logic this article is about the systematic study of the form of arguments for other uses see logic disambiguation part of a series onphilosophyplatokantnietzschebuddhaconfuciusaverroes philosophers aestheticians epistemologists ethicists logicians metaphysicians social and political philosophers traditions african analytic aristotelian buddhist chinese christian continental existentialism hindu jain jewish pragmatism eastern islamic platonic western periods ancient medieval modern contemporary literature aesthetics epistemology ethics logic metaphysics political philosophy branches aesthetics epistemology ethics legal philosophy logic metaphysics political philosophy social philosophy lists index outline years problems publications theories glossary philosophers miscellaneous philosopher philomath philalethes women in philosophy philosophy portalvte logic from the ancient greek translitlogik and is now generally held to consist of the systematic study of the form of valid inference a valid inference is one where there is a specific relation of logical support between the assumptions of the inference and its conclusion in ordinary discourse inferences may be signified by words such as therefore hence ergo and so on there is no universal agreement as to the exact scope and subject matter of logic see rival conceptions below but it has traditionally included the classification of arguments the systematic exposition of the logical form common to all valid arguments the study of inference including fallacies and the study of semantics including paradoxes historically logic has been studied in philosophy since ancient times and mathematics since the mid19th century and recently logic has been studied in computer science linguistics psychology and other fields concepts upon this first and in one sense this sole rule of reason that in order to learn you must desire to learn and in so desiring not be satisfied with what you already incline to capably think there follows one corollary which itself deserves to be inscribed upon every wall of the city of philosophy do not block the way of inquiry charles sanders peirce first rule of logic the concept of logical form is central to logic the validity of an argument is determined by its logical form not by its content traditional aristotelian syllogistic logic and modern symbolic logic are examples of formal logic informal logic is the study of natural language arguments the study of fallacies is an important branch of informal logic since much informal argument is not strictly speaking deductive on some conceptions of logic informal logic is not logic at all see rival conceptions below formal logic is the study of inference with purely formal content an inference possesses a purely formal content if it can be expressed as a particular application of a wholly abstract rule that is a rule that is not about any particular thing or property the works of aristotle contain the earliest known formal study of logic modern formal logic follows and expands on aristotle in many definitions of logic logical inference and inference with purely formal content are the same this does not render the notion of informal logic vacuous because no formal logic captures all of the nuances of natural language symbolic logic is the study of symbolic abstractions that capture the formal features of logical inference symbolic logic is often divided into two main branches propositional logic and predicate logic mathematical logic is an extension of symbolic logic into other areas in particular to the study of model theory proof theory set theory and recursion theory however agreement on what logic is has remained elusive and although the field of universal logic has studied the common structure of logics in 2007 mossakowski et al commented that it is embarrassing that there is no widely acceptable formal definition of a logic logical form logical form logic is generally considered formal when it analyzes and represents the form of any valid argument type the form of an argument is displayed by representing its sentences in the formal grammar and symbolism of a logical language to make its content usable in formal inference simply put to formalize simply means to translate english sentences into the language of logic this is called showing the logical form of the argument it is necessary because indicative sentences of ordinary language show a considerable variety of form and complexity that makes their use in inference impractical it requires first ignoring those grammatical features irrelevant to logic such as gender and declension if the argument is in latin replacing conjunctions irrelevant to logic such as but with logical conjunctions like and and replacing ambiguous or alternative logical expressions any every etc with expressions of a standard type such as all or the universal quantifier second certain parts of the sentence must be replaced with schematic letters thus for example the expression all ps are qs shows the logical form common to the sentences all men are mortals all cats are carnivores all greeks are philosophers and so on the schema can further be condensed into the formula apq where the letter a indicates the judgement all are the importance of form was recognised from ancient times aristotle uses variable letters to represent valid inferences in prior analytics leading jan ukasiewicz to say that the introduction of variables was one of aristotles greatest inventions according to the followers of aristotle such as ammonius only the logical principles stated in schematic terms belong to logic not those given in concrete terms the concrete terms man mortal etc are analogous to the substitution values of the schematic placeholders p q r which were called the matter greek hyle of the inference there is a big difference between the kinds of formulas seen in traditional term logic and the predicate calculus that is the fundamental advance of modern logic the formula apq all ps are qs of traditional logic corresponds to the more complex formula x p x q x displaystyle forall xpxrightarrow qx in predicate logic involving the logical connectives for universal quantification and implication rather than just the predicate letter a and using variable arguments p x displaystyle px where traditional logic uses just the term letter p with the complexity comes power and the advent of the predicate calculus inaugurated revolutionary growth of the subject semantics semantics of logic the validity of an argument depends upon the meaning or semantics of the sentences that make it up aristotles organon especially on interpretation gives a cursory outline of semantics which the scholastic logicians particularly in the thirteenth and fourteenth century developed into a complex and sophisticated theory called supposition theory this showed how the truth of simple sentences expressed schematically depend on how the terms supposit or stand for certain extralinguistic items for example in part ii of his summa logicae william of ockham presents a comprehensive account of the necessary and sufficient conditions for the truth of simple sentences in order to show which arguments are valid and which are not thus every a is b is true if and only if there is something for which a stands and there is nothing for which a stands for which b does not also stand early modern logic defined semantics purely as a relation between ideas antoine arnauld in the port royal logic says that after conceiving things by our ideas we compare these ideas and finding that some belong together and some do not we unite or separate them this is called affirming or denying and in general judging this view psychologism was taken to the extreme in the nineteenth century and is generally held by modern logicians to signify a low point in the decline of logic before the twentieth century modern semantics is in some ways closer to the medieval view in rejecting such psychological truthconditions however the introduction of quantification needed to solve the problem of multiple generality rendered impossible the kind of subjectpredicate analysis that underlies medieval semantics the main modern approach is modeltheoretic semantics based on alfred tarskis semantic theory of truth the approach assumes that the meaning of the various parts of the propositions are given by the possible ways we can give a recursively specified group of interpretation functions from them to some predefined domain of discourse an interpretation of firstorder predicate logic is given by a mapping from terms to a universe of individuals and a mapping from propositions to the truth values true and false modeltheoretic semantics is one of the fundamental concepts of model theory modern semantics also admits rival approaches such as the prooftheoretic semantics that associates the meaning of propositions with the roles that they can play in inferences an approach that ultimately derives from the work of gerhard gentzen on structural proof theory and is heavily influenced by ludwig wittgensteins later philosophy especially his aphorism meaning is use inference inference is not to be confused with implication an implication is a sentence of the form if p then q and can be true or false the stoic logician philo of megara was the first to define the truth conditions of such an implication false only when the antecedent p is true and the consequent q is false in all other cases true an inference on the other hand consists of two separately asserted propositions of the form p therefore q an inference is not true or false but valid or invalid however there is a connection between implication and inference as follows if the implication if p then q is true the inference p therefore q is valid this was given an apparently paradoxical formulation by philo who said that the implication if it is day it is night is true only at night so the inference it is day therefore it is night is valid in the night but not in the day the theory of inference or consequences was systematically developed in medieval times by logicians such as william of ockham and walter burley it is uniquely medieval though it has its origins in aristotles topics and boethius de syllogismis hypotheticis this is why many terms in logic are latin for example the rule that licenses the move from the implication if p then q plus the assertion of its antecedent p to the assertion of the consequent q is known as modus ponens or mode of positing its latin formulation is posito antecedente ponitur consequens the latin formulations of many other rules such as ex falso quodlibet anything follows from a falsehood reductio ad absurdum disproof by showing the consequence is absurd also date from this period however the theory of consequences or of the socalled hypothetical syllogism was never fully integrated into the theory of the categorical syllogism this was partly because of the resistance to reducing the categorical judgment every s is p to the socalled hypothetical judgment if anything is s it is p the first was thought to imply some s is p the second was not and as late as 1911 in the encyclopdia britannica article on logic we find the oxford logician th case arguing against sigwarts and brentanos modern analysis of the universal proposition logical systems formal system a formal system is an organization of terms used for the analysis of deduction it consists of an alphabet a language over the alphabet to construct sentences and a rule for deriving sentences among the important properties that logical systems can have are consistency which means that no theorem of the system contradicts another validity which means that the systems rules of proof never allow a false inference from true premises completeness which means that if a formula is true it can be proven ie is a theorem of the system soundness meaning that if any formula is a theorem of the system it is true this is the converse of completeness note that in a distinct philosophical use of the term an argument is sound when it is both valid and its premises are true some logical systems do not have all four properties as an example kurt gdels incompleteness theorems show that sufficiently complex formal systems of arithmetic cannot be consistent and complete logic and rationality logic and rationality as the study of argument is of clear importance to the reasons that we hold things to be true logic is of essential importance to rationality here we have defined logic to be the systematic study of the form of arguments the reasoning behind argument is of several sorts but only some of these arguments fall under the aegis of logic proper deductive reasoning concerns the logical consequence of given premises and is the form of reasoning most closely connected to logic on a narrow conception of logic see below logic concerns just deductive reasoning although such a narrow conception controversially excludes most of what is called informal logic from the discipline there are other forms of reasoning that are rational but that are generally not taken to be part of logic these include inductive reasoning which covers forms of inference that move from collections of particular judgements to universal judgements and abductive reasoning peirce said that to abduce a hypothetical explanation a displaystyle a from an observed surprising circumstance b displaystyle b is to surmise that a displaystyle a may be true because then b displaystyle b would be a matter of course thus to abduce a displaystyle a from b displaystyle b involves determining that a displaystyle a is sufficient or nearly sufficient but not necessary for b displaystyle b while inductive and abductive inference are not part of logic proper the methodology of logic has been applied to them with some degree of success for example the notion of deductive validity where an inference is deductively valid if and only if there is no possible situation in which all the premises are true but the conclusion false exists in an analogy to the notion of inductive validity or strength where an inference is inductively strong if and only if its premises give some degree of probability to its conclusion whereas the notion of deductive validity can be rigorously stated for systems of formal logic in terms of the wellunderstood notions of semantics inductive validity requires us to define a reliable generalization of some set of observations the task of providing this definition may be approached in various ways some less formal than others some of these definitions may use logical association rule induction while others may use mathematical models of probability such as decision trees rival conceptions rival conceptions of logic logic arose see below 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 logic has been defined history history of logic aristotle 384322bce in europe logic was first developed by aristotle as well as influential terms such as terms predicables syllogisms and propositions in europe during the later medieval period major efforts were made to show that aristotles 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 in 1323 william of ockhams influential summa logicae was released by the 18th century the structured approach to arguments had degenerated and fallen out of favour as depicted in holbergs satirical play erasmus montanus the chinese logical philosopher gongsun long c 325250 bce proposed the paradox one and one cannot become two since neither becomes two in china the tradition of scholarly investigation into logic however was repressed by the qin dynasty following the legalist philosophy of han feizi in india the anviksiki school of logic was founded by medhatithi gautama c 6th century bce in the 20thcentury western philosophers like stanislaw schayer and klaus glashoff have explored indian logic more extensively the syllogistic logic developed by aristotle predominated in the west until the mid19thcentury when interest in the foundations of mathematics stimulated the development of symbolic logic now called mathematical logic in 1854 george boole published an investigation of the laws of thought on which are founded the mathematical theories of logic and probabilities introducing symbolic logic and the principles of what is now known as boolean logic in 1879 gottlob frege published begriffsschrift which inaugurated modern logic with the invention of quantifier notation from 1910 to 1913 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 in 1931 gdel raised serious problems with the foundationalist program and logic ceased to focus on such issues 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 departments often as a compulsory discipline types syllogistic logic aristotelian logic a depiction from the 15th century of the square of opposition which expresses the fundamental dualities of syllogistic the organon was aristotles body of work on logic with the prior analytics constituting the first explicit work in formal logic introducing the syllogistic the parts of syllogistic logic also known by the name term logic are 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 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 aristotles 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 however it was not alone the stoics proposed a system of propositional logic that was studied by medieval logicians also the problem of multiple generality was recognized in medieval times nonetheless problems with syllogistic logic were not seen as being in need of revolutionary solutions today some academics claim that aristotles system is generally seen as having little more than historical value though there is some current interest in extending term logics regarded as made obsolete by the advent of propositional logic and the predicate calculus others use aristotle in argumentation theory to help develop and critically question argumentation schemes that are used in artificial intelligence and legal arguments mwparseroutput templatequoteoverflowhiddenmargin1em 0padding0 40pxmwparseroutput templatequote templatequotecitelineheight15emtextalignleftpaddingleft16emmargintop0i was upset i had always believed logic was a universal weapon and now i realized how its validity depended on the way it was employed propositional logic propositional calculus a propositional calculus or logic also a sentential calculus is a formal system in which formulae representing propositions can be formed by combining atomic propositions using logical connectives and in which a system of formal proof rules establishes certain formulae as theorems an example of a theorem of propositional logic is a b a displaystyle arightarrow brightarrow a which says that if a holds then b implies a predicate logic gottlob freges begriffschrift introduced the notion of quantifier in a graphical notation which here represents the judgement that x f x displaystyle forall xfx is true predicate logic predicate logic is the generic term for symbolic formal systems such as firstorder logic secondorder logic manysorted logic and infinitary logic it provides an account of quantifiers general enough to express a wide set of arguments occurring in natural language for example bertrand russells famous barber paradox there is a man who shaves all and only men who do not shave themselves can be formalised by the sentence x man x y man y shaves x y shaves y y displaystyle exists xtextmanxwedge forall ytextmanyrightarrow textshavesxyleftrightarrow neg textshavesyy using the nonlogical predicate man x displaystyle textmanx to indicate that x is a man and the nonlogical relation shaves x y displaystyle textshavesxy to indicate that x shaves y all other symbols of the formulae are logical expressing the universal and existential quantifiers conjunction implication negation and biconditional whilst aristotelian syllogistic logic specifies a small number of forms that the relevant part of the involved judgements may take predicate logic allows sentences to be analysed into subject and argument in several additional waysallowing predicate logic to solve the problem of multiple generality that had perplexed medieval logicians the development 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 firstorder logic presented in principles of mathematical logic by david hilbert and wilhelm ackermann in 1928 the analytical generality of predicate logic allowed the formalization of mathematics drove the investigation of set theory and allowed the development of alfred tarskis approach to model theory it provides the foundation of modern mathematical logic freges original system of predicate logic was secondorder rather than firstorder secondorder logic is most prominently defended against the criticism of willard van orman quine and others by george boolos and stewart shapiro modal logic modal logic in languages modality deals with the phenomenon that subparts of a sentence may have their semantics modified 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 confusing modality is known as the modal fallacy aristotles logic is in large parts concerned with the theory of nonmodalized logic although there are passages in his work such as the famous seabattle argument in de interpretatione 9 that are now seen as anticipations of modal logic and its connection with potentiality and time the earliest formal system of modal logic was developed by avicenna who ultimately developed a theory of temporally modalized syllogistic 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 axiomatizations 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 and 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 revolutionized the formal technology available to modal logicians and gave a new graphtheoretic way of looking at modality that has driven many applications in computational linguistics and computer science such as dynamic logic informal reasoning and dialectic informal logic and logic and dialectic the motivation for the study of logic in ancient times was clear it is so that one may learn to distinguish good arguments from bad arguments and so become more effective in argument and oratory and perhaps also to become a better person half of the works of aristotles organon treat inference as it occurs in an informal setting side by side with the development of the syllogistic and in the aristotelian school these informal works on logic were seen as complementary to aristotles treatment of rhetoric this ancient motivation is still alive although it no longer takes centre stage in the picture of logic typically dialectical logic forms the heart of a course in critical thinking a compulsory course at many universities dialectic has been linked to logic since ancient times but it has not been until recent decades that european and american logicians have attempted to provide mathematical foundations for logic and dialectic by formalising dialectical logic dialectical logic is also the name given to the special treatment of dialectic in hegelian and marxist thought there have been preformal treatises on argument and dialectic from authors such as stephen toulmin the uses of argument nicholas rescher dialectics and van eemeren and grootendorst pragmadialectics theories of defeasible reasoning can provide a foundation for the formalisation of dialectical logic and dialectic itself can be formalised as moves in a game where an advocate for the truth of a proposition and an opponent argue such games can provide a formal game semantics for many logics argumentation theory is the study and research of informal logic fallacies and critical questions as they relate to every day and practical situations specific types of dialogue can be analyzed and questioned to reveal premises conclusions and fallacies argumentation theory is now applied in artificial intelligence and law mathematical logic mathematical logic mathematical logic comprises 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 the earliest use of mathematics and geometry in relation to logic and philosophy goes back to the ancient greeks such as euclid plato and aristotle one of the boldest attempts to apply logic to mathematics was the logicism pioneered by philosopherlogicians such as gottlob frege and bertrand russell mathematical theories were supposed to be logical tautologies and the programme was to show this by means of a reduction of mathematics to logic the various attempts to carry this out met with failure from the crippling of freges project in his grundgesetze by russells paradox to the defeat of hilberts program by gdels incompleteness theorems both the statement of hilberts program and its refutation by gdel 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 gdels 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 firstorder logical theory freges proof calculus is enough to describe the whole of mathematics though not equivalent to it 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 cantors 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 churchturing thesis philosophical logic philosophical logic philosophical logic deals with formal descriptions of ordinary nonspecialist natural language that is strictly only about the arguments within philosophys other branches most philosophers assume that the bulk of everyday reasoning can be captured in logic if a method or methods to translate ordinary language into that logic can be found philosophical logic is essentially a continuation of the traditional discipline called logic before 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 nonstandard logics eg free logics tense logics as well as various extensions of classical logic eg modal logics and nonstandard semantics for such logics eg kripkes supervaluationism in the semantics of logic logic and the philosophy of language are closely related philosophy of language has to do with the study of how our language engages and interacts with our thinking logic has an immediate impact on other areas of study studying logic and the relationship between logic and ordinary speech can help a person better structure his own arguments and critique the arguments of others many popular arguments are filled with errors because so many people are untrained in logic and unaware of how to formulate an argument correctly computational logic computational logic and logic in computer science a simple toggling circuit is expressed using a logic gate and a synchronous register logic cut to the heart of computer science as it emerged as a discipline alan turings work on the entscheidungsproblem followed from kurt gdels work on the incompleteness theorems the notion of the general purpose computer that came from this work was of fundamental importance to the designers of the computer machinery in the 1940s 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 was 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 today logic is extensively applied in the fields of artificial intelligence and computer science and these fields provide a rich source of problems in formal and informal logic argumentation theory is one good example of how logic is being applied to artificial intelligence the acm computing classification system in particular regards section f3 on logics and meanings of programs and f4 on mathematical logic and formal languages as part of the theory of computer science this work covers formal semantics of programming languages as well as work of formal methods such as hoare logic boolean logic as fundamental to computer hardware particularly the systems section b2 on arithmetic and logic structures relating to operatives and not and or many fundamental logical formalisms are essential to section i2 on artificial intelligence for example modal logic and default logic in knowledge representation formalisms and methods horn clauses in logic programming and description logic furthermore computers can be used as tools for logicians for example in symbolic logic and mathematical logic proofs by humans can be computerassisted using automated theorem proving the machines can find and check proofs as well as work with proofs too lengthy to write out by hand nonclassical logic nonclassical logic the logics discussed above are all bivalent or twovalued that is they are most naturally understood as dividing propositions into true and false propositions nonclassical logics are those systems that reject various rules of classical logic hegel developed his own dialectic logic that extended kants transcendental logic but also brought it back to ground by assuring us that neither in heaven nor in earth neither in the world of mind nor of nature is there anywhere such an abstract eitheror as the understanding maintains whatever exists is concrete with difference and opposition in itself in 1910 nicolai a vasiliev extended the law of excluded middle and the law of contradiction and proposed the law of excluded fourth and logic tolerant to contradiction 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 intuitionistic logic was proposed by lej 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 formalization in mathematics but his student arend heyting studied intuitionistic logic formally as did gerhard gentzen intuitionistic logic is of great interest to computer scientists as it is a constructive logic and sees many applications such as extracting verified programs from proofs and influencing the design of programming languages through the formulaeastypes correspondence modal logic is not truth conditional and so it has often been proposed as a nonclassical logic however modal logic is normally formalized with the principle of the excluded middle and its relational semantics is bivalent so this inclusion is disputable controversies is logic empirical further information is logic empirical 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 another paper of the same name by michael dummett argues that putnams desire for realism mandates the law of distributivity distributivity of logic is essential for the realists 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 antirealism implication strict or material 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 225 which are puzzling because natural language does not support the principle of explosion eliminating this class of paradoxes was the reason for c i lewiss 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 mans 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 paraconsistent logic hegel was deeply critical of any simplified notion of the law of noncontradiction it was based on gottfried wilhelm leibnizs 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 noncontradiction 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 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 anthropomorphismsin 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 this position held by nietzsche however has come under extreme scrutiny for several reasons some philosophers such as jrgen habermas claim his position is selfrefutingand accuse nietzsche of not even having a coherent perspective let alone a theory of knowledge logic portal philosophy portal book logic digital electronics also known as digital logic or logic gates fallacies list of logicians list of logic journals list of logic symbols logic puzzle mathematics list of mathematics articles outline of mathematics metalogic outline of logic philosophy list of philosophy topics outline of philosophy logos reason truth vector logic notes and references bibliography mwparseroutput refbeginfontsize90marginbottom05emmwparseroutput refbeginhangingindentsulliststyletypenonemarginleft0mwparseroutput refbeginhangingindentsullimwparseroutput refbeginhangingindentsdlddmarginleft0paddingleft32emtextindent32emliststylenonemwparseroutput refbegin100fontsize100 barwise j 1982 handbook of mathematical logic 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whitehead alfred north and bertrand russell 1910 principia mathematica cambridge university press cambridge england oclc1041146 logicat wikipedias sister projects definitions from wiktionary media from wikimedia commons news from wikinews quotations from wikiquote texts from wikisource textbooks from wikibooks resources from wikiversity library resources about logic resources in your library resources in other libraries logic at philpapers logic at the indiana philosophy ontology project logic internet encyclopedia of philosophy hazewinkel michiel ed 2001 logical calculus encyclopedia of mathematics springer sciencebusiness media bv kluwer academic publishers isbn9781556080104 an outline for verbal logic introductions and tutorials an introduction to philosophical logic by paul newall aimed at beginners forall x an introduction to formal logic by pd magnus covers sentential and quantified logic logic selftaught a workbook originally prepared for online logic instruction nicholas rescher 1964 introduction to logic st martins press httpsphilpapersorgrecbeclam logic a modern guide by colin beckley an introduction to formal logic essays symbolic logic and the game of logic lewis carroll 1896 math logic the history of formal mathematical logical linguistic and methodological ideas in the dictionary of the history of ideas online tools interactive syllogistic machine a web based syllogistic machine for exploring fallacies figures terms and modes of syllogisms reference material translation tips by peter suber for translating from english into logical notation ontology and history of logic an introduction with an annotated bibliography reading lists the london philosophy study guide offers many suggestions on what to read depending on the students familiarity with the subject logic metaphysics set theory and further logic mathematical logic vtelogic outline history fields argumentation theory axiology critical thinking logic in computer science mathematical logic 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