Logic

Logic (from the Ancient Greek: λογική, translit. logikḗ[1]), originally meaning "the word" or "what is spoken", but coming to mean "thought" or "reason", is a subject concerned with the most general laws of truth,[2] 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 mid-19th century), and recently logic has been studied in computer science, linguistics, psychology, and other fields.

Concepts

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.[3] 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.[4][5] 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'".[6]

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 A(P,Q), 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 Łukasiewicz to say that the introduction of variables was "one of Aristotle's greatest inventions".[7] 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 A(P,Q) (all Ps are Qs) of traditional logic corresponds to the more complex formula in predicate logic, involving the logical connectives for universal quantification and implication rather than just the predicate letter A and using variable arguments 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

The validity of an argument depends upon the meaning or semantics of the sentences that make it up.

Aristotle's 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 extra-linguistic 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." [8]

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.[9] Thus truth and falsity are no more than the agreement or disagreement of ideas. 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.[10] 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 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: an interpretation of first-order 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". 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

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 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'. The first was thought to imply 'some S is P', the second was not, and as late as 1911 in the Encyclopædia Britannica article on Logic, we find the Oxford logician T.H. Case arguing against Sigwart's and Brentano's modern analysis of the universal proposition.

Logical systems

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.[11]
  • Validity, which means that the system's rules of proof never allow a false inference from true premises.
  • Completeness, which means that if a formula is true, it can be proven, i.e. 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).[12]
  • Expressivity, meaning what concepts can be expressed in the system.


Some logical systems do not have all four properties. As an example, Kurt Gödel's incompleteness theorems show that sufficiently complex formal systems of arithmetic cannot be consistent and complete;[5] however, first-order predicate logics not extended by specific axioms to be arithmetic formal systems with equality can be complete and consistent.[13]

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,[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 (1839–1914) first introduced the term as "guessing".[15] Peirce said that to abduce a hypothetical explanation from an observed surprising circumstance is to surmise that may be true because then would be a matter of course.[16] Thus, to abduce from involves determining that is sufficient (or nearly sufficient), but not necessary, for .

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.

Rival conceptions

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."[17]

Logic has been defined[by whom?] as "the study of arguments correct in virtue of their form". 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.[18][page needed]

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