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A formal theory of intermediate quantifiers. (English) Zbl 1176.03011
Summary: The paper provides a logical theory of a specific class of natural language expressions, called intermediate quantifiers (most, a lot of, many, a few, a great deal of, a large part of, a small part of), which can be ranked among generalized quantifiers. The formal frame is fuzzy type theory (FTT). Our main idea lays in the observation that intermediate quantifiers speak about elements taken from a class that is made “smaller” than the original universe in a specific way. Our theory is based on the formal theory of trichotomous evaluative linguistic expressions. Thus, an intermediate quantifier is obtained as a classical quantifier “for all” or “exists” but taken over a class of elements that is determined using an appropriate evaluative expression. In the paper we characterize the behavior of intermediate quantifiers and prove many valid syllogisms that generalize Aristotle’s classical ones.

##### MSC:
 03B52 Fuzzy logic; logic of vagueness 03B65 Logic of natural languages 03C80 Logic with extra quantifiers and operators
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##### References:
 [1] Andrews, P., An introduction to mathematical logic and type theory: to truth through proof, (2002), Kluwer Academic Publishers Dordrecht · Zbl 1002.03002 [2] Běhounek, L.; Cintula, P., Fuzzy class theory, Fuzzy sets and systems, 154, 34-55, (2005) · Zbl 1086.03043 [3] Church, A., A formulation of the simple theory of types, J. symbolic logic, 5, 56-68, (1940) · JFM 66.1192.06 [4] Esteva, F.; Godo, L., Monoidal t-norm based logic: towards a logic for left-continuous t-norms, Fuzzy sets and systems, 124, 271-288, (2001) · Zbl 0994.03017 [5] Glöckner, I., Fuzzy quantifiers: A computational theory, (2006), Springer Berlin · Zbl 1089.03002 [6] Henkin, L., Completeness in the theory of types, J. symbolic logic, 15, 81-91, (1950) · Zbl 0039.00801 [7] Lakoff, G., Hedges: a study in meaning criteria and logic of fuzzy concepts, J. philos. logic, 2, 458-508, (1973) · Zbl 0272.02047 [8] Novák, V., Antonyms and linguistic quantifiers in fuzzy logic, Fuzzy sets and systems, 124, 335-351, (2001) · Zbl 0994.03013 [9] Novák, V., Fuzzy logic deduction with words applied to ancient sea level estimation, (), 301-336 [10] Novák, V., On fuzzy type theory, Fuzzy sets and systems, 149, 235-273, (2005) · Zbl 1068.03019 [11] V. Novák, Fuzzy logic theory of evaluating expressions and comparative quantifiers, in: Proc. 11th Internat. Conf. IPMU, Paris, July 2006, Vol. 2, Éditions EDK, Les Cordeliers, Paris, 2006, pp. 1572-1579. [12] V. Novák, A comprehensive theory of trichotomous evaluative linguistic expressions, in: S. Gottwald, H.P., U. Höhle, E.P. Klement (Eds.), Proc. of 26th Linz Seminar on Fuzzy Set Theory, to appear, Research Report No. 71, $$\langle$$http://irafm.osu.cz/⟩. [13] Novák, V.; Perfilieva, I.; Močkoř, J., Mathematical principles of fuzzy logic, (1999), Kluwer Academic Publishers Boston · Zbl 0940.03028 [14] Peters, S.; WesterstÅhl, D., Quantifiers in language and logic, (2006), Claredon Press Oxford [15] Peterson, P., Intermediate quantifiers. logic linguistics, and Aristotelian semantics, (2000), Ashgate Aldershot [16] Pfeifer, N., Contemporary syllogistics: comparative and quantitative syllogisms, (), 57-71 [17] WesterstÅhl, D., Quantifiers in formal and natural languages, (), 1-131 · Zbl 0875.03079 [18] Zadeh, L.A., A computational approach to fuzzy quantifiers in natural languages, Comput. math., 9, 149-184, (1983) · Zbl 0517.94028
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