Academic journal article Memory & Cognition

Comparing the Meanings of "If" and "All"

Academic journal article Memory & Cognition

Comparing the Meanings of "If" and "All"

Article excerpt

Published online: 8 My 2014

© Psychonomic Society, Inc. 2014

Abstract In this study, we compared the everyday meanings of conditionals ("if p then if") and universally quantified statements ("all p are < ") when applied to sets of elements. The interpretation of conditionals was predicted to be directly related to the conditional probability, such that P("ifp then q") = P(q|p). Quantified statements were assumed to have two interpretations. According to an instance-focused interpretation, quantified statements are equivalent to conditionals, such that P("all p are q") = P(q|p). According to a set-focused interpretation, "all p are q" is true if and only if every instance in set p is an instance of q, so that the statement would be accepted when P(q|p) = 1 and rejected when this probability was below 1. We predicted an instance-focused interpretation of "all" when the relation between p and q expressed a general law for an infinite set of elements. A set-focused interpretation of "all" was predicted when the relation between p and q expressed a coincidence among the elements of a finite set. Participants were given short context stories providing information about the frequency of co-occurrence of cases of p, q, not-p, and not-q in a population. They were then asked to

estimate the probability that a statement (conditional or quantified) would be true for a random sample taken from that population. The probability estimates for conditionals were in accordance with an instance-focused interpretation, whereas the estimates for quantified statements showed features of a set-focused interpretation. The type of the relation between p and q had no effect on this outcome.

Keywords Deductive reasoning · Interpretation · Conditionals · Quantified statements

(ProQuest: ... denotes formulae omitted.)

Given the knowledge that if an animal is a dog, then it barks, does this imply that all dogs bark? There seems to be a close relationship between the meanings of the two assertions, but which relationship exactly? This question is relevant because the meaning that people ascribe to such statements determines the conditions under which they take the statements to be true, and hence the inferences that they are willing to draw from them. The first statement is an instance of a conditional (i.e., an "if-then" structure), the second an instance of an affirma- tive universally quantified statement (i.e., a structure of the form "all ,r are y"). The aim of the present work was to investigate whether people understand parallel statements with "if' and with "all" in the same way.

The truth-functional approach to conditionals

Two main frameworks that have been employed in reasoning research to model people's interpretations of these statements are classical logic and probability theory. In classical logic, "if' is equivalent to "all" when applied to sets of elements: Both would be formalized in the same way. In the example above, they would be represented as V.r(D.r -> B.r), which is read "for all variables .r it holds that if .r is a dog (D), then .r barks (B)." As in classical logic in general, statements of this form have binary truth values (i.e., they are either true or false, with no values in between), and they are truth-functional (i.e., their truth or falsity as a composite statement is a fimction of the truth or falsity of their constituent propositions). The conditional at the core of the statement shares these properties and is called the material conditional. The truth or falsity of the material conditional, as determined by each combination of the truth or falsity of its constituent propositions, is de- scribed in a truth table, shown in the first column of Table 1. Here the conditional is stated as "ifp then qp and its constit- uent propositions p and q stand for statements such as ".r is a dog" or ".r barks." The table shows that the material condi- tional is true when both p and q are true, false when p is true but q is false, and true in the two cases in whichp is false. …

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