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. 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
4
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.
5
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,
6
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
n
1
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
6
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
7
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
n
1
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
n
1
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.
7
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.
0
9 Fact: Let P be 8xLx _ Dx ^:Lx ^ Dx as b efore, and let C , C be 9xLx _
0
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
8
resp ondent.
8
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
8.
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-
9
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.
10
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.
9
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.
10
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
1
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
11
_ _
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
11
_ _
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
12
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,
13
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
12
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 .
13
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
14
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
15
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
14
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.
15
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 .
13
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
0
rst ob ject in the list Am;AAm;::: for which S m;v is not reachable. Hence, v is Av , for
0
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
0
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 :::
0
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
16
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
17
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.
References
Andrews, A. A. 1993. Mental mo dels and tableau logic. Behavioral and Brain Sciences, 16:334.
Bonatti, L. 1994. Prop ositional reasoning by mo del? Psychological Review, 101:725{733.
Bo olos, G. 1987. A Curious Inference. Journal of Philosophical Logic, 16:1{12.
16
In fact Godel Godel, 1986 p ointed out in 1936 that for any pro of system of rst-order logic there are theorems
of that system whichhave drastically shorter pro ofs in second-order logic. For the last twentyyears of his life,
Godel worked on the idea that one could break through the limitations of formal reasoning by using higher-order
concepts. He seems to have b elieved that Husserl gave useful clues to howwe might strengthen our reasoning
powers by creating new concepts. For discussion of Godel's views on all this, see Ho dges, 1997.
17
One example is the work of Godel Godel, 1990 on the mental content of consistency pro ofs for arithmetic,
mentioned at the b eginning of Section 8.2. Intuitionistic logic provides an example of psychological theses spurring
the creation of an entire branch of mathematical logic. And of course the word \natural" in the name \natural
deduction" has psychological overtones. 15
Bo olos, G. S. and Je rey, R. C. 1989. Computability and Logic, Third Edition. Cambridge
University Press, New York.
Braine, M. D. S. 1978. On the relation between the natural logic of reasoning and standard
logic. Psychological Review, 85:1{21.
Braine, M. D. S. 1990. The `natural logic' approachto reasoning. In Overton, W. F., editor,
Reasoning, Necessity, and Logic: Developmental Perspectives. Erlbaum, Hillsdale, NJ.
Braine, M. D. S. and O'Brien, D. P. 1991. A theory of if: Lexical entry, reasoning program,
and pragmatic principles. Psychological Review, 98:182{203.
Byrne, R. and Johnson-Laird, P. N. 1989. Spatial reasoning. Journal of Memory and Language,
28:564{575.
Co ok, S. and Reckhow, R. 1979. The releative eciency of prop ositional pro of systems. Journal
of Symbolic Logic, 44:36{50.
Ebbinghaus, H.-D., Flum, J., and Thomas, W. 1994. Mathematical Logic Second Edition.
Springer-Verlag, New York, NY.
Enderton, H. B. 1972. A Mathematical Introduction to Logic. Academic Press, New York.
Evans, J. S. B. T. and Over, D. E. 1997. Rationality in reasoning: The problem of deductive
comp etence. Current Psychology of Cognition, 161-2:3{38.
Ford, M. 1995. Two mo des of mental representation and problem solution in syllogistic rea-
soning. Cognition, 54:1{71.
Garnham, A. 1993. Anumb er of questions ab out a question of numb er. Behavioral and Brain
Sciences, 16:350{1.
Godel, K. 1986. On the length of pro ofs. In Feferman, S., editor, Kurt Godel, Col lected Works,
Volume I, pages 397{9. Oxford University Press, New York.
Godel, K. 1990. On a hitherto unutilized extension of the nitary standp oint. In Feferman, S.,
editor, Kurt Godel, Col lected Works, Volume II, pages 241{251. Oxford University Press,
New York.
Ho dges, W. 1977. Logic. Penguin Bo oks, New York.
Ho dges, W. 1993. The logical content of theories of deduction. Behavioral and Brain Sciences,
16:353{354.
Ho dges, W. 1997. Turing's philosophical error? In Landau, L. J. and Taylor, J. G., editors,
Concepts for Neural Networks, pages 147{169. Springer-Verlag, Berlin.
Johnson-Laird, P. N. 1983. Mental Models. Harvard University Press, Cambridge MA.
Johnson-Laird, P. N. 1995a. Mental mo dels and probabilistic thinking. Cognition, 50:171{191.
Johnson-Laird, P. N. 1995b. Mental mo dels, deductive reasoning, and the brain. In Gazzaniga,
M. S., editor, The Cognitive Neurosciences. M.I.T. Press, Cambridge MA.
Johnson-Laird, P. N. and Bara, B. G. 1984. Syllogistic inference. Cognition, 16:1{61. 16
Johnson-Laird, P. N., Byrne, R., and Schaeken, W. 1992. Prop ositional reasoning by mo del.
Psychological Review, 99:418{439.
Johnson-Laird, P. N., Byrne, R., and Tab ossi, P. 1989. Reasoning by mo del: the case of
multiple quanti cation. Psychological Review, 96:658{673.
Johnson-Laird, P. N. and Byrne, R. M. J. 1991. Deduction. Erlbaum, Hillsdale, NJ.
Mates, B. 1972. Elementary Logic 2nd Edition. Oxford University Press, New York NY.
Newstead, S. E. 1993. Do mental mo dels provide an adequate account of syllogistic reasoning
p erformance? Behavioral and Brain Sciences, 16:359{60.
Osherson, D. 1974. Logical Inference: Underlying Operations. John Wiley, New York, NY.
Osherson, D. 1975a. Logic and mo dels of logical thinking. In Falmagne, R. J., editor, Reason-
ing: Representation and Process. Erlbaum, Hillsdale, NJ.
Osherson, D. 1975b. Reasoning in Adolescence: Deductive Inference. Lawrence Erlbaum
Asso ciates, Hillsdale NJ.
Partee, B. and Hendriks, H. L. W. 1997. Montague Grammar. In van Benthem, J. and ter
Meulen, A., editors, Handbook of Logic and Language. North-Holland, Amsterdam.
Quine, W. V. O. 1982. Methods of Logic, Fourth Edition. Harvard University Press, Cambridge
MA.
Rips, L. 1994. The Psychology of Proof. MIT Press, Cambridge, MA.
Rips, L. 1995. Deduction and cognition. In Smith, E. E. and Osherson, D., editors, Invitation
To Cognitive Science, Volume 3 second edition, pages 297 { 344. MIT Press, Cambridge
MA.
Rips, L. J. 1983. Cognitive pro cesses in prop ositional reasoning. Psychological Review, 90:38{
71.
Rips, L. J. 1989. The psychology of knights and knaves. Cognition, 31:85{116.
Schaeken, W. S., Johnson-Laird, P. N., and d'Ydewalle, G. 1996. Mental mo dels and temp oral
reasoning. Cognition, 60:205{234.
Schwichtenb erg, H. 1977. Pro of Theory: Some Applications of Cut-Elimination. In Bar-
wise, J., editor, Handbook of Mathematical Logic, pages 867{895. North-Holland Publishing
Company, Amsterdam.
Smullyan, R. 1978. What is the Name of this Book? The Ridd le of Dracula and Other Logical
Puzzles. Prentice-Hall, Englewo o d Cli s NJ.
Supp es, P. 1957. Introduction to Logic. D. Van Nostrand, Princeton NJ.
Tarski, A. 1941. Introduction to Logic and to the Methodology of Deductive Sciences. Oxford
University Press, Oxford, England.
Urquhart, A. 1995. The Complexity of Prop ositional Pro ofs. The Bul letin of Symbolic Logic,
14:425{467. 17
Wetherick, N. E. 1993. More mo dels just means more diculty. Behavioral and Brain Sciences,
16:367{8. 18
8xLx _ Dx ^:Lx ^ Dx
A
_
9xLx _8y Dy
8xLx _ Dx ^:Lx ^ Dx
B
_
9xLx _8y Dy
8xLx _ Dx ^:Lx ^ Dx
C
_
: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.
A
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.
B
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.
C
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
1
m f L=fGeorge g f D =fMario, Peter, Carlg
2
m f L=fGeorge, Mario, Peter f D =fCarlg
3
m f L=; f D =fGeorge, Mario, Peter, Carlg
4
m f L=fGeorge, Mario, Peter, Carlg f D =;
5
m f L=fGeorge, Mariog f D =fGeorge, Mario, Peter, Carlg
6
m f L=fGeorge, Mariog f D =fCarlg
7
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
1
m 22 8 14
2
m 24 6 18
3
m 24 7 17
4
m 24 6 18
5
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
1
two conclusions was individually randomized for each sub ject. 24