Extension and Intension

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Extension and intension ( Latin extensio 'expansion, span, spread' and Latin intensio 'effort, tension, tension') are terms from semantics with which different dimensions of the meaning of linguistic expressions (predicates, sentences) or logical entities (quantities, concepts , Propositions). The pair of terms comes from the context of Aristotelian logic and was established as “étendue de l'idee” , “scope of imagination” and “comprehension de l'idée” , “epitome of imagination” through the logic of Port-Royal . In the philosophy of language , the linguistics , logic and mathematics , extension and intention are often designed differently. For predicates and concepts, the terms scope and content are unproblematic translations.

Conceptual content and scope

The term content is the totality of all characteristics of a term, the term scope is the "totality of the terms subordinate to a term on the same hierarchy level". The scope of the term vehicle, for example, consists of the sub-terms land vehicle , watercraft , aircraft and spacecraft . The term content of the aircraft comprising the features within the earth's atmosphere to fly or to drive ( ballooning ). This distinguishes the aircraft from the spacecraft that is traveling in space . The larger the term, the smaller the scope of the term.

Extension and Intension of Predicates (Terms)


In traditional logic ( conceptual logic ), the extension or scope of a concept was understood to mean the totality of the things to which it extends (which fall under it, which it includes). Accordingly, the extension of the term “ human ” was the totality of all human beings. Since the Pyrrhonic skepticism, however, there have also been doubts about such conceptual potential. With the emergence of empirical sciences, taxonomies came more into the individual scientific areas of responsibility and their philosophical or theological- syllogistic administration became obsolete. In traditional logic, a sufficiently complex ontology has never succeeded to enable practical review and decision-making processes; the only example of this is the widely discussed question of what belongs to the totality of all people and what does not (e.g. deceased people, disabled people , Corpses, future people, only possibly existing people. For the problem see also presentationalism , actualism ). The last defender of such a conceptual logic was Bruno von Freytag-Löringhoff .

In classical logic , terms are often understood as single-digit predicates , that is, as forms of expression with a space. A true statement arises from the statement form "... is a person" if one uses the proper name or the identification of a person in the space. The extension of such a predicate is then the set of referents of all those proper names and labels which, inserted into the space, result in a true statement. The extension is therefore the set of objects to which the property expressed by the predicate belongs. The same applies to multi-digit predicates ( relations ): The extension of the two-digit predicate "... has the same father as ..." consists of the set of all sibling and (paternal) half-sibling pairs.


In logic, opinions differ about what intension and conceptual content are. According to a frequently held view, the intention of a term consists of the totality of the characteristics or properties - the terminology is inconsistent here - which are in fact common to the things it comprises or which constitute the intersection of their necessary characteristics. Accordingly, the intension of the term “man” contains the characteristics animate , mortal , walking on two legs , feathered , rational , producing tools, etc.

Conceptual features mainly appear when defining a term:

  • Humans are unfeathered creatures that walk on two legs .


  • Humans are rational beings .

None of these definitions make use of all the characteristics that are common to all people; both come z. B. without the trait mortal . Nevertheless, they fulfill their purpose, namely to filter out precisely those things from a discourse universe that only includes physical things that fall under the term “human”. If, on the other hand, we were talking about a world in which there is also room for immortals gifted with reason, e.g. B. for the goddesses and gods of Olympus , the second definition would have to be narrowed in order to fulfill this function by adding the characteristic mortal .

The examples also show that terms with different intentions in the same discourse universe can have the same extension: "Unfeathered living beings walking on two legs" and "reasonable living beings" are extensionally the same terms. The reverse does not apply: terms with different extensions always have different intentions in the same discourse universe.

Extensional individuation of concepts

It is well known that many words are ambiguous: the word “bank” can designate a seat or a financial institution. Both meanings are different terms. What constitutes the difference between these concepts and how do you recognize the equality and difference of concepts? A simple attempt to answer this question is called the extensionality thesis, according to which concepts are completely determined by their extensional range. Obviously, the amount of all seating is a different amount than that of all financial institutions.

This extensionality thesis has, among other things, the well-known problem of explaining how it behaves with terms like " evening star " and " morning star ". The extension of both names is identical: Both refer to the planet Venus . Nevertheless, it seems plausible that those who think of the evening star use a different term than those who think of the morning star. The difference, according to the classic formulation of Gottlob Frege , is not in the extension, but in the way in which it refers to the designated object, i.e. the intension. Frege himself does not speak of extension, but of meaning , and not of intension, but of meaning . If one also uses the intension for the individuation of concepts, the extensionality thesis must be rejected.

Inverse relation of intension and extension

If one understands the intension as a set of features and the extension as a set of objects that possess these features, then the intension and extension are obviously opposed to one another in the following way: the more extensive the intension, the smaller the extension and vice versa. According to the Aristotelian ontology, a term like “substance” encompasses everything that exists, a term like “corporeal substance” correspondingly less, and a term like “rational, animated corporeal substance” even fewer objects. Such examples exist in great numbers and suggest the following basic law:

If A and B are two classes within a classification with a uniformly implemented system of characteristics, then the following applies:

When the intension of A (as a subset ) in the intension of B is included, then the extension of is B (as a subset) in the extension of A contained. B is then a subclass (a species) from A , A is a superclass (a Genus) of B .

With the advent of modern logic, the generality of this rule has been challenged in various ways. The reason for this lay in the aforementioned indeterminacy of the concept of intension and in the multitude of possibilities to translate it into the formal language of a logic calculus . Paul Weingartner made the first successful attempt at such a translation . Weingartner was able to show that “with a corresponding definition of intensional abstinence”, the basic rule formulated above represents a theorem of class logic .

The German philosopher Lutz Geldsetzer has also developed a clear "pyramidal" notation for intentional logics and has dealt with the relationship between extension and intention.

The multi-valued so-called Bayesian logic also has dimensional features.


The juxtaposition between extension and intension, whose roots go back to the Aristotelian logic, is shaped in the logic of Port-Royal . A compact formulation by Leibniz can also be cited as an example : “The living being comprises more individuals than humans, but humans contain more ideas or formal properties; one has more copies, the other more realism ; one has more extension, the other more intension. "

In the course of the history of philosophy, the concept of extension and intention has been applied by different authors, whereby one should be extremely careful when equating the pairs of terms, especially since some authors treat them as properties of mental entities (concepts, judgments), others as properties of linguistic expressions. The following table shows some of these names.

Intension Extension author
Intension Extension Leibniz , Carnap , Stegmüller
content scope Kant
comprehension étendue Logic of Jansenism in Port-Royal
connotation and comprehension denotation John Stuart Mill
sense meaning Thank God Frege
meaning denotation Bertrand Russell
sense reference Max Black
meaning reference Willard Van Orman Quine
sense denotation Alonzo Church
intentional meaning (term) extensional meaning (class) Kamlah - Lorenzen
meaning designation Georg Klaus
meaning object Edmund Husserl
meaning reference Franz von Kutschera
meaning Designata Karl-Dieter Opp

It should be noted that with Frege, particular care should be taken to equate the expression “meaning” with the extension. The distinction between extension and intension is basically used for conceptual words ("planet"), while Frege also uses the distinction between meaning and meaning in proper names (where the meaning is the way an object is given, the meaning of the corresponding object) and whole sentences ( the sense here is the thought, the meaning applies the true / false). In addition, there are also differences when applied to terms: While the extension of “planet” includes the planets of the solar system, for Frege the meaning of “planet” is the abstract term “() is a planet”. In addition, in odd contexts or opaque contexts, the original meaning becomes the meaning of the expression. Frege leaves open what takes the place of meaning.

Extension and Intension of Sentences

Extension of a sentence

According to the widespread, controversial view founded by Gottlob Frege , the extension of a statement is its truth value .

“We have seen that a meaning has to be sought for a sentence whenever the meaning of the constituent parts is important; and that is always the case and only then when we ask about the truth value. We are thus urged to recognize the truth value of a sentence as its meaning. "

- Thank God Frege

Intension of a sentence

The intention of a sentence (in Frege: the meaning of a sentence) is, according to the widespread, controversial view, its meaning, content or the expressed (subjective) thought or a proposition; according to Frege, the meaning of a sentence is its thought (in an objective sense). After Rudolf Carnap the intension of a sentence is designated by the set proposition .


Law, jurisprudence and administrative action

It is part of the everyday business of lawyers to link concrete facts with legal norms in which terms, especially indefinite or vague terms, play a central role. On the one hand, it is a question of determining the intention of a term to be used in such a way that clear distinctions can be made in practice, and at the same time indicating the potential extension: Cases with the feature x (determination of intention) belong to the set (determination of extension) of those with the Standardized facts denoted by the term “y”.

Example: Section 242 (Theft) Paragraph 1 of the German Criminal Code reads: "Anyone who takes away a movable property from another person with the intention of illegally appropriating the thing to himself or a third party is punished with imprisonment for up to five years or with a fine." The term “theft” cannot be applied to cases of tapping electrical energy (limited extension of the term “theft”), since electricity is not a “thing” (limited intention of the term “thing”). The loophole in the law that emerged in this way was closed by inserting Section 248c into the StGB, which puts the “withdrawal of electrical energy” under threat of punishment.

Religious Studies and Theology

The question of whether what a concept designates as a linguistic sign exists or not can be treated not only as empirical, but also as ontological or mythological. Then terms like “God”, “Devil”, “Angel” do not simply have a zero extension, but a more complex intention. Gods like Zeus “exist” as part of the term “Greek mythology”.

See also


History of logic

Overview representations
  • Joseph C. Frisch: Extension and Comprehension in Logic. New York 1969.
  • RH Robins: A Short History of Linguistics. Longman, 1967. (4th edition. 1997)
  • Ellen Walther-Klaus: Content and scope. Georg Olms Publishing House. 1987, ISBN 3-487-07829-5 . (comprehensive historical presentation, including on late antiquity, Porphyry, Scholasticism, Petrus Hispanus, Thomas de Vio, Port-Royal, Leibniz, Kant, Erdmann, Peirce, Bolzano and the respective intermediate phases)
  • William T. Parry, Edward A. Hacker: Aristotelian Logic. SUNY, 1991, especially p. 60 ff.
middle Ages
Early modern age
  • Jill Vance Buroker: The Port-Royal semantics of terms. In: Synthesis. 96/3, 1993, pp. 455-475.
  • Wolfgang Lenzen : On the extensional and 'intensional' interpretation of Leibniz's logic. In: Studia Leibnitiana. 15, 1983, pp. 129-148.
  • Chris Swoyer: Leibniz on Intension and Extension. In: Noûs. 29/1, 1995, pp. 96-114.
  • Raili Kauppi: About Leibniz's logic with special consideration of the problem of intension and extension. (= Acta philosphica Fennica. 12). Suomen Filosofinen Yhdistys, Helsinki 1960. ((The Philosophy of Leibniz. 6). Garland, New York / London 1985)
  • A. Heinekamp, ​​F. Schupp: The Intensionale Logic in Leibniz and in the Present. Wiesbaden 1979.


  • Thank God Frege : About meaning and meaning . In: Journal for Philosophy and Philosophical Criticism. NF 100, 1892, pp. 25-50. (Reprints, e.g. In: K. Berka, L. Kreiser (Ed.): Logic Texts. Akademie-Verlag, Berlin 1983, pp. 423–442. Online at gavagai.de )
  • Bertrand Russell : On denoting. In: Mind, New Series. vol. XIV, 1905, pp. 479-493. (Reprinted in: B. Russell: Logic and knowledge. London / New York 1956)
  • Rudolf Carnap : Importance and Necessity. Springer, Berlin / New York 1972. (English: Meaning and Necessity. A Study in Semantics and Modal Logic. 1947. 2nd edition. 1956)
  • Willard Van Orman Quine : Logic and Reification of Universals. New York 1970.
  • Willard Van Orman Quine: Word and Object. Cambridge, Mass. 1960. (German: word and object. Translated by Joachim Schulte and Dieter Birnbacher. Stuttgart 1980)
  • Franz von Kutschera : Philosophy of Language. Fink, Munich 1993, especially p. 66 ff.
  • Clarence Irving Lewis : Notes on the Logic of Intension. In: Structure, Method, and Meaning: Essays in Honor of Henry M. Sheffer. Liberal Arts Press, New York 1951, pp. 25-34.
  • Thomas Bernhard Seiler : Understanding and Understanding. Verlag Allgemeine Wissenschaft, Darmstadt 2001, ISBN 3-935924-00-3 .

Web links

Individual evidence

  1. ^ Antoine Arnauld, Pierre Nicole: La logique ou l'art de penser. I, ch. VI, Paris 1662. (Pierre Claire, Francois Girbal (Hrsg.), Paris 1981, p. 59; German: The Logic or Art of Thinking. 2nd edition. 1994, p. 48) Cf. scope logic / content logic In: Volker Peckhaus: Historical Dictionary of Philosophy . Volume 11, p. 84.
  2. DIN 2342 2011-08, terminology concepts
  3. Eva-Maria Baxmann-Krafft / Gottfried Herzog, Norms for Translators and Technical Authors , 1999, p. 112
  4. See e.g. B. Bruno von Freytag-Löringhoff: Logic I. The system of pure logic and its relationship to logistics. Verlag Kohlhammer, Stuttgart 1955. (5th edition. Stuttgart 1972)
  5. ^ Paul Weingartner: A Predicate Calculus for Intensional Logic. In: Journal of Philosophical Logic. 2, 1973, pp. 220-303.
  6. ^ Paul Weingartner: Theory of Science. Part II, 1: Basic problems of logic and mathematics. Frommann-Holzboog, Stuttgart-Bad Cannstatt 1976, ISBN 3-7728-0324-5 , pp. 159f., Note 191.
  7. ↑ Term content / scope. In: Historical Dictionary of Philosophy . Volume 1, p. 808.
  8. ^ Translation of the French text from Leibniz: Nouveaux Essais. IV, Chapter 17, § 8, In: All writings and letters. Row VI: Philosophical Writings. Volume 6, Berlin 1962, p. 486.
  9. After Herberger, Simon: Wissenschaftstheorie. 1980, p. 243 with further references
  10. Frege: Sense and Meaning. P. 33 f., Quoted in: H. Kaschmieder: Assessable content and thought in the philosophy of Gottlob Freges. Olms, Hildesheim u. a. 1989, p. 61.
  11. ^ Rudolf Carnap: Introduction to Symbolic Logic. 3. Edition. Springer, Vienna / New York 1968, pp. 40, 100.