Models, Rules, and Deductive Reasoning1

Home , Deductive reasoning, Explanation

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

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

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

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

calculus without identity. In order to facilitate translation into natural language, we limit

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.

Mnemonic: read \MMT " as \mental mo dels theory" and \EST " as \elementary semantic theory."

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

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

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

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

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

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. Let reasoner R be given. Supp ose that

just the mo dels m - m of Table 3 come to R's mind when thinking ab out A , and that R

1 7

can determine the truth-values of formulas 1a and 1b in each of these mo dels. Then, in view

of 3b, the subset of m - m satisfying the premises of A is not empty. Moreover, by 3c,

1 7

the ve mo dels satisfying the premise of A also satisfy its conclusion. So, EST predicts that

R will judge the premise of A to logically imply its conclusion.

Intuitively, EST conceives deductive reasoning ab out a given argument as involving three

steps. First, the reasoner R brings to mind a set of mo dels; the particular mo dels evoked

dep end b oth on the argument in question and on particularities of R. Any mo dels for which the

The set S can b e nite or in nite. It is known, however, that nite mo dels suce to characterize validity

within monadic logic Bo olos and Je rey, 1989, Ch. 25.

See, for example Enderton, 1972. 3

truth-value of a premise or conclusion is not clear to R are discarded. If none of the remaining

mo dels satisfy all of the premises, R is left without an intuition ab out the deductive validity

of the argument. Otherwise, R determines whether the conclusion of the argument is true in

every mo del that satis es all of the premises. The argument is judged valid if such is the case,

otherwise invalid.

Theory EST is mo dest in the sense that no constraint is placed on which mo dels are asso ciated

by a given reasoner with a given argument. Indeed, one p ossibility is that the empty set is

uniformly asso ciated with every argument, leading to the prediction that the reasoner has no

deductive judgment at all. On the other hand, EST makes the substantive assumption that

deductive reasoning pro ceeds via the generation of mo dels of the standard kind, illustrated in

Table 3.

2 Two predictions

Thorough test of EST requires op erationalizing the concept \mo del m comes to the mind of

reasoner R." For present purp oses, however, we fo cus on the more transparent concept of

judging a particular argument to be valid or invalid. It is easy to see that EST makes the

following prediction ab out such judgments.

4 Prediction: Supp ose that argument P ::: P ;C is translated into the natural lan-

1 n

;C . Let reasoner R b e given. If R judges that C is a logical ::: P guage argumentP

n 1

, then C is consistent with fP ::: P g, that is, there is at least .. .P consequence of P

1 n

one mo del that satis es all of P ::: P ;C.

1 n

To illustrate, Prediction 4 implies that R will not judge

\Everyone will live tomorrow and everyone will die tomorrow."

to follow logically from

No one will b oth live and die tomorrow."

This is b ecause the conclusion contradicts the premise, so there are no mo dels of b oth.

The derivation of Prediction 4 from EST do es not dep end on the use of standard mo dels.

The same prediction results from anychoice of formal ob ject as mo del, provided only that the

following condition is met.

5 Condition on models: Supp ose that m is a mo del of ' ::: ' , and that the latter

1 n

formulas logically imply . Then m is also a mo del of .

Condition 5 holds for the mo dels guring in EST b ecause the de nition of logical implication

is tailored for this purp ose. The condition also holds for the kinds of non-standard mo dels that

Two limiting cases are worth noting. First, if the conclusion of an argumentisvery long, then R will be

unable to determine its truth-value in any mo del, so the set M of EST will b e empty, and the reasoner will b e

predicted to have no judgment ab out the argument in question. Second, if the premises are formally contradictory,

then R will b e unable to construct a mo del for them, and thus likewise b e left without an intuition ab out the

argument's validity. This second prediction is contrary to logic, but psychologically plausible. 4

app ear in MMT. Indeed, it is precisely for this reason that advo cates of mental mo dels deduce

4 from their own versions of the theory. Thus, Johnson-Laird Johnson-Laird, 1995a, p. 195

writes:

\The theory of reasoning based on mental mo dels makes three principal predictions.

. . . Second, erroneous conclusions will tend to b e consistent with the premises rather

than inconsistent with them. Reasoners will err b ecause they construct some of the

mo dels of the premises | typically, just one mo del of them | and overlo ok other

p ossible mo dels. This prediction can b e tested without knowing the detailed mo dels

p ostulated by the theory; it is necessary only to determine whether or not erroneous

conclusions are consistent with the premises."

A related consequence of EST and of MMT isemb o died in the following prediction.

6 Prediction: Let reasoner R and valid argument P ::: P ;C be given. If R has an

1 n

intuition ab out whether C is a logical consequence of P ... P , then R will judge that

C is in fact a logical consequence of fP ::: P g.

1 n

We derive Prediction 6 from EST as follows. Supp ose R judges that C do es not follow

logically from P ...P . Then clause c of EST implies that the set M of mo dels for R and

P ::: P ;C includes at least one that veri es P ::: P and falsi es C . So P ::: P ;C is

1 n 1 n 1 n

invalid.

3 Exp erimental test of the predictions

In the present section we rep ort exp erimental data that app ear to discon rm Predictions 4

and 6.

3.1 A sp ecial case of the two predictions

Consider arguments B and C in Table 1, along with their translations B , C in Table

2. Let us show:

7 Fact: Argument B is valid whereas the conclusion of C is contradicted by its

premise.

Pro of: Since the conclusion of C is the negation of the conclusion of B it suces

to show that B isvalid. Let m =S ;f b e a mo del of 8xLx _ Dx ^:Lx ^ Dx.

It suces to show that m is a mo del of 9xLx _8y Dy.

Since m satis es 8xLx _ Dx ^:Lx ^ Dx, ff L;fD g is a partition of S .

Supp ose rst that there is a 2 f L. Then a witnesses 9xLxinm. Since f D 6= S ,

8y Dy is false in m. Hence, recalling the truth-conditions for _ , we see that a

witnesses 9xLx _8y Dy in m.

See also Johnson-Laird, 1995b, p. 1007 where the same prediction is formulated. The prediction b ears b oth

on drawing conclusions from given premises and also on evaluating stated conclusions. The latter are referred to

as \putative conclusions" in Johnson-Laird, 1995a. 5

Now supp ose that f L = ;, then f D = S , so m satis es 8y Dy. Cho ose any

a 2S. Then a 62 f L, hence a witnesses 9xLx _8y Dy in m.

So wehave shown that in all cases m satis es 9xLx _8y Dy.

Now supp ose that reasoner R is instructed to designate as valid exactly one of arguments B ,

C in Table 2. Since the arguments translate B , C , resp ectively, Fact 7 and Prediction

4 and 6 imply that R will either cho ose randomly, or else cho ose B . Thus:

8 Specific prediction of EST: Asked to cho ose whichofB , C is the valid argu-

ment, most reasoners will either cho ose randomly or else cho ose B .

We shall shortly describ e exp erimental data that contradict 8. This is a nding that many

readers will have anticipated on the basis of their own intuitions inasmuch as the validity of

B has a paradoxical character. So it might b e useful to disp el one p ossible interpretation of

the faulty prediction 8, namely, that p eople fail to interpret the \or else" construction in B ,

C as exclusive disjunction. According to this idea, the \or else" is interpreted inclusively,

allowing b oth disjuncts to be true. Such a supp osition do es not evade the counter-intuitive

prediction, however, since Fact 7 remains true even if _ is interpreted inclusively,as _. That

is, wehave the following result, proved similarly to b efore.

9 Fact: Let P be 8xLx _ Dx ^:Lx ^ Dx as b efore, and let C , C be 9xLx _

8y Dy, :9xLx _ 8y Dy, resp ectively. Then P implies C and contradicts C .

It follows that 8 remains a consequence of EST even if \or else" in B , C is interpreted

inclusively.

3.2 Exp erimental results

We presented 62 university students in Milan Italy a single page starting with instructions that

can b e translated as follows.

On this page you will nd three \arguments." Each argument consists of a premise

followed by two p otential conclusions. For each argument, please cho ose the con-

clusion that is a logical consequence of the premise. In other words, cho ose the

conclusion whose truth seems to b e guaranteed by the premise.

For each argument, exactly one of the conclusions is in fact a logical consequence of

the premise. So please b e sure to indicate exactly one of the two conclusions in each

case.

The remainder of the page displayed the three arguments each with two conclusions as shown

in Figure 1. The rst two arguments were used as warm-up. The third argumentemb o dies the

sp eci c test of Prediction 8; its two conclusions were individually randomly ordered for each

resp ondent.

The Italian version of the third argument has premise

Ogni persona o vivr a o morir a domani; nessuno vivr a e morir a al lo stesso tempo. 6

Exactly half of the 62 students resp onded correctly to the twowarm-up arguments, cho osing

the rst and second conclusions, resp ectively. These 31 resp ondents may b e considered to b e the

most alert and logically comp etent of our sample. Of the 31 students resp onding correctly to the

rst two arguments, only 6 chose the valid conclusion of the third argument, in conformity with

8. Twenty- ve resp ondents chose the contradictory conclusion. The same pattern emerges for

the 31 students who resp onded incorrectly to one or b oth of the rst two arguments. Seven

chose the valid conclusion of the third argument, in conformity with 8. Twenty-four chose the

contradictory conclusion.

Suchchoices are clearly not random, and they contradict 8. Hence, they contradict EST,

and also MMT since, as wesaw, the latter seems committed to Predictions 4 and 6 hence to

3.3 Replication with variant wording

The \such that" wording app earing in the crucial argument of Figure 1 strikes us as a natural

means of expressing existential quanti cation. Indeed, the lo cution app ears prominently in clas-

sic logic texts as well as in contemp orary discussions of quanti ers in linguistics. It nonetheless

seemed prudentto rep eat the exp eriment of Section 3.2 using an alternative wording. For this

purp ose we replaced the last pair of conclusions shown in Figure 1 with the following:

10 a There is someone with the following prop erty: either he lives tomorrow or else

everyone dies tomorrow.

b There is no one with the following prop erty: either he lives tomorrow or else every-

one dies tomorrow.

All other asp ects of the stimuli and pro cedure were the same as b efore. Forty-nine students

participated in the variant exp eriment; none had participated in the rst version. Thirty-four

students resp onded correctly to the two warm-up arguments. Of these 34, only 8 chose the

valid conclusion of the third argument, in conformity with 8. Twenty-six resp ondents chose

the contradictory conclusion. For the remaining 15 students resp onding incorrectly to one

or b oth of the rst two arguments, only 3 chose the valid conclusion of the third argument,

in conformity with 8. Twelve chose the contradictory conclusion. As b efore, these results

contradict Prediction 8 of EST and MMT. They also suggest that lo cutions involving \such

that" compared to \following prop erty" are equally interpretable.

and alternative conclusions:

C' e qualcuno tale che o lui vivr a domani o altrimenti tutti moriranno domani.

Non c' e nessuno tale che o lui vivr a domani o altrimenti tutti moriranno domani.

For \such that" in logic texts, see Tarski, 1941, p. 9, Supp es, 1957, p. 48, Quine, 1982, p. 143, among

many other works. In linguistics it app ears, for example in Partee and Hendriks, 1997, p. 52.

The Italian version of the argument has the same premise as b efore, and alternative conclusions:

C' e qualcuno con la seguente propriet a: o lui vivr a domani o altrimenti tutti moriranno domani.

Non c' e nessuno con la seguente propriet a: o lui vivr a domani o altrimenti tutti moriranno domani. 7

4 Isolating the defect in EST

In explaining the data just rep orted it is tempting to assert that the students failed to grasp the

intended logical forms of the arguments they evaluated. According to this idea, neither 2c nor

10a were mentally asso ciated with the meaning expressed in the formula 9xLx _8y Dy.

There is danger of vacuity, however, in this kind of explanation inasmuch as the principal

criterion for grasping a logical form is the abilityto recognize its valid consequences. To avoid

circularity, a theory that exploits logical notation | like EST and MMT | must sp ecify how

formulas are mapp ed onto natural language. The exp eriments rep orted ab ove relied on two of

the simplest mappings p ossible. A prop onentofmental mo dels who wishes to defend the theory

by questioning the latter translations will ultimately need to prop ose some alternative scheme

for converting formulas into English.

Let us now consider a di erent explanation for the data rep orted ab ove. Within the mental

mo dels p ersp ective, it is natural to sp eculate that the resp ondents to ok the contradictory con-

clusion to b e true in the mo dels of the premise, and the valid conclusion to b e false. We tested

this hyp othesis by presenting students with mo dels m - m in Table 3, and requesting them to

1 5

select which of the two sentences 2c,d is true in each. From the p oint of view of logic, since

m - m satisfy the premise 2a, each satisfy its logical consequence 2c and not the negation

1 5

2d.

To determine the students' view of the matter, for each of the mo dels m - m we prepared

1 5

a page with the following instructions.

On this page you will nd a question concerning the truth or falsityofsentences in

a given situation. The question describ es a situation involving a few p eople, along

with a pair of sentences. Exactly one of the two sentences is true in the situation

describ ed. Please indicate which sentence is true of the situation.

The remainder of the page displayed the mo del and the two conclusions 2c, 2d. Figure

2 illustrates using m . The order of the conclusions was individually randomized for each

participant.

The ve problems were distributed to one hundred and seventeen students from the same

p opulation as b efore no student participated in either of the earlier studies. Each participant

resp onded to just one problem, as indicated in Table 4. Also shown in Table 4 is the choice

between the two conclusions. It can be seen that in each of the ve mo dels investigated, our

resp ondents manifested a strong tendency to reject the true sentence 2c in favor of its false

negation 2d.

What explains the misleading character of the two sentences? One factor may b e the emb ed-

_ _

ded quanti cation present in their formulas 9xLx _8y Dy and :9xLx _8y Dy. We see

therein that the universal quanti cation 8y Dy o ccurs within the scop e of the existential quan-

ti er 9x. A contrast is provided by 9xLx _8y Dy which underlies the sentence: \Someone

will live tomorrow or else everyone will die tomorrow." Here there is no emb edded quanti cation,

and an informal survey of opinion revealed little dicultyinevaluating its truth within m - m .

1 5

_ _

Observe, however, that 9xLx _8y Dy is not logically equivalentto 9xLx _8y Dy. So

the latter formula is not simply a confusing means of expressing the former. Therefore nothing

_ _

Sp eci cally, m of Table 3 satis es 9xLx _8y Dy but not 9xLx _8y Dy .

6 8

excludes the p ossibility that the greater diculty asso ciated with 9xLx _8y Dy stems from

its particular meaning, rather than from quanti er emb edding per se.

5 Discussion of EST and MMT

Whatever the role of quanti er emb edding in explaining our results, the fact remains that EST

makes a false prediction, and must therefore be revised. One p otential revision is to abandon

the assumption that p eople p erceive the truth-values of sentences within mo dels in the standard

way. Condition 5 on mo dels would then no longer be a feature of the theory, thereby lifting

Predictions 4 and 6. With such mo di cation it is p ossible to retain the view that the validity

of an argument is evaluated by verifying the \truth" of the conclusion in every mo del of the

premises that the argument brings to mind | but now it is truth-according-to-the-reasoner that

matters. To illustrate, Table 4 suggests that the resp ondents in our study consider 2d rather

than 2c as true in the mo dels of 2a. On the revised theory this is enough to explain their

preference for the contradictory argumentC inTable 2 over its valid counterpart B .

A drawback to the foregoing prop osal is the reduction it entails in EST's predictivepower.

Without detailed information ab out which sentences a reasoner R takes to be true in a given

mo del, the theory is mute ab out R's judgment of logical consequence. Moreover, relaxing the

satisfaction relation b etween sentences and standard mo dels may not b e enough to x what is

ailing EST. We questioned students ab out the kind of situations that arguments B and C

bring to mind, and were struckby the richness of the resp onses. Prop erties nowhere mentioned

in the premise and conclusion often to ok center stage, for example, the \control" that one p erson

exerts over others, or his \p ower" to in uence the course of future events. If such a wide array

of mentally represented situations participate in logical judgment, and if even the simplest of

them do not verify and falsify sentences standardly, then it b ecomes particularly challenging to

mo dify EST in a way that is at once clear, predictively nonvacuous, and accurate.

Finally, let us consider the b earing of EST's predictive inaccuracies on MMT . It was noted

earlier that MMT is apparently committed to Condition 5, hence implies Predictions 4 and

6. The results of our rst exp eriment are thus as discon rming for MMT as they are for EST.

However, the mental mo dels envisioned by MMT are not the standard ones of EST, so there

might be a natural way of mo difying them so as to avoid unwanted application of Condition

5. Alternatively,itmay prove p ossible to formulate a supplementary hyp othesis of non-mo del-

theoretic character to explain the attractiveness of argument C compared to B e.g., an

hyp othesis concerning quanti er emb edding. To b e empirically adequate, of course, any new

version of MMT must predict not only logical error of the kind do cumented here, but also the

accurate judgment triggered by a wide range of other arguments.

6 The rule-based p ersp ective on deductive reasoning

The principal rival to MMT is the rule-based theory prop osed in Osherson, 1974; Braine,

1978, and develop ed with great sophistication and p ersuasiveness in Rips, 1994. Rule-

based theories claim that deductive reasoning ab out an argument is the covert attempt to

In a similar spirit, pragmatic principles are invoked in Johnson-Laird and Bara, 1984 to explain facts ab out

syllogistic reasoning that would otherwise b e at variance with MMT .

See also Osherson, 1975a; Osherson, 1975b; Rips, 1983 and Braine, 1990; Braine and O'Brien, 1991. 9

construct a derivation of the conclusion from the premises, relying for this purp ose on mentally

represented inference rules. Thus, rule-theorists psychologize the proof-theoretic p ersp ectiveon

validity instead of the semantic p ersp ective that inspires the p ostulation of mental mo dels.

There is an apparent shift of topic in the transition from MMT to rule theories. The argu-

ments of concern to mental mo del theorists are typically simple in structure, and their p erceived

validity/invalidity is supp osed to arise after brief re ection, p erhaps partly unconscious. In

contrast, rule-theorists like Rips conceive their theory as embracing extended reasoning ab out

stated information in view of detecting its logical consequences. The more-or-less immediate

intuition of argument validity is conceived as the sp ecial case of a brief derivation. We take

another sp ecial case of deductive reasoning to b e the ability to follow someone else's pro of of the

validity of an argument. For such a pro of to b e comprehensible and p ersuasive, each step should

be veri able using inference schemata that participate in the detection of validity in simpler

settings.

Our goal in the present section is to call attention to two features of rule theories, and to

argue that no theory with b oth of them is likely to b e true. To describ e the rst feature, let us

say that a theory is comprehensive if it claims to cover all cases where human b eings make or

verify deductive inferences of a rst-order character. As noted ab ove, this includes not only

the detection of validity in given arguments, but also cases where a p erson is \solving problems

that have traditionally come under the heading of deduction" Rips, 1994, p. 5. Recognizing

the correctness of a putative derivation is an activity that falls traditionally under the heading

of deduction, so it also falls within the purview of comprehensive theories at least, according

to our use of the term.

Regarding the second feature, we note that the inference rules app earing in virtually all

rule-theories yet prop osed can b e easily validated using a standard system of rst-order logic.

Such inference rules include Modus Ponens A; A ! B B and Universal Instantiation

8x'x 't=x for any term t. A theory whose inference rules are justi able in this way will

henceforth b e called normal. For a precise de nition, we select some familiar pro of calculus for

rst-order logic, let us say, the one presented in Benson Mates' p opular text Mates, 1972. We

then consider another calculus to be normal if the length of its shortest derivation for a given

inference is always comparable to that in the standard system. To b e de nite, we stipulate that

for a pro of system S to b e normal it must b e the case that for every inference I for which Mates'

system provides a pro of P , there must be a pro of of I within S that requires no more than a

trillion times the numberofsymb ols o ccurring in P . It must also b e the case that S provide no

derivations for invalid inferences.

We are not certain to what extent rule theorists haveembraced the hyp othesis that there is a

comprehensive and normal theory of deductive reasoning. Perhaps the most explicit endorsement

can b e found in Braine, 1978, p. 20, where a close relationship is evoked between \standard

logic" and \the entailments of the natural system." Although the author says his theory is

only intended to cover prop ositional reasoning, he sp eculates that \there must presumably b e

natural logics" to cover other areas of deduction. See also Rips, 1995, Sec. 9.4. Whether or

not accepted in explicit form by psychologists, comprehensiveness and normality are attractive

See Rips, 1994, pp. 5-14, Rips, 1995, p. 324, and Rips, 1989. The latter pap er b ears on complex

deductive reasoning ab out \Knights and Knaves" problems as develop ed in Smullyan, 1978.

First-order logic is the familiar \predicate calculus" studied in college. See, for example, Mates, 1972; Quine,

1982; Ho dges, 1977. The quanti ers in rst-order logic range over variables for individuals in the domain of

discourse. In contrast, second-order logic allows quanti ers that range over subsets of the domain, third-order

logic allows quanti ers for sets of sets, etc. 10

assumptions ab out deductive comp etence, and are likely to be held implicitly by numerous

students of reasoning. It is for this reason that we wish to examine whether the two assumptions

are jointly tenable.

As a preliminary, let us raise and set aside one kind of ob jection to the theories at issue.

Any comprehensive normal theory of deductive comp etence seems to run into trouble when it

is applied to mathematical reasoning. Many mathematical theorems can be phrased as rst-

order inferences; for example nearly every result in set theory can be written as a rst-order

inference from the Zermelo-Fraenkel axioms. But at rst sight it is not at all plausible that

mathematicians make their deductions by using no more than the rules enco ded in rst-order

logic certainly, their pro ofs often deploy a wider arsenal of inference principles, for example,

mathematical induction. However, it seems p ossible to disarm this ob jection by refusing to

classify sophisticated mathematical reasoning as a basic human skill since it rests up on arti cial

techniques which p eople require years to master. So it is sensible to exclude fancy mathematical

reasoning even from a \comprehensive" theory.

For this reason, we prop ose to discuss an example of deductive reasoning that cannot b e set

aside on grounds of mathematical sophistication, yet nonetheless is not feasibly provable within

any normal theory. The example is due to the late George Bo olos Bo olos, 1987, who quali ed

it as a \curious inference." Let us quote the conclusion to his study.

\The fact that we so readily recognize the validity of [the curious inference] would

seem to provide as strong a pro of as could b e asked for that no standard rst-order

logical system can be taken to be a satisfactory idealization of the psychological

mechanisms or pro cesses, whatever they might b e, wherebywe recognize rst-order!

logical consequences." Bo olos, 1987, p. 5

We shall try to explain the character and signi cance of Bo olos' example.

7 Bo olos' curious inference

It will b e a serious challenge to comprehensive, normal theories of reasoning if we can exhibit

an inference with the following prop erties:

11 a The inference is valid in rst-order logic.

b There is a pro of of the inference that can b e recognized as correct byatypical alb eit

determined human reasoner with no particular mathematical skill or knowledge.

c No standard pro of calculus for rst-order logic could b e used to prove the inference

in less than a lifetime.

We claim that the following example, drawn from Bo olos, 1987, meets conditions 11a-c.

12 Premises:

a P m

b 8x [ P x ! P Ax ]

c 8x [ S x; m =Am ] 11

d 8x [ S m;Ax=AAS m ;x]

e 8x8y [ S Ax; Ay =S x; S Ax; y ]

Conclusion:

P S AAAAm ; AAAAm

In this inference, S is a binary function symb ol, A is a unary function symb ol, P is a unary

predicate, and m is an individual constant symb ol.

It may b e helpful to have a natural-language example of 12. Supp ose we discover a Martian

sp ecies that can pro duce either asexually or sexually. We isolate a particular genome of the

sp ecies and wonder whether its descendents would prosp er in Earth's environment. Let us use

the following abbreviations.

Ax = the result of asexual repro duction by x.

S x; y = the result of sexual repro duction by x and y .

P x x can prosp er in Earth's environment.

m = the particular genome we discovered.

Then, the rst premise says that m can prosp er on Earth. The second premise says that

this prop erty is passed along via asexual repro duction. The third premise says that sexual

repro duction between m and anyone else yields the same creature as asexual repro duction by

m. The remaining premises are more complex, but p erfectly comprehensible. The conclusion

concerns what happ ens if we take two copies of the fourth asexual descendantofm, and mate

them with each other. The conclusion states that the resulting creature can prosp er on Earth.

8 Do es the curious inference meets conditions 11a-c?

We consider our three claims ab out argument 12. It will not b e necessary to argue separately

for 11a since 12 is clearly a rst-order argument, and its validity follows from 11b.

8.1 In favor of 11c.

For every standard pro of calculus of rst-order logic there are two algorithms whichwe can call

the normalizing algorithm and the bounding algorithm. The normalizing algorithm converts any

pro of in the calculus into a normal form known as \a cut-free sequent pro of." The b ounding

algorithm reveals how many more steps are needed in the cut-free sequent pro of of a given infer-

ence compared to the original. These are well-known to ols of pro of theory see Schwichtenb erg,

1977, and they apply straightforwardly to Mates' calculus our reference system for de ning

the class of normal theories.

Bo olos shows that every cut-free sequent pro of of 12 contains a tremendous number n of

symb ols. Moreover, given a normal system S , the normalizing and b ounding algorithms can b e

used to show that any pro of P of 12 within S will require only \slightly" fewer than n symb ols.

So, the numb er of symb ols in P is still tremendous. For example, it dwarfs the numb er of protons

that could be densely packed into the visible universe by even the most generous estimates.

There is thus no short pro of of 12 in S | for example, no pro of that could b e written down 12

at the rate of one trillion symb ols p er nanosecond in the time it has taken the human race to

evolve from sea creatures.

To b e sure, this argument for 11c do es not extend b eyond normal systems of logic. More-

over, the unsolved problem NP = co-NP of complexity theory is closely related to the size

of derivations in prop ositional logic Co ok and Reckhow, 1979, so gives witness that there are

fundamental facts still unknown ab out pro of systems, even in prop ositional logic see Urquhart,

1995 for discussion. A rule-based theory of human reasoning based on a nonnormal pro of sys-

tem, however, would b e radically di erent from anything on o er to day, so it is safe to assume

that rule-theorists in psychology have some kind of normal theory in mind.

8.2 In favor of 11b.

To demonstrate 11b, wemust prove 12 by a comprehensible argument that relies on no sp ecial

mathematical skill or background. The pro of o ered in Bo olos, 1987 do es not serve our purp ose

since understanding it requires exp erience with the technicalities of second-order logic. To make

it plausible that a simpler pro of is p ossible, we note that 12 lies within the class of inferences

that Kurt Godel claimed humans can make by mere mental insp ection \Anschauung" of the

relevant concepts, without needing any formal pro of see Godel, 1990. But p erhaps what

Godel found obvious would not be obvious to the rest of us! So we attempt to give our own

informal but convincing argument for 12. In fact it is an example of the typ e of \argumentby

descent" that Pierre Fermat used in the seventeenth century,hundreds of years b efore predicate

logic was formalized.

Pro of of the validity of 12

Assume premises 12a-e. Let us say that an ob ject is \reachable" if it is m or Am or AAm

or . . . allowing any nite number of A's in front of m. Now 12a says that P holds of m.

And 12b tells us that if P holds of m then it holds of Am; and that if it holds of Am then

it holds of AAm; and so on. So 12a and 12b together tell us that P holds of any reachable

ob ject. Therefore, to deduce the conclusion, we need only show that S AAAAm; AAAAm is

reachable, in other words, that S AAAAm; AAAAm =m,orS AAAAm; AAAAm=Am,or

S AAAAm; AAAAm=AAm etc. For this purp ose, it suces to show the following, stronger

fact:

14 For every reachable ob ject u, if ob ject v is also reachable then S u; v is reachable.

So we prove 14. If 14 is true when u is m, and when u is Am, and when u is AAm, and so

on forever, then 14 is true for all reachable u. Therefore if 14 is not true for all reachable u,

there must b e some rst u in the order m;Am;AAm ::: for which it is not true. We will show

that this is imp ossible, thereby proving 14, and hence 12.

Supp ose rst that 14 is false when u is m. Then by the same reasoning as b efore, there

must be a rst v in the order m;Am;AAm ::: for which S u; v is not reachable. This v is

not m, since premise 12c tells us that S m; m is Am hence reachable. So, take v to b e the

rst ob ject in the list Am;AAm;::: for which S m;v is not reachable. Hence, v is Av , for

some v written with one fewer A's than v is. But this is imp ossible, as wenow show. Premise

0 0 0

12d tells us that if v = Av , then S m;v=AAS m;v , and S m;v is reachable since v is 13

supp osed to b e rst in the order Am;AAm;::: for which S m;v is not reachable. So, for some

number of A's, S m;v = A Am, hence S m;v = AAA Am, so S m;v is reachable

after all, contradicting our choice of v . But now we've exhausted the p ossible choices of v for

which S m;v is not reachable. It follows that our putative counterexample u to 14 cannot

be m.

Hence, the counterexample u must b e one of Am;AAm :::. Again, let us take u to b e the rst

0 0

such counterexample in the list. So, u can b e written as Au , where u has one fewer A than u

do es. Since u is a counterexample to 14, there must b e some rst v in the list m;Am;AAm :::

for which S u; v is not reachable. As b efore, 12c shows that v is not m, soitmust b e Av , for

0 0

v with one fewer A than app ears in v . By assumption S u; v is reachable, and since 14 was

0 0

true for all reachable ob jects earlier than u, S u ;Su; v must also be reachable inasmuch

0 0 0

as u comes b efore u. But 12e says that S u; v is precisely this same S u ;Su; v , hence

reachable as well. So the counterexample u cannot b e anyof Am;AAm :::, which in conjunction

with the preceding demonstration that u is not m shows that there is no counterexample at all.

End of pro of

The foregoing demonstration will be convincing to virtually anyone who agrees to some

mental e ort; an hour's worth is largely sucient. Atworst, the pro of needs tinkering to clinch

the matter of its accessibility. It thus shows the curious inference to enjoy prop erty 11b.

8.3 The curious inference and deductive reasoning

Our stalking horse in this section is the class of comprehensive, normal theories of logical com-

p etence. Such theories represent the ability to discover and verify pro ofs of validity in rst order

contexts as issuing from a system of derivation that resembles standard ones like Mates'. We

claim, with Bo olos, that the curious inference emb o dies a decisive ob jection to such theories. On

the one hand, the validityof the inference can be checked by a simple and convincing demon-

stration; indeed, the demonstration is so simple that even its sp ontaneous invention is likely

op en to many untrained reasoners. On the other hand, the validityof the inference cannot be

demonstrated feasibly within any pro of system that remotely resembles Mates'.

It is consistent with these facts to conclude that deductive comp etence relies on rst-order

inference rules that are harnessed in some non-normal way, sp eci cally, within a pro of-system

that escap es the complexity b ounds on all systems considered standard to day. This would be

a capital discovery. Alternatively and more plausibly,it seems to us, the accessibility of our

pro of may b e taken to illustrate the recourse that reasoners have to inference principles that go

beyond rst-order logic, even in the context of a purely rst-order argument like 12. Indeed,

the pro of pivots on an app eal to the ellipsis \..." in the de nition of reachable ob jects, and

then exploits the way such ob jects are \written" to prove a fact ab out them all. The ellipsis,

and similar expressions like \and so on" and \etc", allow us to de ne the set of ob jects that

can be constructed from some xed ob ject by applying a given function any nite number of

times. In contrast, it follows from the so-called compactness property that such sets cannot be

de ned within the language of rst-order logic see Ebbinghaus et al., 1994, Ch. 9. Similarly,

evo cation of the way ob jects are written suggests the ability to ascend to a metalinguistic p er-

sp ective on the argument under scrutiny. It seems plausible to us that the deductive comp etence

of nonsp ecialists can make ready use of such devices to validate putative pro ofs and to invent 14

them, even when a purely rst-order argument could be given in principle, as in the case of

12. This is just what is denied by normal theories.

Let it b e clear that there are several ways normality could b e violated by deductive reasoning.

On the one hand, principles of second-order logic might b e applied to arguments of sup er cially

rst-order syntax. As is the case for 12, the passage to second-order reasoning can shorten

the pro ofs of some rst-order validities. Another path to nonnormality is to implicitly add

substantive assumptions to the premises of an argument, and then pro ceed via purely rst-order

reasoning. For example, by adding the axioms of set-theory to the premises, our argument for

12 could b e paraphrased into rst-order reasoning. We shall resist the temptation to sp eculate

ab out how normality breaks down in ordinary reasoning since we wish to maintain fo cus on

the fundamental p oint raised in this section, namely, that the class of normal, comprehensive

theories is not an adequate representation of deductive comp etence.

9 Concluding remarks

Two approaches to characterizing deductive comp etence have b een discussed in this pap er.

One adopts a semantic stance by p ositing the manipulation of mental mo dels in the search for

counterexamples to a putatively valid argument. The other adopts a pro of-theoretic stance by

p ositing the generation of covert derivations of conclusions from premises using inference rules

of a sort familiar from standard texts in logic. In each case we have tried to de ne a precise

class of theories while acknowledging that contemp orary psychological hyp otheses do not fall

strictly within either of them. Still, the ob jections wehave raised might help to condition further

psychological theorizing of b oth mo del-theoretic and pro of-theoretic character.

There can b e no doubt ab out the dicultyofuntangling these issues empirically, esp ecially

given the narrow range of b ehavioral data available to psychologists. So it is worth emphasizing

the imp ortance of the enterprise. The analysis of logical intuition has more than once b een a

stimulus to the development of mathematical logic even if the latter sub ject now has wider

motivation. So it may b e hop ed that progress in understanding the psychology of deductive

reasoning will suggest new ideas ab out logical formalisms, their interpretation, and ecient

means of assessing implication within them.

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8xLx _ Dx ^:Lx ^ Dx

9xLx _8y Dy

8xLx _ Dx ^:Lx ^ Dx

9xLx _8y Dy

8xLx _ Dx ^:Lx ^ Dx

:9xLx _8y Dy

Table 1: Three, one-premise arguments in the monadic predicate calculus without identity. 19

Every p erson will either live tomorrow or die tomorrow;

no one will b oth live and die tomorrow.

Someone will live tomorrow or else everyone

will die tomorrow.

Every p erson will either live tomorrow or die tomorrow;

no one will b oth live and die tomorrow.

There is someone such that either he lives tomorrow

or else everyone dies tomorrow.

Every p erson will either live tomorrow or die tomorrow;

no one will b oth live and die tomorrow.

There is no one such that either he lives tomorrow

or else everyone dies tomorrow.

Table 2: Translation into natural language of the arguments A, B , C shown in Table 1. 20

m f L=fGeorge, Mariog f D =fPeter, Carlg

m f L=fGeorge g f D =fMario, Peter, Carlg

m f L=fGeorge, Mario, Peter f D =fCarlg

m f L=; f D =fGeorge, Mario, Peter, Carlg

m f L=fGeorge, Mario, Peter, Carlg f D =;

m f L=fGeorge, Mariog f D =fGeorge, Mario, Peter, Carlg

m f L=fGeorge, Mariog f D =fCarlg

Table 3: Various mo dels that apply to the arguments shown in Table

1. Each has domain fGeorge, Mario, Peter, Carlg and interprets the two

predicates L and D . 21

Number of Correct Incorrect

Model respondents choice of 2c choice of 2d

m 23 7 16

m 22 8 14

m 24 6 18

m 24 7 17

m 24 6 18

total: 117 34 83

Table 4: Number of resp ondents for each of the problems corresp onding

to mo dels m - m . Also shown in each case is the numb er of resp ondents

1 5

who judged 2c versus 2d to b e true in the corresp onding situation. 22

On the musical tastes of doctors:

No do ctor listens to b oth Mozart and Brahms.

Each do ctor either do es not listen to Mozart or do es not listen to Brahms.

Some do ctor do es not listen to Mozart.

On the games preferred by lawyers:

All lawyers either playchess or monop oly.

Some lawyer plays monop oly.

Every lawyer that do es not playchess plays monop oly.

On the prospects of living or dying tomorrow:

Every p erson will either live tomorrow or die tomorrow; no one will b oth live and die tomorrow.

There is someone such that either he lives tomorrow or else everyone dies tomorrow.

There is no one such that either he lives tomorrow or else everyone dies tomorrow.

Figure 1: The three items app earing in the inference questionnaire. The

order of conclusions in the third argument was individually randomized

for each sub ject. 23

Situation:

There are four p eople, named George, Mario, Peter and Carl.

George and Mario will live tomorrow.

Peter and Carl will die tomorrow.

Sentences:

There is someone such that either he lives tomorrow or else everyone dies tomorrow.

There is no one such that either he lives tomorrow or else everyone dies tomorrow.

Figure 2: The problem corresp onding to m in Table 3. The order of the

two conclusions was individually randomized for each sub ject. 24