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

Abstract

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 (1903Leibniz ( / 1988 lines as external representations and show them to support the predictions.

Preadolescents Solve Natural Syllogisms Proficiently

“…There is one characteristic of the natural task that may give rise to a more serious objection against the origin of the improvement in performance. Sentences universally quantified by all are indeterminate between strict and non‐strict inclusion (Politzer et al., ). However, in natural syllogisms the implicit minor premise expresses a strict inclusion.…”

“…We assume that the mental representation of the four sentences is based on generalized quantification (Politzer, Van der Henst, Delle Luche, & Noveck, ): all P are Q and some P are not Q are represented as the inclusion and the non‐inclusion of P in Q, respectively; and no P are Q and some P are Q are represented as an empty and a non‐empty intersection of P and Q, respectively. We now examine the ecthetic procedure in some detail, applying it first to the artificial case, and showing that it may be more or less sophisticated depending on the logical form of the syllogism.…”

The processes of inference

“…124-125;Khemlani and Johnson-Laird in press). Other psychological theories likewise treat quantifiers as relations between sets; some of them make use of diagrammatic representations to handle relations (Ceraso and Provitera 1971;Erickson 1974;Ford 1995), and others rely on formal rules of inference (Stenning and Yule 1997;Guyote and Sternberg 1981;Geurts 2003;Politzer, van der Henst, Luche, and Noveck 2006). What distinguishes the model theory is that it relies on models of individual entities and properties to represent sets.…”

“…Researchers often place heuristics at the forefront of theories of reasoning (see also Ford 1995;Stenning and Yule 1997;Chater and Oaksford 1999;Politzer et al 2006), but until now proponents of the model theory have downplayed their use. In an effort to bridge the two approaches, mReasoner embodies heuristics in its system 1 processes.…”

A process model of immediate inferences

“…Braine & O'Brien, 1998;Rips, 1994), though they differ in the way they treat the relations between the sets. For instance, some theorists make use of diagrammatic representations to handle relations (Ceraso & Provitera, 1971;Erickson, 1974;Ford, 1995;Newell, 1981), others rely on formal rules of inference (Geurts, 2003;Guyote & Sternberg, 1981;Politzer, van der Henst, Luche, & Noveck, 2006;Stenning & Yule, 1997); and yet others analyze sets in terms of simulated possibilities, i.e., mental models (Bucciarelli & Johnson-Laird, 1999;Johnson-Laird & Byrne, 1991;Polk & Newell, 1995). The psychological systems can all account for how individuals make valid deductions, however few of them can account for the differences in relative difficulty between various reasoning problems.…”