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45-651: According to the Routledge Encyclopedia of Philosophy : "For the Islamic philosophers, logic included not only the study of formal patterns of inference and their validity but also elements of the philosophy of language and even of epistemology and metaphysics . Because of territorial disputes with the Arabic grammarians, Islamic philosophers were very interested in working out the relationship between logic and language, and they devoted much discussion to

90-521: A natural language such as English contains, at any given time, a finite number of words, any comprehensive list of definitions must either be circular or rely upon primitive notions . If every term of every definiens must itself be defined, "where at last should we stop?" A dictionary, for instance, insofar as it is a comprehensive list of lexical definitions , must resort to circularity . Many philosophers have chosen instead to leave some terms undefined. The scholastic philosophers claimed that

135-556: A rabbit (a class) is, by pointing at several and expecting another to understand. The process of ostensive definition itself was critically appraised by Ludwig Wittgenstein . An enumerative definition of a concept or a term is an extensional definition that gives an explicit and exhaustive listing of all the objects that fall under the concept or term in question. Enumerative definitions are only possible for finite sets (and only practical for small sets). Divisio and partitio are classical terms for definitions. A partitio

180-408: A real sense and analogical reasoning in a metaphorical sense. On the other hand, al-Ghazali (1058–1111) and Ibn Qudamah al-Maqdisi (1147-1223) argued that Qiyas refers to analogical reasoning in a real sense and categorical syllogism in a metaphorical sense. Other Islamic scholars at the time, however, argued that the term Qiyas refers to both analogical reasoning and categorical syllogism in

225-445: A "square" is a member of both genera (the plural of genus ): the genus "rectangle" and the genus "rhombus". One important form of the extensional definition is ostensive definition . This gives the meaning of a term by pointing, in the case of an individual, to the thing itself, or in the case of a class, to examples of the right kind. For example, one can explain who Alice (an individual) is, by pointing her out to another; or what

270-460: A concept. For naming the object of a definition mathematicians can use either a neologism (this was mainly the case in the past) or words or phrases of the common language (this is generally the case in modern mathematics). The precise meaning of a term given by a mathematical definition is often different from the English definition of the word used, which can lead to confusion, particularly when

315-431: A definition is something, typically expressed in words, that attaches a meaning to a word or group of words. The word or group of words that is to be defined is called the definiendum , and the word, group of words, or action that defines it is called the definiens . For example, in the definition "An elephant is a large gray animal native to Asia and Africa" , the word "elephant" is the definiendum , and everything after

360-401: A definition was taken to be a statement of the essence of a thing. Aristotle had it that an object's essential attributes form its "essential nature", and that a definition of the object must include these essential attributes. The idea that a definition should state the essence of a thing led to the distinction between nominal and real essence—a distinction originating with Aristotle. In

405-477: A different approach in medicine. Avicenna contributed inventively to the development of inductive logic , which he used to pioneer the idea of a syndrome . In his medical writings, Avicenna was the first to describe the methods of agreement, difference and concomitant variation which are critical to inductive logic and the scientific method . Ibn Hazm (994–1064) wrote the Scope of Logic , in which he stressed on

450-454: A large grey animal], "is an Asian elephant." ); and precising definitions , which reduce the vagueness of a word, typically in some special sense ( "'Large', among female Asian elephants, is any individual weighing over 5,500 pounds." ). An intensional definition , also called a connotative definition, specifies the necessary and sufficient conditions for a thing to be a member of a specific set . Any definition that attempts to set out

495-406: A metalogic). On the other hand, lambda-calculi are a kind of logic where the definitions are included as the feature of the formal system itself. Authors have used different terms to classify definitions used in formal languages like mathematics. Norman Swartz classifies a definition as "stipulative" if it is intended to guide a specific discussion. A stipulative definition might be considered

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540-461: A real sense. The first original Arabic writings on logic were produced by al-Kindi (Alkindus) (805–873), who produced a summary on earlier logic up to his time. The first writings on logic with non-Aristotelian elements was produced by al-Farabi (Alfarabi) (873–950), who discussed the topics of future contingents , the number and relation of the categories , the relation between logic and grammar , and non-Aristotelian forms of inference . He

585-417: A temporary, working definition, and can only be disproved by showing a logical contradiction. In contrast, a "descriptive" definition can be shown to be "right" or "wrong" with reference to general usage. Swartz defines a precising definition as one that extends the descriptive dictionary definition (lexical definition) for a specific purpose by including additional criteria. A precising definition narrows

630-431: A term by pointing out examples. A term may have many different senses and multiple meanings, and thus require multiple definitions. In mathematics , a definition is used to give a precise meaning to a new term, by describing a condition which unambiguously qualifies what the mathematical term is and is not. Definitions and axioms form the basis on which all of modern mathematics is to be constructed. In modern usage,

675-417: A term is only needed to avoid misunderstanding. Locke and Mill also argued that individuals cannot be defined. Names are learned by connecting an idea with a sound, so that speaker and hearer have the same idea when the same word is used. This is not possible when no one else is acquainted with the particular thing that has "fallen under our notice". Russell offered his theory of descriptions in part as

720-521: A thing must have in order that the name apply to it. This leads to a corresponding distinction between nominal and real definitions. A nominal definition is the definition explaining what a word means (i.e., which says what the "nominal essence" is), and is definition in the classical sense as given above. A real definition, by contrast, is one expressing the real nature or quid rei of the thing. This preoccupation with essence dissipated in much of modern philosophy. Analytic philosophy , in particular,

765-405: A way of defining a proper name, the definition being given by a definite description that "picks out" exactly one individual. Saul Kripke pointed to difficulties with this approach, especially in relation to modality , in his book Naming and Necessity . There is a presumption in the classic example of a definition that the definiens can be stated. Wittgenstein argued that for some terms this

810-411: Is a statement of the meaning of a term (a word , phrase , or other set of symbols ). Definitions can be classified into two large categories: intensional definitions (which try to give the sense of a term), and extensional definitions (which try to list the objects that a term describes). Another important category of definitions is the class of ostensive definitions , which convey the meaning of

855-453: Is also credited for categorizing logic into two separate groups, the first being "idea" and the second being " proof ". Averroes (1126–98) was the last major logician from al-Andalus , who wrote the most elaborate commentaries on Aristotelian logic. Avicenna (980–1037) developed his own system of logic known as "Avicennian logic" as an alternative to Aristotelian logic. By the 12th century, Avicennian logic had replaced Aristotelian logic as

900-575: Is critical of attempts to elucidate the essence of a thing. Russell described essence as "a hopelessly muddle-headed notion". More recently Kripke's formalisation of possible world semantics in modal logic led to a new approach to essentialism . Insofar as the essential properties of a thing are necessary to it, they are those things that it possesses in all possible worlds. Kripke refers to names used in this way as rigid designators . A definition may also be classified as an operational definition or theoretical definition . A homonym is, in

945-455: Is not possible since it is not known who the future prime ministers will be (even though all prime ministers from the past and present can be listed). A genus–differentia definition is a type of intensional definition that takes a large category (the genus ) and narrows it down to a smaller category by a distinguishing characteristic (i.e. the differentia ). More formally, a genus–differentia definition consists of: For example, consider

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990-461: Is simply an intensional definition. A divisio is not an extensional definition, but an exhaustive list of subsets of a set, in the sense that every member of the "divided" set is a member of one of the subsets. An extreme form of divisio lists all sets whose only member is a member of the "divided" set. The difference between this and an extensional definition is that extensional definitions list members , and not subsets . In classical thought,

1035-479: Is the capacity for a sign (such as a word , phrase , or symbol ) to have multiple meanings (that is, multiple semes or sememes and thus multiple senses ), usually related by contiguity of meaning within a semantic field . It is thus usually regarded as distinct from homonymy , in which the multiple meanings of a word may be unconnected or unrelated. In mathematics, definitions are generally not used to describe existing terms, but to describe or characterize

1080-520: Is to create or alter rights, duties, or crimes. A recursive definition , sometimes also called an inductive definition, is one that defines a word in terms of itself, so to speak, albeit in a useful way. Normally this consists of three steps: For instance, we could define a natural number as follows (after Peano ): So "0" will have exactly one successor, which for convenience can be called "1". In turn, "1" will have exactly one successor, which could be called "2", and so on. The second condition in

1125-599: The Illuminationist school , founded by Shahab al-Din Suhrawardi (1155–1191), who developed the idea of "decisive necessity ", an important innovation in the history of logical philosophical speculation. Another systematic refutation of Greek logic was written by Ibn Taymiyyah (1263–1328), the Ar-Radd 'ala al-Mantiqiyyin ( Refutation of Greek Logicians ), where he argued against the usefulness, though not

1170-470: The Posterior Analytics , he says that the meaning of a made-up name can be known (he gives the example "goat stag") without knowing what he calls the "essential nature" of the thing that the name would denote (if there were such a thing). This led medieval logicians to distinguish between what they called the quid nominis , or the "whatness of the name", and the underlying nature common to all

1215-611: The 7th century with the process of Qiyas , before the Arabic translations of Aristotle's works. Later during the Islamic Golden Age , there was a logical debate among Islamic philosophers, logicians and theologians over whether the term Qiyas refers to analogical reasoning, inductive reasoning or categorical syllogism. Some Islamic scholars argued that Qiyas refers to inductive reasoning, which Ibn Hazm (994-1064) disagreed with, arguing that Qiyas does not refer to inductive reasoning, but refers to categorical syllogism in

1260-607: The definition itself refers to natural numbers, and hence involves self-reference . Although this sort of definition involves a form of circularity , it is not vicious , and the definition has been quite successful. In the same way, we can define ancestor as follows: Or simply: an ancestor is a parent or a parent of an ancestor. In medical dictionaries , guidelines and other consensus statements and classifications , definitions should as far as possible be: Certain rules have traditionally been given for definitions (in particular, genus-differentia definitions). Given that

1305-484: The development of a strict science of citation , the isnad or "backing", and the development of a scientific method of open inquiry to disprove claims, the ijtihad , which could be generally applied to many types of questions. Early forms of analogical reasoning , inductive reasoning and categorical syllogism were introduced in Fiqh (Islamic jurisprudence), Sharia (Islamic law) and Kalam (Islamic theology) from

1350-661: The dominant system of logic in the Islamic world. The first criticisms of Aristotelian logic were written by Avicenna, who produced independent treatises on logic rather than commentaries. He criticized the logical school of Baghdad for their devotion to Aristotle at the time. He investigated the theory of definition and classification and the quantification of the predicates of categorical propositions , and developed an original theory on " temporal modal " syllogism. Its premises included modifiers such as "at all times", "at most times", and "at some time". While Avicenna often relied on deductive reasoning in philosophy, he used

1395-429: The essence of something, such as that by genus and differentia , is an intensional definition. An extensional definition , also called a denotative definition, of a concept or term specifies its extension . It is a list naming every object that is a member of a specific set . Thus, the " seven deadly sins " can be defined intensionally as those singled out by Pope Gregory I as particularly destructive of

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1440-481: The focal point of logic. Even poetics was considered as a syllogistic art in some fashion by most of the major Islamic Aristotelians." Important developments made by Muslim logicians included the development of "Avicennian logic" as a replacement of Aristotelian logic. Avicenna 's system of logic was responsible for the introduction of hypothetical syllogism , temporal modal logic and inductive logic . Other important developments in early Islamic philosophy include

1485-513: The following genus–differentia definitions: Those definitions can be expressed as a genus ("a plane figure") and two differentiae ("that has three straight bounding sides" and "that has four straight bounding sides", respectively). It is also possible to have two different genus–differentia definitions that describe the same term, especially when the term describes the overlap of two large categories. For instance, both of these genus–differentia definitions of "square" are equally acceptable: Thus,

1530-572: The highest genera (called the ten generalissima ) cannot be defined, since a higher genus cannot be assigned under which they may fall. Thus being , unity and similar concepts cannot be defined. Locke supposes in An Essay Concerning Human Understanding that the names of simple concepts do not admit of any definition. More recently Bertrand Russell sought to develop a formal language based on logical atoms . Other philosophers, notably Wittgenstein , rejected

1575-514: The importance of sense perception as a source of knowledge. Al-Ghazali (Algazel) (1058–1111) had an important influence on the use of logic in theology, making use of Avicennian logic in Kalam . Fakhr al-Din al-Razi (b. 1149) criticised Aristotle's " first figure " and developed a form of inductive logic , foreshadowing the system of inductive logic developed by John Stuart Mill (1806–1873). Systematic refutations of Greek logic were written by

1620-411: The life of grace and charity within a person, thus creating the threat of eternal damnation. An extensional definition, on the other hand, would be the list of wrath, greed, sloth, pride, lust, envy, and gluttony. In contrast, while an intensional definition of " prime minister " might be "the most senior minister of a cabinet in the executive branch of parliamentary government", an extensional definition

1665-516: The meanings are close. For example, a set is not exactly the same thing in mathematics and in common language. In some case, the word used can be misleading; for example, a real number has nothing more (or less) real than an imaginary number . Frequently, a definition uses a phrase built with common English words, which has no meaning outside mathematics, such as primitive group or irreducible variety . In first-order logic definitions are usually introduced using extension by definition (so using

1710-399: The need for any undefined simples. Wittgenstein pointed out in his Philosophical Investigations that what counts as a "simple" in one circumstance might not do so in another. He rejected the very idea that every explanation of the meaning of a term needed itself to be explained: "As though an explanation hung in the air unless supported by another one", claiming instead that explanation of

1755-429: The pair stalk (part of a plant) and stalk (follow/harass a person) and the pair left (past tense of leave) and left (opposite of right). A distinction is sometimes made between "true" homonyms, which are unrelated in origin, such as skate (glide on ice) and skate (the fish), and polysemous homonyms, or polysemes , which have a shared origin, such as mouth (of a river) and mouth (of an animal). Polysemy

1800-510: The question of the subject matter and aims of logic in relation to reasoning and speech. In the area of formal logical analysis, they elaborated upon the theory of terms , propositions and syllogisms as formulated in Aristotle's Categories, De interpretatione and Prior Analytics. In the spirit of Aristotle, they considered the syllogism to be the form to which all rational argumentation could be reduced, and they regarded syllogistic theory as

1845-404: The set of things that meet the definition. C.L. Stevenson has identified persuasive definition as a form of stipulative definition which purports to state the "true" or "commonly accepted" meaning of a term, while in reality stipulating an altered use (perhaps as an argument for some specific belief). Stevenson has also noted that some definitions are "legal" or "coercive" – their object

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1890-407: The strict sense, one of a group of words that share the same spelling and pronunciation but have different meanings. Thus homonyms are simultaneously homographs (words that share the same spelling, regardless of their pronunciation) and homophones (words that share the same pronunciation, regardless of their spelling). The state of being a homonym is called homonymy . Examples of homonyms are

1935-413: The things it names, which they called the quid rei , or the "whatness of the thing". The name " hobbit ", for example, is perfectly meaningful. It has a quid nominis , but one could not know the real nature of hobbits, and so the quid rei of hobbits cannot be known. By contrast, the name "man" denotes real things (men) that have a certain quid rei . The meaning of a name is distinct from the nature that

1980-534: The validity, of the syllogism and in favour of inductive reasoning . Routledge Encyclopedia of Philosophy Too Many Requests If you report this error to the Wikimedia System Administrators, please include the details below. Request from 172.68.168.226 via cp1108 cp1108, Varnish XID 216988157 Upstream caches: cp1108 int Error: 429, Too Many Requests at Thu, 28 Nov 2024 07:53:20 GMT Definition A definition

2025-489: The word "is" is the definiens . The definiens is not the meaning of the word defined, but is instead something that conveys the same meaning as that word. There are many sub-types of definitions, often specific to a given field of knowledge or study. These include, lexical definitions , or the common dictionary definitions of words already in a language; demonstrative definitions , which define something by pointing to an example of it ( "This," [said while pointing to

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