Some proof theoretical remarks on quantification in ordinary language

This paper surveys the common approach to quantification and generalised quantification in formal linguistics and philosophy of language. We point out how this general setting departs from empirical linguistic data, and give some hints for a different view based on proof theory, which on many aspects gets closer to the language itself. We stress the importance of Hilbert's oper- ator epsilon and tau for, respectively, existential and universal quantifications. Indeed, these operators help a lot to construct semantic representation close to natural language, in particular with quantified noun phrases as individual terms. We also define guidelines for the design of the proof rules corresponding to generalised quantifiers.

[1]  Klaus von Heusinger,et al.  Definite Descriptions and Choice Functions , 1997 .

[2]  J. Hintikka.  On denoting what? , 2005, Synthese.

[3]  Dov M. Gabbay,et al.  Quantification in Nonclassical Logic , 2009 .

[4]  Peter Geach,et al.  Reference and Generality: An Examination of Some Medieval and Modern Theories , 1965 .

[5]  Stanley Peters,et al.  Quantifiers in language and logic , 2006 .

[6]  Benson Mates,et al.  The Development of Logic. , 1986 .

[7]  Mark Steedman,et al.  Taking Scope - The Natural Semantics of Quantifiers , 2011 .

[8]  Ivan Boh.  Reference and Generality: An Examination of Some Medieval and Modern Theories , 1987 .

[9]  Grigori Mints,et al.  Cut elimination for a simple formulation of epsilon calculus , 2008, Ann. Pure Appl. Log..

[10]  K. Heusinger.  Choice Functions and the Anaphoric Semantics of Definite NPs , 2004 .

[11]  Uwe Reyle,et al.  From discourse to logic , 1993 .

[12]  Christian Retoré,et al.  Variable types for meaning assembly: a logical syntax for generic noun phrases introduced by most , 2012, ArXiv.

[13]  Jakub Szymanik,et al.  Comprehension of Simple Quantifiers: Empirical Evaluation of a Computational Model , 2010, Cogn. Sci..

[14]  P. Bernays,et al.  Grundlagen der Mathematik , 1934 .

[15]  Klaus von Heusinger,et al.  The Epsilon Operator and E-Type Pronouns , 1995 .

[16]  Martha Kneale,et al.  The development of logic , 1963 .

[17]  Edward L. Keenan,et al.  Erratum to: Stanley Peters and Dag Westerståhl: Quantifiers in language and logic , 2011 .