An extensional statement is a non-intensional statement. Substitution of co-extensive expressions into it always preserves logical value. A language is intensional if it contains intensional statements, and extensional otherwise. All natural languages are intensional.
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Lambda calculus is usually interpreted as using intensional equality. There are potential problems with the interpretation of results because of the difference between the intensional and extensional definition of equality.
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Materna, Pavel (2014) "Is Transparent Intensional Logic a non-classical logic?". Logic and Logical Philosophy 23 (1): 47–55. ISSN 2300-9802. Duží, Marie, Bjørn Jespersen & Pavel Materna (2010) Procedural Semantics for Hyperintensional Logic – Foundations and Applications of Transparent Intensional Logic.
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Already in 1951, Alonzo Church had developed an intensional calculus. The semantical motivations were explained expressively, of course without those tools that we know in establishing semantics for modal logic in a formal way, because they had not been invented then: Church has not provided formal semantic definitions. Later, possible world approach to semantics provided tools for a comprehensive study in intensional semantics. Richard Montague could preserve the most important advantages of Church's intensional calculus in his system.
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Pavel Materna (born 21 April 1930) is a Czech philosopher, logician and key representative of transparent intensional logic.
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An intensional statement-form is a statement-form with at least one instance such that substituting co-extensive expressions into it does not always preserve logical value. An intensional statement is a statement that is an instance of an intensional statement-form. Here co-extensive expressions are expressions with the same extension. That is, a statement-form is intensional if it has, as one of its instances, a statement for which there are two co-extensive expressions (in the relevant language) such that one of them occurs in the statement, and if the other one is put in its place (uniformly, so that it replaces the former expression wherever it occurs in the statement), the result is a (different) statement with a different logical value.
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Intensional logic is an approach to predicate logic that extends first-order logic, which has quantifiers that range over the individuals of a universe (extensions), by additional quantifiers that range over terms that may have such individuals as their value (intensions). The distinction between intensional and extensional entities is parallel to the distinction between sense and reference.
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He questioned the "intensional isomorphism" concept of Rudolf Carnap.Avrum Stroll, Twentieth-century Analytic Philosophy, Stroll, New York: Columbia University Press, 2000, p. 83.
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During 1985–1988G. Japaridze, "The polymodal logic of provability". Intensional Logics and Logical Structure of Theories. Metsniereba, Tbilisi, 1988, pages 16-48 (Russian).
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Amsterdam: John Benjamins, 191–205. . Materna, Pavel (1985) "“Linguistic constructions” in the transparent intensional logic". Prague Bulletin of Mathematical Linguistics 43: 5–24. ISSN 0032-6585.
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In logic, philosophy, and mathematics, extensional and intensional definitions are two key ways in which the object(s) or concept(s) a term refers to can be defined.
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The first higher-dimensional models of intensional type theory were constructed by Steve Awodey and his student Michael Warren in 2005 using Quillen model categories. These results were first presented in public at the conference FMCS 2006Foundational Methods in Computer Science, University of Calgary, June 7th - 9th, 2006 at which Warren gave a talk titled "Homotopy models of intensional type theory", which also served as his thesis prospectus (the dissertation committee present were Awodey, Nicola Gambino and Alex Simpson). A summary is contained in Warren's thesis prospectus abstract. At a subsequent workshop about identity types at Uppsala University in 2006Identity Types - Topological and Categorical Structure, Workshop, Uppsala, November 13-14, 2006 there were two talks about the relation between intensional type theory and factorization systems: one by Richard Garner, "Factorisation systems for type theory",Richard Garner, Factorisation axioms for type theory and one by Michael Warren, "Model categories and intensional identity types".
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The needs of thoroughly intensional theories such as untyped lambda calculus have been met in denotational semantics. Topos theory has long looked like a possible 'master theory' in this area.
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An intensional statement, then, is an instance of such a form; it has the same form as a statement in which substitution of co-extensive terms fails to preserve logical value.
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Two lambda terms are equal if they are alpha convertible. The extensional definition of function equality is that two functions are equal if they perform the same mapping; : f = g \iff (\forall x f\ x = g\ x) One way to describe this is that extensional equality describes equality of functions, where as intensional equality describes equality of function implementations. The extensional definition of equality is not equivalent to the intensional definition. This can be seen in the example below.
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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 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.
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Bertrand Russell thought that this demonstrated the failure of substitutivity of identicals in intensional contexts. In "A Puzzle about Belief,"Kripke, Saul. "A Puzzle about Belief." First appeared in, Meaning and Use. ed.
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As becomes clear, intensional definitions are best used when something has a clearly defined set of properties, and they work well for terms that have too many referents to list in an extensional definition. It is impossible to give an extensional definition for a term with an infinite set of referents, but an intensional one can often be stated concisely – there are infinitely many even numbers, impossible to list, but the term "even numbers" can be defined easily by saying that even numbers are integer multiples of two. Definition by genus and difference, in which something is defined by first stating the broad category it belongs to and then distinguished by specific properties, is a type of intensional definition. As the name might suggest, this is the type of definition used in Linnaean taxonomy to categorize living things, but is by no means restricted to biology.
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This inequivalence is created by considering lambda terms as values. In typed lambda calculus this problem is circumvented, because built-in types may be added to carry values that are in a canonical form and have both extensional and intensional equality.
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Pavel Tichý (; 18 February 1936, Brno, Czechoslovakia - 26 October 1994, Dunedin, New Zealand) was a Czech logician, philosopher and mathematician. He worked in the field of intensional logic and founded Transparent Intensional Logic, an original theory of the logical analysis of natural languages – the theory is devoted to the problem of saying exactly what it is that we learn, know and can communicate when we come to understand what a sentence means. He spent roughly 25 years working on it. His main work is a book The Foundations of Frege's Logic, published by Walter de Gruyter in 1988.
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At one time the idea that types in intensional type theory with their identity types could be regarded as groupoids was mathematical folklore. It was first made precise semantically in the 1998 paper of Martin Hofmann and Thomas Streicher called "The groupoid interpretation of type theory", in which they showed that intensional type theory had a model in the category of groupoids. This was the first truly "homotopical" model of type theory, albeit only "1-dimensional" (the traditional models in the category of sets being homotopically 0-dimensional). Their paper also foreshadowed several later developments in homotopy type theory.
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However, this doesn't prevent extensional type theory from being a basis for a practical tool, for example, NuPRL is based on extensional type theory. In contrast in intensional type theory type checking is decidable, but the representation of standard mathematical concepts is somewhat more cumbersome, since intensional reasoning requires using setoids or similar constructions. There are many common mathematical objects, which are hard to work with or can't be represented without this, for example, integer numbers, rational numbers, and real numbers. Integers and rational numbers can be represented without setoids, but this representation isn't easy to work with.
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Arthur Norman Prior (4 December 1914 – 6 October 1969), usually cited as A. N. Prior, was a New Zealand–born logician and philosopher. Prior (1957) founded tense logic, now also known as temporal logic, and made important contributions to intensional logic, particularly in Prior (1971).
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A fundamental distinction is extensional vs intensional type theory. In extensional type theory definitional (i.e., computational) equality is not distinguished from propositional equality, which requires proof. As a consequence type checking becomes undecidable in extensional type theory because programs in the theory might not terminate.
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An intensional definition gives the meaning of a term by specifying necessary and sufficient conditions for when the term should be used. In the case of nouns, this is equivalent to specifying the properties that an object needs to have in order to be counted as a referent of the term. For example, an intensional definition of the word "bachelor" is "unmarried man". This definition is valid because being an unmarried man is both a necessary condition and a sufficient condition for being a bachelor: it is necessary because one cannot be a bachelor without being an unmarried man, and it is sufficient because any unmarried man is a bachelor.
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For example, an intensional definition of square number can be "any number that can be expressed as some integer multiplied by itself". The rule—"take an integer and multiply it by itself"—always generates members of the set of square numbers, no matter which integer one chooses, and for any square number, there is an integer that was multiplied by itself to get it. Similarly, an intensional definition of a game, such as chess, would be the rules of the game; any game played by those rules must be a game of chess, and any game properly called a game of chess must have been played by those rules.
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Typically (although it depends on the type theory used), the axiom of choice will hold for functions between types (intensional functions), but not for functions between setoids (extensional functions). The term "set" is variously used either as a synonym of "type" or as a synonym of "setoid".
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Lewis gives Marcus special recognition in his "Notes on the Logic of Intension", originally printed in Structure, Method, and Meaning: Essays in Honor of Henry M. Sheffer (New York, 1951). Here Lewis recognizes Barcan Marcus as the first logician to extend propositional logic as a higher order intensional logic.
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Another influential philosopher, Pavel Tichý initiated Transparent Intensional Logic, an original theory of the logical analysis of natural languages—the theory is devoted to the problem of saying exactly what it is that we learn, know and can communicate when we come to understand what a sentence means.
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In at least one source,"From Aristotle to EA: a type theory for EA" a class is a set in which an individual member can be recognized in one or both of two ways: a) it is included in an extensional definition of the whole set (a list of set members) b) it matches an Intensional definition of one set member. By contrast, a "type" is an intensional definition; it is a description that is sufficiently generalized to fit every member of a set. Philosophers sometimes distinguish classes from types and kinds. We can talk about the class of human beings, just as we can talk about the type (or natural kind), human being, or humanity.
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His research interests include categorical logic, domain theory and Martin-Löf type theory. In joint work with Martin Hofmann he constructed a model for intensional Martin-Löf type theory where identity types are interpreted as groupoids. This was the first model with non-trivial identity types, i.e. other than sets.
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Cook, Roy T. "Intensional Definition". In A Dictionary of Philosophical Logic. Edinburgh: Edinburgh University Press, 2009. 155. This is the opposite approach to the extensional definition, which defines by listing everything that falls under that definition – an extensional definition of bachelor would be a listing of all the unmarried men in the world.
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In philosophical logic, the masked-man fallacy (also known as the intensional fallacy and the epistemic fallacy) is committed when one makes an illicit use of Leibniz's law in an argument. Leibniz's law states that if A and B are the same object, then A and B are indiscernible (that is, they have all the same properties). By modus tollens, this means that if one object has a certain property, while another object does not have the same property, the two objects cannot be identical. The fallacy is "epistemic" because it posits an immediate identity between a subject's knowledge of an object with the object itself, failing to recognize that Leibniz's Law is not capable of accounting for intensional contexts.
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If-then arguments posit logical sequences that sometimes include objects of the mind. For example, a counterfactual argument proposes a hypothetical or subjunctive possibility which could or would be true, but might not be false. Conditional sequences involving subjunctives use intensional language, which is studied by modal logic,Modal Logic. Springer Online Reference Works.
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This is similar to an ostensive definition, in which one or more members of a set (but not necessarily all) are pointed out as examples. The opposite approach is the intensional definition, which defines by listing properties that a thing must have in order to be part of the set captured by the definition.
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" (1909 p. 66) Propositional functions: "The characteristic of a class concept, as distinguished from terms in general, is that "x is a u" is a propositional function when, and only when, u is a class-concept." (1903:56) Extensional versus intensional definition of a class: "71. Class may be defined either extensionally or intensionally.
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On December 3, 2019, lawyer Jordan Merson filed a lawsuit in New York on behalf of nine anonymous accusers (Jane Does 1-9) and against Epstein’s estate for battery, assault, and intensional emotional distress. The claims date from 1985 through the 2000s, and include individuals who were 13, 14, and 15 when they first encountered Epstein.
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OptimJ associative arrays are very handy when associated to their specific initialization syntax. Initial values can be given in intensional definition, as in: int[String] age = { "Stephan" -> 37, "Lynda" -> 29 }; or can be given in extensional definition, as in: int[String] length[String name : names] = name.length(); Here each of the entries `length[i]` is initialized with `names[i].length()`.
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The disquotational principle is a philosophical principle which holds that a rational speaker will accept "p" if and only if he or she believes p. The quotes indicate that the statement p is being treated as a sentence, and not as a proposition. This principle is presupposed by claims that hold that substitution fails in certain intensional contexts.
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Bickenbach, Jerome E., and Jacqueline M. Davies. Good reasons for better arguments: An introduction to the skills and values of critical thinking. Broadview Press, 1996. p. 49 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).
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Intensional attributes can resemble, but are not identical to, the properties perceived by the five senses. Attributes are names of properties. When, even partially, the properties of a thing match the attributes of that thing in the mind of the one making the judgment, the thing will be said to have "value". When they completely correspond, the thing will be called "good".
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A philosopherStanford University primer, 2006. might characterize this state of affairs thus: "Although the concepts of bipartiteness and absence of odd cycles differ in intension, they have identical extension."Meanings, in this sense, are often called intensions, and things designated, extensions. Contexts in which extension is all that matters are, naturally, called extensional, while contexts in which extension is not enough are intensional.
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Ludlow's PhD dissertation defended a proposal dating back to the medieval logician Jean Buridan, and revived by W.V.O. Quine in philosophy and James McCawley in linguistics, according to which so-called "intensional transitive verbs" like "seek" and "want" are really propositional attitudes in disguise. He has subsequently developed these ideas in collaboration with the linguists Richard Larson and Marcel den Dikken.
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Intensional languages cannot be given an adequate semantics in terms of the extensions of expressions in them, since the extensions themselves do not suffice to determine a logical value. (If they did, then one could not change the logical value by substituting co-extensive expressions.) On the other hand, for the first half of the 20th century the only known systems of formal semantics worked by assigning extensions to expressions and used a Tarski- style truth-definition of statements constructed from the primitive expressions of the language under analysis. Hence, these semantical methods were pathetically inadequate for understanding the semantics of any but a few small artificial languages or mutilated fragments of natural languages. This situation changed in the 1960s with the invention of possible-world or "intensional" semantics, the main form of which is due to Saul Kripke.
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Harnad, Stevan. "On Fodor on Darwin on Evolution." arXiv preprint arXiv:0904.1888 (2009). Fodor's position on this is that the breeder who, on this picture, makes up the mechanism of selection, has a mind which necessarily supplies the required intensional causal explanation; is sensitive to the relevant counterfactuals. However, without a breeder with mental states natural selection loses the power to support the relevant counterfactuals.
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Often in mathematics, when one defines an equivalence relation on a set, one immediately forms the quotient set (turning equivalence into equality). In contrast, setoids may be used when a difference between identity and equivalence must be maintained, often with an interpretation of intensional equality (the equality on the original set) and extensional equality (the equivalence relation, or the equality on the quotient set).
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Transparent intensional logic (frequently abbreviated as TIL) is a logical system created by Pavel Tichý. Due to its rich procedural semantics TIL is in particular apt for the logical analysis of natural language. From the formal point of view, TIL is a hyperintensional, partial, typed lambda calculus. TIL applications cover a wide range of topics from formal semantics, philosophy of language, epistemic logic, philosophical, and formal logic.
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Divisio and partitio are classical terms for definitions. A partitio 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.
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Mining object behavior with ADABU. In Proceedings of the 2006 international workshop on Dynamic systems analysis (WODA '06). ACM, New York, NY, USA, 17-24Carlo Ghezzi, Andrea Mocci, and Mattia Monga. 2009. Synthesizing intensional behavior models by graph transformation. In Proceedings of the 31st International Conference on Software Engineering (ICSE '09). IEEE Computer Society, Washington, DC, USA, 430-440Mark Gabel and Zhendong Su. 2008.
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Modal logic is historically the earliest area in the study of intensional logic, originally motivated by formalizing "necessity" and "possibility" (recently, this original motivation belongs to alethic logic, just one of the many branches of modal logic). Modal logic can be regarded also as the most simple appearance of such studies: it extends extensional logic just with a few sentential functors: these are intensional, and they are interpreted (in the metarules of semantics) as quantifying over possible worlds. For example, the Necessity operator (the 'square') when applied to a sentence A says 'The sentence "('square')A" is true in world i if it is true in all worlds accessible from world i'. The corresponding Possibility operator (the 'diamond') when applied to A asserts that "('diamond')A" is true in world i iff A is true in some worlds (at least one) accessible to world i.
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This note included a syntactic definition of "equality types" that were claimed to be interpreted in the model by path-spaces, but did not consider Per Martin-Löf's rules for identity types. It also stratified the universes by homotopy dimension in addition to size, an idea that later was mostly discarded. On the syntactic side, Benno van den Berg conjectured in 2006 that the tower of identity types of a type in intensional type theory should have the structure of an ω-category, and indeed a ω-groupoid, in the "globular, algebraic" sense of Michael Batanin. This was later proven independently by van den Berg and Garner in the paper "Types are weak omega-groupoids" (published 2008), and by Peter Lumsdaine in the paper "Weak ω-Categories from Intensional Type Theory" (published 2009) and as part of his 2010 Ph.D. thesis "Higher Categories from Type Theories".
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If, however, a set is defined intensionally, then it is a set of things that meet some requirement to be a member. Thus, such a set can be seen as creating a type. Note that it also creates a class from the extension of the intensional set. A type always has a corresponding class (though that class might have no members), but a class does not necessarily have a corresponding type.
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With a few others, he was responsible for introducing analytic philosophy to the French-speaking community. From philosophy, he widened his investigations to the formal semantics of natural language that required an expertise in linguistics as well as in modal and intensional logics. Later on, Gochet shifted naturally with the trend toward applications in computer science and artificial intelligence. In particular, this led to his long- standing interest in epistemic logic.
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What is justice? Aside from being immutable, timeless, changeless, and one over many, the Forms also provide definitions and the standard against which all instances are measured. In the dialogues Socrates regularly asks for the meaning – in the sense of intensional definitions – of a general term (e. g. justice, truth, beauty), and criticizes those who instead give him particular, extensional examples, rather than the quality shared by all examples.
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Dunn was born in Fort Wayne, Indiana in 1941. He went to high school in Lafayette, Indiana, where he worked in Purdue Biology laboratories after school and summers. He was the first in his family to go to college. He has an A.B. in Philosophy from Oberlin College and a Ph.D. in Philosophy (Logic) from the University of Pittsburgh, where he wrote his dissertation, The Algebra of Intensional Logics.
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Another way the authors put the same point is as follows: If the mechanism of natural selection (as it is currently formulated) is correct, then it is a paradigm example of intensional causation. Intensional causation requires either (1) there be a mind involved in the causal process, or (2) the causal mechanism has access to nomological laws. Since there is neither (1) nor (2) at the biological level, the theory of natural selection cannot be correct. In a response, published by the London Review of Books in November 2007, to "Why Pigs Don't Have Wings", Tim Lewens states that Elliott Sober gave the following solution to the problem of appealing to metaphors such a "Mother Nature" in 1984: Lewens continues: Elliott Sober argues against Fodor with an analogy: imagine a toy tube that contained several balls of different colour and size, with the two traits as locally coextensive; balls of the same color have the same size and balls of the same size have the same color.
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Materna, Pavel (1998) Concepts and Objects. Helsinki: The Philosophical Society of Finland. . Materna, Pavel & Petr Kolář (1993) "On the Nature of Facts" IN Philosophie und Logik. Perspectives in Analytical Philosophy. Berlin: Walter De Gruyter, 77–96. . Materna, Pavel, Eva Hajičová & Petr Sgall (1987) "Redundant answers and topic-focus articulation". Linguistics and Philosophy 10 (1): 101–113. ISSN 0165-0157. Svoboda, Aleš & Pavel Materna (1987) "Functional sentence perspective and intensional logic" IN Functionalism in Linguistics.
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Intensional Logics and Logical Structure of Theories. Metsniereba, Tbilisi, 1988, pages 16-48 (Russian). Japaridze proved the arithmetical completeness of this system, as well as its inherent incompleteness with respect to Kripke frames. GLP has been extensively studied by various authors during the subsequent three decades, especially after Lev Beklemishev, in 2004,L. Beklemishev, "Provability algebras and proof-theoretic ordinals, I". Annals of Pure and Applied Logic 128 (2004), pages 103-123.
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The Austrian constitution uses the term "supreme executive organ" multiple times but provides neither an intensional nor an extensional definition. Article 19 of the Federal Constitutional Law contains what appears to be a taxative enumeration of supreme executive organs, but this list is universally dismissed as specious and useless. The defining characteristics of supreme executive organs and the consequences of membership to the class have been established by case law and academic scholarship.
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Material implication allows a true consequent to follow from a false antecedent. Lewis proposed to replace material implication with strict implication, such that a (contingently) false antecedent does not always strictly imply a (contingently) true consequent. This strict implication was not primitive, but defined in terms of negation, conjunction, and a prefixed unary intensional modal operator, \Diamond. If X is a formula with a classical bivalent truth value, then \DiamondX can be read as "X is possibly true".
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Another difficulty for the interpretation of lambda calculus as a deductive system is the representation of values as lambda terms, which represent functions. The untyped lambda calculus is implemented by performing reductions on a lambda term, until the term is in normal form. The intensional interpretation of equality is that the reduction of a lambda term to normal form is the value of the lambda term. This interpretation considers the identity of a lambda expression to be its structure.
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Suppose one defines a miniskirt as "a skirt with a hemline above the knee". It has been assigned to a genus, or larger class of items: it is a type of skirt. Then, we've described the differentia, the specific properties that make it its own sub-type: it has a hemline above the knee. Intensional definition also applies to rules or sets of axioms that define a set by describing a procedure for generating all of its members.
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A possible world is a complete and consistent way the world is or could have been. They are widely used as a mathematical tool in logic, philosophy, and linguistics in order to provide a semantics for intensional and modal logic. Their metaphysical status has been a subject of controversy in philosophy, with modal realists such as David Lewis arguing that they are literally existing alternate realities, and others such as Robert Stalnaker arguing that they are not.
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Possible worlds play a central role in the work of both linguists and philosophers working in formal semantics. Contemporary formal semantics is couched in formal systems rooted in Montague grammar, which is itself built on Richard Montague's intensional logic. Contemporary research in semantics typically uses possible worlds as formal tools without committing to a particular theory of their metaphysical status. The term possible world is retained even by those who attach no metaphysical significance to them.
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Schirn completed his doctoral degree at the University of Freiburg in 1974 with a thesis on identity and synonymy in logic and semantics and subsequently taught at University of Oxford, University of Cambridge and Michigan State University. Schirn’s research during this time focused on theories of meaning and intensional semantics. He continued working in this area at the University of California at Berkeley, St. John’s College (Oxford), Harvard University and at Wolfson College (Oxford). In 1985, Schirn was awarded his habilitation at the University of Regensburg.
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Glue analyses within other syntactic formalisms have also been proposed; besides LFG, glue analyses have been proposed within HPSG, context-free grammar, categorial grammar, and tree-adjoining grammar. Glue is a theory of the syntax–semantics interface which is compatible not only with various syntactic frameworks, but also with different theories of semantics and meaning representation. Semantic formalisms that have been used as the meaning languages in glue semantics analyses include versions of discourse representation theory, intensional logic, first-order logic, and natural semantic metalanguage.
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Constructions, and the entities they construct, are organized into a ramified type theory incorporating a simple type theory. The semantics is tailored to the hardest case, as constituted by hyperintensional contexts, and generalized from there to intensional and extensional contexts. The underlying logic is a Frege-style function/argument one, treating functions, rather than relations or sets, as primitive, together with a Church-style logic, centred on the operations of functional abstraction and application. Key constraints informing TIL approach to semantic analysis are compositionality and anti-contextualism.
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One approach rejects the law of excluded middle and consequently reductio ad absurdum.Morgenstern, L. (1986), 'A First Order Theory of Planning, Knowledge and Action', in Halpern, J. (ed.), Theoretical Aspects of Reasoning about Knowledge: Proceedings of the 1986 Conference, Morgan Kaufmann, Los Altos, pp. 99–114. Another approach upholds reductio ad absurdum and thus accepts the conclusion that (K) is both not known and known, thereby rejecting the law of non-contradiction.Priest, G. (1991), 'Intensional Paradoxes', Notre Dame Journal of Formal Logic 32, pp. 193–211.
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The 1984 type theory was extensional while the type theory presented in the book by Nordström et al. in 1990, which was heavily influenced by his later ideas, intensional, and more amenable to being implemented on a computer. Martin-Löf's intuitionistic type theory developed the notion of dependent types and directly influenced the development of the calculus of constructions and the logical framework LF. A number of popular computer-based proof systems are based on type theory, for example NuPRL, LEGO, Coq, ALF, Agda, Twelf, Epigram, and Idris.
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SNePS is a knowledge representation, reasoning, and acting (KRRA) system developed and maintained by Stuart C. Shapiro and colleagues at the State University of New York at Buffalo. SNePS is simultaneously a logic-based, frame-based, and network-based KRRA system. It uses an assertional model of knowledge, in that a SNePS knowledge base (KB) consists of a set of assertions (propositions) about various entities. Its intended model is of an intensional domain of mental entities—the entities conceived of by some agent, and the propositions believed by it.
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Tagg is probably best known for his work in the field of music analysis. Using mainly pieces of popular music as analysis objects, he stresses the importance of non-notatable parameters of expression and of vernacular perception in understanding "how music communicates what to whom with what effect" in today's world. He has adapted Charles Seeger's notion of the museme to demonstrate how combinations of such units are used to create both syncritic (intensional) structures inside the extended present, and diatactical (extensional) ones over time.See chapters 11–12 in Philip Tagg, Music’s Meanings, 2013.
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TIL provides an overarching semantic framework for all sorts of discourse, whether colloquial, scientific, mathematical or logical. The semantic theory is a procedural one, according to which sense is an abstract, pre-linguistic procedure detailing what operations to apply to what procedural constituents to arrive at the product (if any) of the procedure. TIL procedures, known as constructions, are hyperintensionally individuated. Construction is the single most important notion of transparent intensional logic, being a philosophically well- motivated and formally worked-out conception of Frege’s notion of mode of presentation.
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Possible worlds are one of the foundational concepts in modal and intensional logics. Formulas in these logics are used to represent statements about what might be true, what should be true, what one believes to be true and so forth. To give these statements a formal interpretation, logicians use structures containing possible worlds. For instance, in the relational semantics for classical propositional modal logic, the formula \Diamond P (read as "possibly P") is actually true iff P is true at some world which is accessible from the actual world.
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Intuitionistic type theory (also known as constructive type theory, or Martin- Löf type theory) is a type theory and an alternative foundation of mathematics. Intuitionistic type theory was created by Per Martin-Löf, a Swedish mathematician and philosopher, who first published it in 1972. There are multiple versions of the type theory: Martin-Löf proposed both intensional and extensional variants of the theory and early impredicative versions, shown to be inconsistent by Girard's paradox, gave way to predicative versions. However, all versions keep the core design of constructive logic using dependent types.
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Lexical meaning composition is based on morphosemantic composition functions whose arguments are (n-tuples of) concepts and whose values are again concepts. Such semantic functions occur in the semantic content of morphological functions such as morphological complement, modifier, and nucleus, and operate on the basis of (morphological or semantic) application conditions. In Integrational Sentence Semantics, sentence meanings are construed as intensional relations between potential utterances and potential speakers. For any syntactic unit that has a sentence meaning, the meanings of the unit jointly represent a necessary condition for successful utterances of the unit.
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Richard Milton Martin (1916, Cleveland, Ohio - 22 November 1985, Milton, Massachusetts) was an American logician and analytic philosopher. In his Ph.D. thesis written under Frederic Fitch, Martin discovered virtual sets a bit before Quine, and was possibly the first non-Pole other than Joseph Henry Woodger to employ a mereological system. Building on these and other devices, Martin forged a first-order theory capable of expressing its own syntax as well as some semantics and pragmatics (via an event logic), all while abstaining from set and model theory (consistent with his nominalist principles), and from intensional notions such as modality.
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Marcus proposed the view in the philosophy of language according to which proper names are what Marcus termed mere "tags" ("Modalities and Intensional Languages" (Synthese, 1961) and elsewhere). According to her tag theory of names (a direct reference theory), these "tags" are used to refer to an object, which is the bearer of the name. The meaning of the name is regarded as exhausted by this referential function. This view contrasts for example with Bertrand Russell's description theory of proper names as well as John Searle's cluster description theory of namesCraig, E. (ed.), Routledge Encyclopedia of Philosophy, vol.
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He also suggested that despite Carnap's claim that every designation refers to both an intension and an extension, his system "provides only for the designation of intensional entities". The philosopher Dagfinn Føllesdal wrote that while Carnap admitted that he ignored complications with his proposed system of modal logic, he failed to explain what they were. He criticized Carnap for this, and suggested that Carnap was unaware of some of the problems with his views. The philosopher E. J. Lowe wrote that Meaning and Necessity was "important and influential", and laid the foundations of much subsequent work in the semantics of modal logic.
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Circular definition of "musicality" Does the definition of music determine its aspects, or does the combination of certain aspects determine the definition of music? For example, intensional definitions list aspects or elements that make up their subject. Some definitions refer to music as a score, or a composition (; ; ): music can be read as well as heard, and a piece of music written but never played is a piece of music notwithstanding. The process of reading music, at least for trained musicians, involves a process, called "inner hearing" or "audiation" by Gordon, where the music is heard in the mind as if it were being played .
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In his article "Quantifying In" (1968), Kaplan discusses issues in intensional and indirect (Ungerade, or oblique) discourse, such as substitution failure, existential generalization failure, and the distinction between de re / de dicto propositional attitude attributions. Such issues were made salient primarily by W. V. Quine in his "Quantifiers and Propositional Attitudes" (1956). The phrase "quantifying in" comes from Quine's discussion of what he calls "relational" constructions of an existential statement. In such cases, a variable bound by an anterior variable-binding operator occurs within a non-extensional context such as that created by a 'that' clause, or, alternatively, by propositional attitude or modal operators.
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He has examined the class versus individual interpretations of species and clades in light of his work on phylogenetic definitions of taxon names, proposing that contrary to how those interpretations are commonly presented, they are not mutually exclusive, which suggests that the same is true of ostensive and intensional definitions. He has argued that the philosopher Karl Popper’s concept of degree of corroboration is analogous to the likelihood ratio of nested hypotheses and that in phylogenetics the probability of the evidence given the background knowledge in the absence of the hypothesis of interest (a critical component of Popper’s "Degree of Corroboration") is represented by the likelihood of a star tree.
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So how are the three seemingly irreconcilable principles above resolved? Davidson distinguishes causal relations, which are an extensional matter and not influenced by the way they are described, from law-like relations, which are intensional and dependent on the manner of description. There is no law of nature under which events fall when they are described according to the order in which they appeared on the television news. When the earthquake caused the Church of Santa Maria dalla Chiesa to collapse, there is surely some physical law(s) which explains what happened, but not under the description in terms of the event on Channel 7 at six p.m.
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Some philosophers have decided, however, that it is important to exclude certain predicates (or purported predicates) from the principle in order to avoid either triviality or contradiction. An example (detailed below) is the predicate that denotes whether an object is equal to x (often considered a valid predicate). As a consequence, there are a few different versions of the principle in the philosophical literature, of varying logical strength—and some of them are termed "the strong principle" or "the weak principle" by particular authors, in order to distinguish between them. Willard Van Orman Quine thought that the failure of substitution in intensional contexts (e.g.
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For instance, the terms rantans or brillig have no intension and hence no meaning. Such terms may be suggestive, but a term can be suggestive without being meaningful. For instance, ran tan is an archaic onomatopoeia for chaotic noise or din and may suggest to English speakers a din or meaningless noise, and brillig though made up by Lewis Carroll may be suggestive of 'brilliant' or 'frigid'. Such terms, it may be argued, are always intensional since they connote the property 'meaningless term', but this is only an apparent paradox and does not constitute a counterexample to the claim that without intension a word has no meaning.
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Katz's theory, to take this example, is based on the fundamental idea that sense should not have to be defined in terms of, nor determine, referential or extensional properties but that it should be defined in terms of, and determined by, all and only the intensional properties of names. He illustrates the way a metalinguistic description theory can be successful against Kripkean counterexamples by citing, as one example, the case of "Jonah." Kripke’s Jonah case is very powerful because in this case the only information that we have about the Biblical character Jonah is just what the Bible tells us. Unless we are fundamentalist literalists, it is not controversial that all of this is false.
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For Aristotle and his scholastic followers, the notion of essence is closely linked to that of definition (ὁρισμός horismos).S. Marc Cohen, "Aristotle's Metaphysics", Stanford Encyclopedia of Philosophy, accessed 20 April 2008. In the history of western thought, essence has often served as a vehicle for doctrines that tend to individuate different forms of existence as well as different identity conditions for objects and properties; in this logical meaning, the concept has given a strong theoretical and common-sense basis to the whole family of logical theories based on the "possible worlds" analogy set up by Leibniz and developed in the intensional logic from Carnap to Kripke, which was later challenged by "extensionalist" philosophers such as Quine.
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Thus, the "seven deadly sins" can be defined intensionally as those singled out by Pope Gregory I as particularly destructive of 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 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).
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However though 'iron is a metal' may be implied by 'cats lay eggs' it doesn't seem to be relevant to it the way in which 'cats are mammals' and 'mammals give birth to living young' are relevant to each other. If one states "I love ice cream," and another person responds "I have a friend named Brad Cook," then these statements are not relevant. However, if one states "I love ice cream," and another person responds "I have a friend named Brad Cook who also likes ice cream," this statement now becomes relevant because it relates to the first person's idea. More recently a number of theorists have sought to account for relevance in terms of "possible world logics" in intensional logic.
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Constraints on the search space are allowed, as are predicates that are defined on a rule rather than on a set of examples (called intensional predicates); most importantly a potentially incorrect hypothesis is allowed as an initial approximation to the predicate to be learned. The main goal of FOCL is to incorporate the methods of explanation-based learning (EBL) into the empirical methods of FOIL. Even when no additional knowledge is provided to FOCL over FOIL, however, it utilizes an iterative widening search strategy similar to depth-first search: first FOCL attempts to learn a clause by introducing no free variables. If this fails (no positive gain), one additional free variable per failure is allowed until the number of free variables exceeds the maximum used for any predicate.
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Ludlow's earliest work in semantics was an attempt to combine work in the theory of meaning with contemporary work in generative linguistics, but using resources that are more parsimonious than those typically used in semantic theory—for example without using the higher-order functions and intensional objects deployed in Montague grammar. The resources were largely limited to primitives like truth and reference to individuals. His subsequent work has explored ways of formalizing alternative approaches to semantic theory—including the possibility of formalizing a Wittgensteinian use theory or expressivist semantics for natural language, which is to say a theory in which the building blocks of a semantic theory are expressions of attitudes rather than primitives like truth and reference.See chapter 5 of his The Philosophy of Generative Linguistics, Oxford University Press, 2011.
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Symposium held in Kirchberg/Wechsel, Austria, August 14–21, 1988. Since later in the 1990s, Ruy de Queiroz has been engaged, jointly with Dov Gabbay, in a program of providing a general account of the functional interpretation of classical and non-classical logics via the notion of labeled natural deduction. As a result, novel accounts of the functional interpretation of the existential quantifier, as well as the notion of propositional equality, were put forward, the latter allowing for a recasting of Richard Statman's notion of direct computation, and a novel approach to the dichotomy "intensional versus extensional" accounts of propositional equality via the Curry–Howard correspondence. Since the early 2000s, Ruy de Queiroz has been investigating, jointly with Anjolina de Oliveira, a geometric perspective of natural deduction based on a graph-based account of Kneale's symmetric natural deduction.
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The "quantifying in" idiom captures the notion that the variable-binding operator (for example, the existential quantifier 'something') reaches into, so to speak, the non- extensional context to bind the variable occurring within its scope. For example, (using a propositional attitude clause), if one quantifies into the statement "Ralph believes that Ortcutt is a spy," the result is (partly formalized): :(Ǝx) (Ralph believes that x is a spy) :["There is someone Ralph believes is a spy"] In short, Kaplan attempts (among other things) to provide an apparatus (in a Fregean vein) that allows one to quantify into such intensional contexts even if they exhibit the kind of substitution failure that Quine discusses. If successful, this shows that Quine is wrong in thinking that substitution failure implies existential generalization failure for (or inability to quantify into) the clauses that exhibit such substitution failure.
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If n > 1, the attributes are n-place intensional relations between real-world entities, and the extension is the set of n-tuples of real- world entities among which the n-place relations in the intension hold. Such 'relational concepts' typically occur as lexical meanings of verbs and adpositions (prepositions, etc.) but also of other kinds of relational words. The only concept for which the notions of intension and extension are not defined is the (0-place) 'empty concept,' occurring as the meaning component of lexical words such as auxiliaries and modal particles, and of all affixes, that is, of linguistic entities whose contribution to meaning composition is not based on lexical meanings. Given the notion of empty concept, the IL conception of concepts is both flexible and powerful enough to assign meanings to lexical words of any kind.
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The basic goal of PLN is to provide reasonably accurate probabilistic inference in a way that is compatible with both term logic and predicate logic, and scales up to operate in real time on large dynamic knowledge bases. The goal underlying the theoretical development of PLN has been the creation of practical software systems carrying out complex, useful inferences based on uncertain knowledge and drawing uncertain conclusions. PLN has been designed to allow basic probabilistic inference to interact with other kinds of inference such as intensional inference, fuzzy inference, and higher-order inference using quantifiers, variables, and combinators, and be a more convenient approach than Bayesian networks (or other conventional approaches) for the purpose of interfacing basic probabilistic inference with these other sorts of inference. In addition, the inference rules are formulated in such a way as to avoid the paradoxes of Dempster-Shafer theory.
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