The Interpretation of Classically Quantified Sentences: A Set-Theoretic Approach

We present a set-theoretic model of the mental representation of classically quantified sentences (All P are Q, Some P are Q, Some P are not Q, and No P are Q). We take inclusion, exclusion, and their negations to be primitive concepts. We show that although these sentences are known to have a diagrammatic expression (in the form of the Gergonne circles) that constitutes a semantic representation, these concepts can also be expressed syntactically in the form of algebraic formulas. We hypothesized that the quantified sentences have an abstract underlying representation common to the formulas and their associated sets of diagrams (models). We derived 9 predictions (3 semantic, 2 pragmatic, and 4 mixed) regarding people's assessment of how well each of the 5 diagrams expresses the meaning of each of the quantified sentences. We report the results from 3 experiments using Gergonne's (1817) circles or an adaptation of Leibniz (1903/1988) lines as external representations and show them to support the predictions.

[1]  K. Drozd.  Language acquisition and conceptual development: Children's weak interpretations of universally quantified questions , 2001 .

[2]  Laurence R. Horn,et al.  On the semantic properties of logical operators in english' reproduced by the indiana university lin , 1972 .

[3]  I. Noveck,et al.  Characterizing the time course of an implicature: An evoked potentials study , 2003, Brain and Language.

[4]  G. Politzer.  Comparison of Deductive Abilities Across Language , 1991 .

[5]  Gennaro Chierchia,et al.  Meaning and Grammar: An Introduction to Semantics , 1990 .

[6]  J. Piaget,et al.  The early growth of logic in the child : classification and seriation , 1965 .

[7]  A. Newell.  Reasoning, problem solving, and decision processes: the problem space as a fundamental category , 1993 .

[8]  S. Levinson,et al.  Language Acquisition and Conceptual Development , 2001 .

[9]  Laura Macchi,et al.  Reasoning and pragmatics , 2000 .

[10]  Jonathan Evans,et al.  Reasoning about necessity and possibility : a test of the mental model theory of deduction , 1999 .

[11]  Leonhard Euler,et al.  Lettres à une princesse d'Allemagne sur divers sujets de physique & de philosophie , 1812 .

[12]  H. Grice.  Logic and conversation , 1975 .

[13]  Christine Alm-Arrius.  An Introduction to Semantics , 1999 .

[14]  Laurence R. Horn.  A Natural History of Negation , 1989 .

[15]  Willem J. M. Levelt,et al.  Advances in psycholinguistics , 1970 .

[16]  Stephen E. Newstead,et al.  Drawing inferences from quantified statements: a study of the square of opposition , 1983 .

[17]  Keith Stenning,et al.  Reconnecting interpretation to reasoning through individual differences , 2006, Quarterly journal of experimental psychology.

[18]  K. Stenning.  Seeing Reason: Image and language in learning to think , 2002 .

[19]  P. Johnson-Laird,et al.  Psychology of Reasoning: Structure and Content , 1972 .

[20]  S. Newstead,et al.  Interpretational errors in syllogistic reasoning , 1989 .

[21]  Anthony J. Sanford,et al.  Communicating quantities: a review of psycholinguistic evidence of how expressions determine perspectives , 2000 .

[22]  Ira A. Noveck,et al.  Pragmatic Inferences Related to Logical Terms , 2004 .

[23]  L. Goble.  The Blackwell guide to philosophical logic , 2001 .

[24]  K. Tanaka.  The Blackwell Guide to Philosophical Logic , 2002 .

[25]  Kurt F. Leidecker,et al.  Concise history of logic , 1961 .

[26]  Ian Begg,et al.  On the Interpretation of Syllogisms , 1982 .

[27]  Guy Politzer,et al.  Reasoning, Judgement and Pragmatics , 2004 .

[28]  J. M. Bocheński,et al.  A history of formal logic , 1961 .

[29]  D L Fisher.  A three-factor model of syllogistic reasoning: The study of isolable stages , 1981, Memory & cognition.

[30]  C. Hanlon.  Acquisition of set-relational quantifiers in early childhood , 1987 .

[31]  M. Braine,et al.  What do children know about the universal quantifiers all and each? , 1996, Cognition.

[32]  L. Couturat,et al.  Opuscules Et Fragments Inédits , 1903 .

[33]  Gergonne.  Variétés. Essai de dialectique rationnelle , 1817 .

[34]  I. Begg,et al.  Empirical reconciliation of atmosphere and conversion interpretations of syllogistic reasoning errors , 1969 .

[35]  R. Revlin,et al.  Understanding quantified categorical expressions , 1980, Memory & cognition.

[36]  Robyn Carston,et al.  Relevance Theory and the saying/implicating distinction , 2008 .

[37]  Marcel Adam Just,et al.  Comprehending quantified sentences: The relation between sentence-picture and semantic memory verification , 1974 .

[38]  I. Noveck.  When children are more logical than adults: experimental investigations of scalar implicature , 2001, Cognition.

[39]  N. Wetherick.  Psychology and syllogistic reasoning: Further considerations , 1993 .

[40]  S. Amano,et al.  Psychology of Reasoning , 1923 .

[41]  G. Miller,et al.  Cognitive science. , 1981, Science.

[42]  S. Murray.  Structure and Content , 2004 .

[43]  H. Gaifman,et al.  Symbolic Logic , 1881, Nature.

[44]  D. Meyer.  On the representation and retrieval of stored semantic information , 1970 .

[45]  Guy Politzer.  Some precursors of current theories of syllogistic reasoning , 2005 .

[46]  Carol L. Smith.  Quantifiers and Question Answering in Young Children. , 1980 .

[47]  Wilma Bucci,et al.  The interpretation of universal affirmative propositions , 1978, Cognition.

[48]  Jeanette K. Gundel,et al.  Topic and Focus , 2008 .

[49]  Gennaro Chierchia,et al.  Meaning and grammar (2nd ed.): an introduction to semantics , 2000 .

[50]  Processing Artificial Set Inclusion Relations: Educing the Appropriate Schema. , 1982 .

[51]  木村 和夫.  Pragmatics , 1997, Language Teaching.

[52]  Lewis Bott,et al.  Some utterances are underinformative: The onset and time course of scalar inferences , 2004 .

[53]  K. Stenning,et al.  Image and Language in Human Reasoning: A Syllogistic Illustration , 1997, Cognitive Psychology.

[54]  Jon Oberlander,et al.  A Cognitive Theory of Graphical and Linguistic Reasoning: Logic and Implementation , 1995, Cogn. Sci..

[55]  D. Sperber,et al.  Relevance: Communication and cognition, 2nd ed. , 1995 .

[56]  Gennaro Chierchia,et al.  Meaning and grammar , 1990 .