power; coins which have lost their pictures and now matter only as metal, no longer as coins."[64] His rejection of truth did not lead him to reject the idea of either inference or logic completely, but rather suggested that "logic [came] into existence in man's head [out] of illogic, whose realm originally must have been immense. Innumerable beings who made inferences in a way different from ours perished".[65] Thus there is the idea that logical inference has a use as a tool for human survival, but that its existence does not support the existence of truth, nor does it have a reality beyond the instrumental: "Logic, too, also rests on assumptions that do not correspond to anything in the real world".[66] This position held by Nietzsche however, has come under extreme scrutiny for several reasons. Some philosophers, such as Jürgen Habermas, claim his position is self-refuting—and accuse Nietzsche of not even having a coherent perspective, let alone a theory of knowledge.[67] Georg Lukács, in his book The Destruction of Reason, asserts that, "Were we to study Nietzsche's statements in this area from a logico-philosophical angle, we would be confronted by a dizzy chaos of the most lurid assertions, arbitrary and violently incompatible."[68] Bertrand Russell described Nietzsche's irrational claims with "He is fond of expressing himself paradoxically and with a view to shocking conventional readers" in his book A History of Western Philosophy 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.[1][2][3][4][5] The logical form of an argument in a natural language can be represented in a symbolic formal language, and independently of natural language formally defined "arguments" can be made in math and computer science. Logic is the study of the forms of reasoning in arguments and the development of standards and criteria to evaluate arguments.[6] Deductive arguments can be valid or sound: in a valid argument, premisses necessitate the conclusion, even if one or more of the premisses is false and the conclusion is false; in a sound argument, true premisses necessitate a true conclusion. Inductive arguments, by contrast, can have different degrees of logical strength: the stronger or more cogent the argument, the greater the probability that the conclusion is true, the weaker the argument, the lesser that probability.[7] The standards for evaluating non-deductive arguments may rest on different or additional criteria than truth—for example, the persuasiveness of so-called "indispensability claims" in transcendental arguments,[8] the quality of hypotheses in retroduction, or even the disclosure of new possibilities for thinking and acting.[9] Contents 1 Etymology 2 Formal and informal 3 Standard types 3.1 Deductive arguments 3.2 Inductive arguments 4 Deductive 4.1 Validity 4.2 Soundness 5 Inductive 6 Defeasible arguments and argumentation schemes 7 By analogy 8 Other kinds 8.1 In informal logic 8.1.1 Logical status 8.2 World-disclosing 9 Explanations 10 Fallacies and non-arguments 11 See also 12 Notes 13 References 14 Further reading 15 External links Etymology The Latin root arguere (to make bright, enlighten, make known, prove, etc.) is from Proto-Indo-European argu-yo-, suffixed form of arg- (to shine; white).[10] Formal and informal Further information: Informal logic and Formal logic Informal arguments as studied in informal logic, are presented in ordinary language and are intended for everyday discourse. Conversely, formal arguments are studied in formal logic (historically called symbolic logic, more commonly referred to as mathematical logic today) and are expressed in a formal language. Informal logic may be said to emphasize the study of argumentation, whereas formal logic emphasizes implication and inference. Informal arguments are sometimes implicit. That is, the rational structure – the relationship of claims, premises, warrants, relations of implication, and conclusion – is not always spelled out and immediately visible and must sometimes be made explicit by analysis. Standard types Argument terminology There are several kinds of arguments in logic, the best-known of which are "deductive" and "inductive." An argument has one or more premises but only one conclusion. Each premise and the conclusion are truth bearers or "truth-candidates", each capable of being either true or false (but not both). These truth values bear on the terminology used with arguments. Deductive arguments A deductive argument asserts that the truth of the conclusion is a logical consequence of the premises. Based on the premises, the conclusion follows necessarily (with certainty). For example, given premises that A=B and B=C, then the conclusion follows necessarily that A=C. Deductive arguments are sometimes referred to as "truth-preserving" arguments

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