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The analysis of self-reference and diagonalization has motivated the application of algebraic and topological techniques: consider Scott’s models for extensional lambda calculus (Scott 1972) and their subsequent categorical understanding.
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(A correspondent of Lambert, Georg von Holland, had experimented with an extensional theory, and in 1839 the English writer Thomas Solly presented an extensional logic in A Syllabus of Logic, though not an algebraic one.)
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Bolzano's system of extensional propositional logic is closely connected with his extensional logic of ideas.
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For, we shall see, even principles of classical logic involving only the extensional, truth-functional connectives can fail at open worlds: their openness consists in their not being closed under any non-trivial logical consequence principle.
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A function concept may now be called extensional if and only if it requires functions that are extensionally equivalent at the actual world to be identical.
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We label this stronger theory EM, for Extensional Mereology, the attribute ‘extensional’ being justified precisely by the exclusion of countermodels that, like the one in Figure 4, contain distinct objects with the same proper parts.
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But an extensional theory is, of course, not thereby free of problems.
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A very different variant of the Extensional approach was proposed by C.D.
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Among the variety of objects in the extensional canon, sets are distinguished by their almost complete absence of structure.
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However, a large debate remained unsettled, and that was the choice between the two versions of type theory that Martin-Löf presented: the earlier intensional theory, and the extensional theory first proposed in CMCP.
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., the box and diamond of necessity and possibility, behave in a non-standard fashion are less deviant than worlds where also the extensional operators, like classical conjunction and disjunction, do.
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Extensional.
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Connectability of laws of different theories requires meaning invariance in the form of extensional equivalence between descriptions, with bridge principles between coextensive but distinct terms in different theories.
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A program is then seen in an extensional way: it is a binary relation between pairs of states of an LTS.
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Of the two main parts of Bolzano's formal logic, the extensional logic of ideas (the logic of classes) and the extensional logic of propositions, the first part comes from a long tradition beginning with Boethius (and derived from Aristotelian syllogistic) and ending — in Bolzano's times — with Gergonne.
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Extensional Model: our episodes of experiencing are themselves temporally extended, and are thus able to incorporate change and persistence in a quite straightforward way.
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He takes reference statements to quantify implicitly over senses, that is, functions of expressions, and then defines an extensional relation of material or extensional equivalence between senses (see SM III §63: 84).
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Bolzano's system of relations between propositions is constructed from the extensional relations between ideas as defined in the Maass-Bolzano logic of classes.
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Popper captured the crucial difference between material (extensional) generalizations and subjunctive (intensional) generalizations as follows:
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One gets the axiom of choice from the axioms of cardinalities and extensional codings; the details are technical.
non-extensional
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