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Propositional logic Main article: 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 {\displaystyle A\rightarrow B\rightarrow A}{\displaystyle A\rightarrow B\rightarrow A}, which says that if A holds, then B implies A.
that Aristotle's 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.



Predicate logic Gottlob Frege's Begriffschrift introduced the notion of quantifier in a graphical notation, which here represents the judgement that {\displaystyle \forall x.F(x)}{\displaystyle \forall x.F(x)} is true. Main article: Predicate logic Predicate logic is the generic term for symbolic formal systems such as first-order logic, second-order logic, many-sorted 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 Russell's famous barber paradox, "there is a man who shaves all and only men who do not shave themselves" can be formalised by the sentence {\displaystyle (\exists x)({\text{man}}(x)\wedge (\forall y)({\text{man}}(y)\rightarrow ({\text{shaves}}(x,y)\leftrightarrow \neg {\text{shaves}}(y,y))))}(\exists x)({\text{man}}(x)\wedge (\forall y)({\text{man}}(y)\rightarrow ({\text{shaves}}(x,y)\leftrightarrow \neg {\text{shaves}}(y,y)))), using the non-logical predicate {\displaystyle {\text{man}}(x)}{\displaystyle {\text{man}}(x)} to indicate that x is a man, and the non-logical relation {\displaystyle {\text{shaves}}(x,y)}{\displaystyle {\text{shaves}}(x,y)} 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 ways—allowing 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 analytic philosophy, but the formulation of predicate logic most often used today is the first-order 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 Tarski's approach to model theory. It provides the foundation of modern mathematical logic. Frege's original system of predicate logic was second-order, rather than first-order. Second-order logic is most prominently defended (against the criticism of Willard Van Orman Quine and others) by George Boolos and Stewart Shapiro. Modal logic Main article: Modal logic In languages, modality deals with the phenomenon that sub-parts 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. Aristotle's logic is in large parts concerned with the theory of non-modalized logic. Although, there are passages in his work, such as the famous sea-battle 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.[43][dead link] While the study of necessity and possibility remained important to philosophers, little logical innovation happened until the landmark investigations of C. I. 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 graph-theoretic way of looking at modality that has driven many applications in computational linguistics and computer science, such as dynamic logic. Informal reasoning and dialectic Main articles: Informal logic, Dialogical 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 Aristotle's 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 Aristotle's 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 pre-formal treatises on argument and dialectic, from authors such as Stephen Toulmin (The Uses of Argument), Nicholas Rescher (Dialectics),[44][45][46] and van Eemeren and Grootendorst (Pragma-dialectics). 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 Main article: 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.[47] 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.[48] Many other ancient and medieval philosophers applied mathematical ideas and methods to their philosophical claims.[49] One of the boldest attempts to apply logic to mathematics was the logicism pioneered by philosopher-logicians 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.[11] The various attempts to carry this out met with failure, 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. 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.[50] 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. 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.[51] 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.[52] 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.[53] Philosophical logic Main article: Philosophical logic Philosophical logic deals with formal descriptions of ordinary, non-specialist ("natural") language, that is strictly only about the arguments within philosophy's 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 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. Kripke's 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.[54][55] Computational logic Main articles: 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 Turing's work on the Entscheidungsproblem followed from Kurt Gödel's 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 mimics the problem-solving skills of a human being. This was more difficult than expected because of the complexity of human reasoning. In the summer of 1956, John McCarthy, Marvin Minsky, Claude Shannon and Nathan Rochester organized a conference on the subject of what they called "artificial intelligence" (a term coined by McCarthy for the occasion). Newell and Simon proudly presented the group with the Logic Theorist and were somewhat surprised when the program received a lukewarm reception. 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 field of artificial intelligence, and this field 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 F.3 on "Logics and meanings of programs" and F.4 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 system's section B.2 on "Arithmetic and logic structures", relating to operatives AND, NOT, and OR; Many fundamental logical formalisms are essential to section I.2 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 computer-assisted. Using automated theorem proving, the machines can find and check proofs, as well as work with proofs too lengthy to write out by hand. Non-classical logic Main article: Non-classical logic The logics discussed above are all "bivalent" or "two-valued"; that is, they are most naturally understood as dividing propositions into true and false propositions. Non-classical logics are those systems that reject various rules of Classical logic. Hegel developed his own dialectic logic that extended Kant's 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 'either–or' as the understanding maintains. Whatever exists is concrete, with difference and opposition in itself".[56] 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.[57] In the early 20th century Jan Lukasiewicz investigated the extension of the traditional true/false values to include a third value, "possible" (or an indeterminate, a hypothesis) so inventing ternary logic, the first multi-valued logic in the Western tradition.[58] A minor modification of the ternary logic was later introduced in a sibling ternary logic model proposed by Stephen Cole Kleene. Kleene's system differs from the Lukasiewicz's logic with respect to an outcome of the implication. The former assumes that the operator of implication between two hypotheses produces a hypothesis. 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.[59] 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 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 formulae-as-types correspondence. Modal logic is not truth conditional, and so it has often been proposed as a non-classical 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?"[60] Hilary Putnam, building on a suggestion of W. V. Quine, argued that in general the facts of propositional logic


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