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Multi-valued logics depart from classicality by rejecting the principle of bivalence which requires all propositions to be either true or false. Fuzzy logics are multivalued logics that have an infinite number of “degrees of truth”, represented by a real number between 0 and 1. 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. Walton understands a dialogue as a game between two players.
Dialogues are games of persuasion: each player has the goal of convincing the opponent of their own 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. For this reason, it is normally necessary to formulate a sequence of arguments as intermediary steps, each of which brings the opponent a little closer to one’s intended conclusion.
Besides these positive arguments leading one closer to victory, there are also negative arguments preventing the opponent’s victory by denying their conclusion. Fallacies , on the other hand, are violations of the standards of proper argumentative rules.
The epistemic approach to informal logic, on the other hand, focuses on the epistemic role of arguments. They achieve this by linking justified beliefs to beliefs that are not yet justified. Degrees of belief are understood as subjective probabilities in the believed proposition, i. Bad or irrational reasoning, on the other hand, violates these laws. Logic is studied in various fields. In many cases, this is done by applying its formal method to specific topics outside its scope, like to ethics or computer science.
This can happen in diverse ways, like by investigating the philosophical presuppositions of fundamental logical concepts, by interpreting and analyzing logic through mathematical structures, or by studying and comparing abstract properties of formal logical systems. Philosophy of logic is the philosophical discipline studying the scope and nature of logic. It studies the application of logical methods to philosophical problems in fields like metaphysics , ethics, and epistemology.
Mathematical logic is the study of logic within mathematics. Major subareas include model theory , proof theory , set theory , and computability theory. Research in mathematical logic commonly addresses the mathematical properties of formal systems of logic. However, it can also include attempts to use logic to analyze mathematical reasoning or to establish logic-based foundations of mathematics.
Mathematical theories were supposed to be logical tautologies , and the programme was to show this by means of a reduction of mathematics to logic. 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.
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. In computer science, logic is studied as part of the theory of computation. Key areas of logic that are relevant to computing include computability theory , modal logic , and category theory.
Early computer machinery was based on ideas from logic such as the lambda calculus. For instance, modern artificial intelligence builds on logicians’ work in argumentation theory , while automated theorem proving can assist logicians in finding and checking proofs. In logic programming languages such as Prolog , a program computes the consequences of logical axioms and rules to answer a query. Formal semantics is a subfield of both linguistics and philosophy which uses logic to analyze meaning in natural language.
It is an empirical field which seeks to characterize the denotations of linguistic expressions and explain how those denotations are composed from the meanings of their parts.
The field was developed by Richard Montague and Barbara Partee in the s, and remains an active area of research. Central questions include scope , binding , and linguistic modality. What is the epistemological status of the laws of logic? What sort of argument is appropriate for criticizing purported principles of logic?
In an influential paper entitled ” Is Logic Empirical? Quine , argued that in general 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.
Another paper of the same name by Michael Dummett argues that Putnam’s desire for realism mandates the law of distributivity. In this way, the question, “Is Logic Empirical? 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.
Logic arose from a concern with correctness of argumentation. Modern logicians usually wish to ensure that logic studies just those arguments that arise from appropriately general forms of inference.
For example, Thomas Hofweber writes in the Stanford Encyclopedia of Philosophy that logic “does not, however, cover good reasoning as a whole. That is the job of the theory of rationality. Rather it deals with inferences whose validity can be traced back to the formal features of the representations that are involved in that inference, be they linguistic, mental, or other representations. The idea that logic treats special forms of argument, deductive argument, rather than argument in general, has a history in logic that dates back at least to logicism in mathematics 19th and 20th centuries and the advent of the influence of mathematical logic on philosophy.
A consequence of taking logic to treat special kinds of argument is that it leads to identification of special kinds of truth, the logical truths with logic equivalently being the study of logical truth , and excludes many of the original objects of study of logic that are treated as informal logic. Robert Brandom has argued against the idea that logic is the study of a special kind of logical truth, arguing that instead one can talk of the logic of material inference in the terminology of Wilfred Sellars , with logic making explicit the commitments that were originally implicit in informal inference.
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, sometimes 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 ” Innumerable beings who made inferences in a way different from ours perished”. This position held by Nietzsche however, has come under extreme scrutiny for several reasons. Logic was developed independently in several cultures during antiquity. One major early contributor was Aristotle , who developed term logic in his Organon and Prior Analytics.
Inferences are expressed by means of syllogisms that consist of two propositions sharing a common term as premise, and a conclusion that is a proposition involving the two unrelated terms from the premises. Aristotle’s monumental insight was the notion that arguments can be characterized in terms of their form. Aristotelian logic was highly regarded in classical and medieval times, both in Europe and the Middle East.
It remained in wide use in the West until the early 19th century. Ibn Sina Avicenna — CE was the founder of Avicennian logic , which replaced Aristotelian logic as the dominant system of logic in the Islamic world ,  and also had an important influence on Western medieval writers such as Albertus Magnus  and William of Ockham. In Europe during the later medieval period, major efforts were made to show that Aristotle’s ideas were compatible with Christian faith.
During the High Middle Ages , logic became a main focus of philosophers, who would engage in critical logical analyses of philosophical arguments, often using variations of the methodology of scholasticism. Initially, medieval Christian scholars drew on the classics that had been preserved in Latin through commentaries by such figures such as Boethius , later the work of Islamic philosophers such as Ibn Sina and Ibn Rushd Averroes — CE were drawn on, which expanded the range of ancient works available to medieval Christian scholars since more Greek work was available to Muslim scholars that had been preserved in Latin commentaries.
In , William of Ockham ‘s influential Summa Logicae was released. By the 18th century, the structured approach to arguments had degenerated and fallen out of favour, as depicted in Holberg ‘s satirical play Erasmus Montanus. The Chinese logical philosopher Gongsun Long c. By the 16th century, it developed theories resembling modern logic, such as Gottlob Frege ‘s “distinction between sense and reference of proper names” and his “definition of number”, as well as the theory of “restrictive conditions for universals” anticipating some of the developments in modern set theory.
The syllogistic logic developed by Aristotle predominated in the West until the midth century, when interest in the foundations of mathematics stimulated the development of symbolic logic now called mathematical logic. In , George Boole published The Laws of Thought ,  introducing symbolic logic and the principles of what is now known as Boolean logic. In , Gottlob Frege published Begriffsschrift , which inaugurated modern logic with the invention of quantifier notation, reconciling the Aristotelian and Stoic logics in a broader system, and solving such problems for which Aristotelian logic was impotent, such as the problem of multiple generality.
From to , Alfred North Whitehead and Bertrand Russell published Principia Mathematica  on the foundations of mathematics, attempting to derive mathematical truths from axioms and inference rules in symbolic logic. The development of logic since Frege, Russell, and Wittgenstein had a profound influence on the practice of philosophy and the perceived nature of philosophical problems see analytic philosophy and philosophy of mathematics.
Logic, especially sentential logic, is implemented in computer logic circuits and is fundamental to computer science. Logic is commonly taught by university philosophy, sociology, advertising and literature departments, often as a compulsory discipline. From Wikipedia, the free encyclopedia. Study of correct reasoning. This article is about the study of correct reasoning. For other uses, see Logic disambiguation. For the school of Chinese philosophy, see School of Names.
Plato Kant Nietzsche. Buddha Confucius Averroes. Ancient Medieval Modern Contemporary. Aestheticians Epistemologists Ethicists Logicians Metaphysicians Social and political philosophers Women in philosophy. Further information: premise. Further information: logical truth. Further information: argument and inference. Main article: Formal system.
Main article: Formal language. Main article: Proof system. Main article: Semantics of logic. Main articles: Soundness , completeness logic , and metalogic. Main article: Propositional calculus. Main article: First-order logic. Further information: Modal logic. Further information: Higher-order logic. Main articles: Philosophy of logic and Philosophical logic. Main article: Mathematical logic. Main articles: Computational logic and Logic in computer science. Main article: Formal semantics natural language.
Further information: Is Logic Empirical? Main article: Paraconsistent logic. Main article: Conceptions of logic. Main article: History of logic. Philosophy portal. ISBN Josephson, John R. Abductive Inference: Computation, Philosophy, Technology. New York: Cambridge University Press. Bunt, H. Amsterdam: John Benjamins. ISBN , In the philosophical literature, the term is sometimes further restricted to refer only to particular logic-based formal systems such as those which are complete or motivated by intuitions close to those which motivated classical logic.
Belnap, Nuel. Boston: Reidel; Jayatilleke, K. University of Hawaii Press. The first part deals with Frege’s distinction between sense and reference of proper names and a similar distinction in Navya-Nyaya logic. In the second part we have compared Frege’s definition of number to the Navya-Nyaya definition of number.
In the third part we have shown how the study of the so-called ‘restrictive conditions for universals’ in Navya-Nyaya logic anticipated some of the developments of modern set theory. Encyclopedia of Religion. Retrieved 29 December Retrieved 21 November Creation Tool.
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