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Models, Rules, and Deductive Reasoning1

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Models, Rules, and Deductive Reasoning1 1 Mo dels, rules, and deductive reasoning Daniel Osherson Laura Macchi Wilfrid Ho dges Rice University Universit a di Milano University of London January 14, 1999 1 Corresp onding author: D. Osherson, Psychology Dept. MS-25, Rice University, 6100 Main St., Houston TX 77005 e-mail: [email protected]. Abstract We formulate a simple theory of deductive reasoning based on mental mo dels. One prediction of the theory is exp erimentally tested and found to b e incorrect. The b earing of our results on contemp orary theories of mental mo dels is discussed. We then consider a p otential ob jection to current rule-theories of deduction. Such theories picture deductive reasoning as the successive application of inference-schemata from rst-order logic. Relying on a theorem due to George Bo olos, we show that under weak hyp otheses rst-order schemata cannot account for many p eople's abilitytoverify the validity of rst-order arguments. The hyp othesis that deductive reasoning is mediated by the construction of mental mo dels has enjoyed predictive success across several studies. It has also proven to be a fertile source of ideas ab out other kinds of judgment, for example, temp oral, spatial, and probabilistic. At the same time, the theory has su ered from p ersistent criticism for ambiguity ab out the details of mo del construction and assignment. A defect of this character, however, is to be exp ected in such an original and dynamic thesis; it should not discourage attempts at clari cation and 1 extension. In view of assisting the latter pro cess, the present pap er de nes an elementary theory of deductive reasoning that emb o dies a clear version of the mental mo dels hyp othesis. The new theory applies to a class of arguments that seem not yet to have b een examined within the mental mo dels framework. For this reason we do not claim that our theory is strictly implied by the more general thesis illustrated in Johnson-Laird, 1983 or Johnson-Laird and Byrne, 1991. Toavoid confusion, we denote the theory de ned in the latter articles by MMT, and the 2 theory stated b elow by EST. Although EST is not a formal corollary of MMT , it preserves some imp ortant features of the latter. It will b e seen that these common features lead EST to an incorrect exp erimental prediction. The question thus arises whether MMT is committed to the same mistake. Subsequentto examining MMT we consider its principal rival, namely, rule-based theories. A theorem of pro of-theory rep orted in Bo olos, 1987 is exploited for the purp ose of raising a p otential ob jection to a large class of theories of this kind. Our discussion pro ceeds as follows. Section 1 presents EST. Two empirical consequences of EST are derived in Section 2. Exp erimental data inconsistent with the consequences are rep orted in Section 3. In Section 4 we describ e a follow-up exp eriment designed to isolate the defect in EST. Section 5 discusses the b earing of our ndings on MMT. Remarks ab out rule-theories b egin in Section 6. We conclude in Section 9. 1 The theory EST Description of our theory b egins by sp ecifying the kind of reasoning problem to which it applies. For this purp ose we x a formal language and consider the class of arguments that arise from translating its formulas into natural language. 1.1 Language For our formal language, we cho ose a simple fragment of predicate logic, namely, the monadic 3 calculus without identity. In order to facilitate translation into natural language, we limit 1 For exp erimental evidence supp orting the mental mo dels approach to deductive reasoning, see Johnson-Laird and Byrne, 1991; Johnson-Laird et al., 1992; Johnson-Laird et al., 1989. Extension of the theory to other domains is rep orted in Schaeken et al., 1996; Byrne and Johnson-Laird, 1989; Johnson-Laird, 1995a. Concerns ab out the clarityofMental Mo dels Theory are voiced in Ford, 1995, pp. 7-8, Bonatti, 1994, Ho dges, 1993, Andrews, 1993, Garnham, 1993, Newstead, 1993, Wetherick, 1993 Evans and Over, 1997, p. 27, Rips, 1994, p. 359. 2 Mnemonic: read \MMT " as \mental mo dels theory" and \EST " as \elementary semantic theory." 3 The monadic calculus without identity includes only 1-place predicates like \is red" and \is happy" rather than higher-arity predicates like \loves." The 2-ary relation of identity \=" is likewise excluded. A formula is said to b e closed if all of the variables o ccuring in it are b ound by quanti ers. For background, see Quine, 1982. 1 _ attention to closed formulas whose sentential connectives include no more than :, ^, _, and _ . The latter denotes exclusive disjunction, which holds b etween two sentences just in case exactly one of them is true. Our language thus excludes the conditional ! and the biconditional $, source of much confusion when passing from logic to natural language. Some formulas that will b e of interest in the sequel are the following. 1 a 8xLx _ Dx ^:Lx ^ Dx _ b 9xLx _8y Dy _ c 9xLx _8y Dy _ d :9xLx _8y Dy 1.2 Translation into natural language The translation of a formula into natural language is achieved via standard lo cutions, as discussed in many texts, e.g., Ho dges, 1977. Toavoid hyp erformalization we assume that a few examples suce to convey the idea. Thus, in the domain of p eople, if D means \will die tomorrow" and L means \will live tomorrow", then the formulas of 1 are carried into: 2 a Every p erson will either live tomorrow or die tomorrow; no one will b oth live and die tomorrow. b Someone will live tomorrow or else everyone will die tomorrow. c There is someone such that either he lives tomorrow or else everyone dies tomorrow. d There is no one such that either he lives tomorrow or else everyone dies tomorrow. Only translations of such straightforward character will be at issue in what follows, and it is assumed that there is no ambiguity in recovering the translated formula. Wehave restricted our formal language to make the latter assumption tenable. In particular, the absence of predicates of greater arity eliminates the scop e ambiguity asso ciated with multiple quanti cation. Of course, the clarity of our translation pro cedure do es not guarantee that logically untutored reasoners understand sentences like 2a-d in the same way we do. It will thus b e necessary in the sequel to return to issues of interpretation. 1.3 Arguments By an argument is meant a nonempty, nite sequence of at least two formulas. The last formula is called the argument's conclusion, the remaining formulas are its premises. For example, writing premises ab ove conclusions, three arguments are shown in Table 1. Natural language counterparts to the arguments are provided in Table 2. 1.4 Mo dels By a model for a given argument is meant a pair S ;f, where S is a nonempty set, and f is a function that maps each predicate app earing in the argumentinto a subset p ossibly empty of 2 4 S . We illustrate with some mo dels for arguments involving just the predicates L and D . Let S b e the set fGeorge, Mario, Peter, Carlg. Then seven mo dels for the arguments in Table 1 are shown in Table 3. Supp ose that we are given a mo del S ;f and a formula ' whose predicates b elong to the domain of f . Then we rely on the standard de nition of truth-in-a-mo del to interpret the idea 5 that \S ;f satis es '." . We illustrate with the mo dels in Table 3. 3 a Neither of the mo dels m , m satisfy the formula 1a. 6 7 b All of the mo dels m - m satisfy the formula 1a. 1 5 c All of the mo dels m - m satisfy the formula 1b. 1 5 d All of the mo dels m - m satisfy the formula 1c. 1 5 e None of the mo dels m - m satisfy the formula 1d. 1 5 1.5 The theory EST Our theory of deductive reasoning maynow b e formulated as follows. The Theory EST: Supp ose that argument P ::: P ;C is translated into the 1 n ;C . Let reasoner R b e given. Then there is ::: P natural language argumentP n 1 a set M of mo dels for P ::: P ;C with the following prop erties. 1 n a M are all the mo dels m such that: i. m comes to the mind of R when R attempts to determine the validityof ;C , and P ::: P n 1 ii. R is able to determine in m the resp ective truth values of P ::: P ;C. 1 n b If no mo del in M satis es eachofP ::: P , then R has no rm judgment ab out 1 n . ::: P whether C follows logically from P n 1 if every mo del in c Otherwise, R judges C to follow logically from P ::: P n 1 M that satsi es each of P ::: P also satis es C . R judges C not to follow 1 n logically from P ::: P if any of the mo dels in M that satisfy eachofP ::: P 1 n n 1 fails to satisfy C . We illustrate with argument A of Table 2. Observe that the premise of A translates formula 1a whereas the conclusion translates 1b.
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