Among the important properties that logical systems can have are:. Some logical systems do not have all four properties. 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 , [13] which is a form of inference that goes from observation to a hypothesis that accounts for the reliable data observation and seeks to explain relevant evidence.

The American philosopher Charles Sanders Peirce â€” first introduced the term as "guessing". 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 well-understood 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.

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.

## Foundations of Logic-Based Trust Management - IEEE Conference Publication

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 [ by whom? This has not been the definition taken in this article, but 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.

Logic comes from the Greek word logos , originally meaning "the word" or "what is spoken", but coming to mean "thought" or "reason". In the Western World, logic was first developed by Aristotle , who called the subject 'analytics'. There was also the rival Stoic logic. 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.

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. In India, the Anviksiki school of logic was founded by Medhatithi Gautama c.

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 [4] 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.

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Logic is commonly taught by university philosophy, sociology, advertising and literature departments, often as a compulsory discipline. The Organon was Aristotle 's body of work on logic, with the Prior Analytics constituting the first explicit work in formal logic, introducing the syllogistic. Aristotle's 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.

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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 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. I 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. 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".

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. 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 waysâ€”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 analytical 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 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.

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.

While the study of necessity and possibility remained important to philosophers, little logical innovation happened until the landmark investigations of Clarence Irving Lewis in , 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.

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## Logic And Foundations Of Mathematics

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 , [31] [32] [33] 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 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.