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Sets and Extensions in the Twentieth Century. A History of its Central Concepts. The Rise of Modern Logic: Greek, Indian and Arabic Logic. Studies in Logic and the Foundations of Mathematics Book How to write a great review. The review must be at least 50 characters long. The title should be at least 4 characters long. Your display name should be at least 2 characters long. At Kobo, we try to ensure that published reviews do not contain rude or profane language, spoilers, or any of our reviewer's personal information.
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Close Report a review At Kobo, we try to ensure that published reviews do not contain rude or profane language, spoilers, or any of our reviewer's personal information. 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 suggests obvious difficulties, leading Locke to distinguish between 'real' truth, when our ideas have 'real existence' and 'imaginary' or 'verbal' truth, where ideas like harpies or centaurs exist only in the mind. Modern semantics is in some ways closer to the medieval view, in rejecting such psychological truth-conditions. However, the introduction of quantification , needed to solve the problem of multiple generality , rendered impossible the kind of subject-predicate analysis that underlies medieval semantics.
The main modern approach is model-theoretic semantics , based on Alfred Tarski 's 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: Model-theoretic semantics is one of the fundamental concepts of model theory. Modern semantics also admits rival approaches, such as the proof-theoretic 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 Wittgenstein 's later philosophy, especially his aphorism "meaning is use".
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: 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: 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 Aristotle's 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 so-called ' 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 so-called hypothetical judgment 'if anything is S, it is P'. Case arguing against Sigwart's and Brentano's modern analysis of the universal proposition. 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:.
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 , [14] 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 — 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. 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.
In Europe, logic was first developed by Aristotle. 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. 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. Logic is commonly taught by university philosophy 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.
However, it was not alone: 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 had always believed logic was a universal weapon, and now I realized how its validity depended on the way it was employed.
Quantification in Nonclassical Logic - 1st Edition - ISBN: , View all volumes in this series: Studies in Logic and the Foundations of Mathematics . 1.) Basic Propositional Logic 2.) Basic Predicate Logic 3.) Kripke Semantics 4. Quantification in Nonclassical Logic. Volume 1. D.M. Gabbay, V.B. Shehtman and D.P. Skvortsov. Volume , Pages ii-xx, (). Previous volume.
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 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 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.
The motivation for the study of logic in ancient times was clear: 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 , [32] [33] [34] and van Eemeren and Grootendorst Pragma-dialectics. Handbook of Logic and Language. Logic-Based Program Synthesis and Transformation. Proof, Computation and Agency. Formal Aspects of Component Software. Strict Finitism and the Logic of Mathematical Applications. The Logic of Time. Industrial Deployment of System Engineering Methods.
Language and Automata Theory and Applications. Grand Timely Topics in Software Engineering. Algorithmic Aspects of Cloud Computing. Computational Complexity of Solving Equation Systems. Tools and Algorithms for the Construction and Analysis of Systems. Duality Theories for Boolean Algebras with Operators.
Trends in Artificial Intelligence: Observation's Solution to the Riddle of Existence. Visionary Mathematician, Scientist and Neohumanist Scholar. Analysis and Control of Boolean Networks. Reasoning and Unification over Conceptual Graphs. Handbook of Philosophical Logic. A Study in Formal Pragmatics. British Logic in the Nineteenth Century.
Mediaeval and Renaissance Logic. A Practical Logic of Cognitive Systems. Approaches to Legal Rationality.