Introduction

A habitual phenomenon in human cognition seems to prove that our reasoning is not
absolutely logical, at least if the criterion assumed to check that is classical
logic. In particular, the problem is a rule that not only is valid in this last
logic, but that it can even considered as a basic schema in that logic. Indeed,
^{Deaño (1999: 154)} shows that, in a
framework based on ^{Gentzen’s (1934}-^{1935)} work, that is, a framework whose
structures must be well-formed formulae of classical logic, the aforementioned rule
has that status. The rule is the disjunction introduction rule (from now on, DIR),
which is often expressed in this way:

p

-----------

So p ∨ q

Where ‘∨’ represents disjunction.

Or, if preferred,

q

-----------

So p ∨ q

This schema causes difficulties because most of the time people do not apply it or think that it is incorrect. It is accepted only in certain specific cases. Of course, this fact can lead one to assume, as said, that the human inferential activity is not logical -or that that activity is not necessarily coherent with standard logic- and to think that, if that were so, the rule should be used. However, what is important for this paper in this way is that, obviously, any framework proposed with the intention to explain reasoning cannot ignore this phenomenon. Such a framework has to account for it and predicts the situations in which the rule will be applied and the circumstances in which it will not.

The mental models theory (from now on, MMT) appears to be able to do that. Indeed,
^{Orenes and Johnson-Laird (2012)} give an
explanation following this last approach that proposes the reasons why people only
use DIR on certain particular occasions and the usual behavior is to reject it.
Nevertheless, it can be thought that ^{Orenes and
Johnson-Laird’s (2012)} account of DIR is not completely coherent with the
general framework of MMT, the reason being that this theory includes certain
assumptions that can lead to predictions absolutely different from those raised by
them. Thus, possible problems that can be indicated are the difference between
Mental Models and Fully Explicit Models that MMT provides, and the fact that further
developments of this very approach (e.g., ^{Khemlani,
Orenes, & Johnson-Laird, 2014}) have made it evident that the habitual
way people understand disjunction does not match that described by ^{Orenes and Johnson-Laird (2012)}. However, my
main aim here is to show that such problems are not real from ^{Orenes and Johnson-Laird’s (2012)} framework and that hence
their arguments are absolutely compatible with the general approach of MMT.

In this way, to achieve that goal, firstly, I will comment on Orenes and ^{Johnson-Laird’s (2012)} proposal about DIR. Then
I will address the points of MMT that are apparently in conflict with their
explanation to argue that there are not really contradictions between such points
and that explanation. Accordingly, I begin focusing on essentially ^{Orenes and Johnson-Laird’s (2012)} work.

1. DIR in Orenes and Johnson-Laird’s (2012) paper

Actually, the study carried out by ^{Orenes and
Johnson-Laird (2012)}, which not only considers dis-junction but also the
conditional, has been analyzed and reviewed in several different papers (e.g., ^{López-Astorga, 2015}). Nonetheless, I will try
to keep to just what is claimed in Orenes and Johnson-Laird’s original text about
disjunction here. In this regard, it can be said that their basic idea is that, as
indicated by MMT (see, e.g., ^{Johnson-Laird, 2012:
138, Table 9.2}), a disjunction such as p ∨ q refers to a model with three
semantic possibilities or possible scenarios:

[1]: p & q

[2]: p & ¬q

[3]: ¬p & q

Where ‘¬’ expresses negation and ‘&’ can be understood as conjunction.

This means that p ∨q can describe [1], i.e., a situation in which both of the disjuncts are true, [2], i.e., a situation in which only the first disjunct (p) is true, or [3], i.e., a situation in which only the second disjunct (q) is true. So, if q is the premise, the second possibility [2] is precisely the problem with DIR, since, as it can be noticed, it indicates that q is false. One example can be helpful to explain this in a clearer way. Let us think about this inference with the formal structure of DIR:

“Viv is here.

Therefore, Pat is here or Viv is here, or both” (Orenes & ^{Johnson-Laird, 2012: 362}).

As pointed out and explicitly shown by Orenes and ^{Johnson-Laird (2012: 362)}, the disjunction embedded into the conclusion
can be true in three cases:

[1]: Pat & Viv

[2]: Pat & ¬Viv

[3]: ¬Pat & Viv

Clearly, [2] is incompatible with the premise *Viv is here*, as it
provides that Viv is not here. Thus, this is the cause that people tend not to
consider DIR to be correct, since its conclusion includes a possibility that is
absolutely inconsistent with the previous information, that is, the premise.

But, as said, there are also cases in which individuals accept the rule and, if a
theory wants to exactly describe the way the human deductive activity really works,
it should provide an account of them too. As also mentioned, Orenes and ^{Johnson-Laird (2012)} present such an account as
well, which is supported by an important thesis of MMT. According to this last
theory, when people are identifying the models of a sentence, *modulation
processes* can happen. Such processes are related to semantics and
pragmatics and can modify the content of a particular model (by adding or
eliminating possibilities) by virtue of the meaning of the words that the sentence
includes and the circumstances in which it is expressed (in addition to ^{Orenes & Johnson-Laird (2012)}, see, e.g.,
^{Johnson-Laird (2012)}, ^{Johnson-Laird & Byrne (2002)}, or ^{Johnson-Laird, Khemlani, & Goodwin (2015)}.
This can be seen in an obvious way by means of another example with the same
structure as the previous one:

“Lucia wore jewelry.

Therefore, Lucia wore the bracelet or she wore jewelry” (^{Orenes & Johnson-Laird, 2012: 363}).

Now, a modulation process occurs and its result is the elimination of [2] as a possibility for the dis-junction in the conclusion:

[1]: Bracelet & Jewelry

[3]: ¬Bracelet & Jewelry

And the reason is not hard to understand: if pragmatics and the exact meanings of the
words are taken into account, according to MMT, it can be thought that it is not
possible wearing a bracelet and not wearing jewelry, as the former pragmatically and
semantically implies the latter. So Orenes and ^{Johnson-Laird (2012)} claim that, in cases such as this one, which are
called by them cases of *modulated* inference, people should consider
DIR to be valid, since there is no contradiction between the premise and the
possibilities of the conclusion (neither [1] nor [3] are in conflict with the
premise). And, certainly, their second experiment appeared to confirm their
predictions. In that experiment they used tasks of the two types commented on and
the results were clear: most of their participants thought that the non-modulated
inferences were not correct and that the modulated ones were valid.

Of course, there are works that criticize this account and propose an alternative
explanation for the results presented by Orenes and ^{Johnson-Laird (2012)}, for example, that of ^{López-Astorga (2015)}, which also gives a commentary on their
proposal akin to the one just provided above. I will briefly refer to ^{López-Astorga’s (2015)} paper below but, as far
as my aims here are concerned, what should be done now is to indicate the reasons
why certain ^{Orenes and Johnson-Laird’s (2012)}
arguments that can seem problematic assuming the basic framework of MMT are not
truly so. I do that in the next section.

2. The exclusive character of disjunction and the difference between Mental Models and Fully Explicit Models

A first difficulty in Orenes and ^{Johnson-Laird’s
(2012)} explanation can be that, if later papers supporting MMT are taken
into account, it appears that their account needs to be reviewed. Indeed, for
example, the results of one of the experiments included in ^{Khemlani et al.’s (2014)} research make it evident that people
usually interpret disjunctions as exclusive. In a condition of that experiment, they
presented to their participants a disjunction such as p ∨ q and asked them to
respond with which of theses possibilities that disjunction was compatible:

[1]: p & q

[2]: p & ¬q

[3]: ¬p & q

[4]: ¬p & ¬q

The majority response was that [2] and [3] were the possibilities linked to p ∨ q,
[1] and [4] being ignored. [4] is not a problem, since it was not considered by
Orenes and ^{Johnson-Laird (2012) }.
Nevertheless, [1] does be a difficulty because they included it in their comments on
their examples. In this way, if we pay attention to the results of this experiment,
which was carried out with thematic content, and not, as my description here, with
just formal structures, it must be stated that the real possibilities corresponding
to the example of non-modulated inference above that should have been mentioned by
Orenes and ^{Johnson-Laird (2012)} are just:

[2]: Pat & ¬Viv

[3]: ¬Pat & Viv

Likewise, in the case of the modulated inference, only one scenario should have been taken into account:

[3]: ¬Bracelet & Jewelry

And this is so because, as indicated, these are the actual possibilities that,
following ^{Khemlani et al.’s (2014)} results,
people could identify if faced to disjunctions such as those embedded into the
conclusions of the examples cited. Of course, it must be acknowledged that the fact
that, as pointed out, the expression *or both* appears in the
non-modulated version can lead individuals to notice that the disjunction is
inclusive and that [1] is a possible scenario. Nevertheless, in the same way, it
must also be said that, in the materials that were really used in their experiment,
the non-modulated inferences did not have such an expression (see Orenes & ^{Johnson-Laird, 2012: 375}), that the example of
modulated inference, as shown, did not include it either, and that the actual
modulated inferences presented to their participants lacked it as well (see also
^{Orenes & Johnson-Laird, 2012:
375}).

Thus, it can be thought that ^{Khemlani et al.’s
(2014)} experimental results are already enough to undermine Orenes and
^{Johnson-Laird’s (2012)} explanation.
However, this is not truly so. On the one hand, that its disjunction is exclusive
does not have an influence on the non-modulated inference at all. [2] continues to
be a possible scenario and, as said, it denies the premise. On the other hand,
although one might think that the situation is different in the modulated inference,
MMT has the necessary resources to argue in favor of ^{Orenes and Johnson-Laird’s (2012)} account regardless of whether the
disjunction is inclusive or exclusive. The case in which the disjunction is regarded
as inclusive has already been commented on above. As far as the alternative in which
it is considered as exclusive is concerned, modulation can solve the problem as well
and show why the inference must be accepted. It is true that firstly it can be
thought that the prediction should also be that people would reject the inference.
And this is so because the only possibility of the conclusion is [3], which provides
that the bracelet is not worn, that is, an information that cannot be derived just
from the premise, i.e., from the fact that jewelry is worn. However, this could only
be actually thought after a superficial reading of the theses raised by MMT.

As explained, [2] is removed because modulation reveals that it is not possible a
scenario with the bracelet and without jewelry. Nevertheless, that very modulation
process can make the disjunction inclusive, since it shows that [2] is not possible
because, as also commented on, whenever there is a bracelet, there is jewelry as
well. And this clearly leads to [1] as another possibility. Certainly, modulation
does not only eliminate possible scenarios. It can also modify the elements of a
particular scenario (^{Johnson-Laird et al., 2015:
202}). Hence it can be admitted that, in the case of the modulated
inference, modulation does not just removes [2], but that it transforms [2] into [1]
(by virtue of general knowledge, it is known that it is not possible having a
bracelet without having jewelry, but it is also known that, if a bracelet is had,
jewelry is had too). Accordingly, [1] and [3] can be the possibilities again. Thus,
it can be interpreted that what the disjunction in the conclusion really means is
that, whether the bracelet is worn [1] or not [3], as indicated in the premise,
jewelry is always worn (which is true in both [1] and [3]).

So, given that the possibilities can be the same if the disjunction is understood as
exclusive, this is not a great problem for Orenes and ^{Johnson-Laird’s (2012)} proposal, as their general arguments can
continue to be raised. In this way, what ^{Khemlani et
al.’s (2014)} study actually implies is not that the explanation given by
^{Orenes and Johnson-Laird (2012)} does not
hold, but only that that explanation needs to be updated by means of the information
achieved with more current research.

Nonetheless, another point that appears to be a true difficulty for that explanation
and allow questioning it is the fact that, from the beginning, MMT distinguishes
between Mental Models and Fully Explicit Models (see, e.g., ^{Johnson-Laird, 2004}, ^{2006},
^{2012}; ^{Johnson-Laird et al., 2015}; ^{Oakhill
& Garnham, 1996}). The distinction is based on an essential principle
of the theory: *the principle of truth*. According to it, the Mental
Models “…represent only what is possible given the truth of assertions, not what is
impossible…” (^{Johnson-Laird, 2012: 138}). And
this implies that only secondly individuals can identify the Fully Explicit Models,
that is, the sets of possibilities with the complete scenarios standing for both
what is true in them and what is false in them. Thus, although transforming Mental
Models into Fully Explicit Models is possible, it is also an activity that can be
hard and require deep reflection. For this reason, at first most people tend to only
detect the Mental Models of each sentence.

For the case of an exclusive disjunction such as p ∨ q (where ‘∨’ expresses exclusive disjunctive relationship), this means that the initial models, that is, the Mental Models, considered by most individuals are not even [2] and [3], which are the Fully Explicit Models, but simply:

[5]: p

[6]: q

As it can be checked, [5] and [6] correspond to [2] and [3], the difference being that, certainly, in the former what is denied, that is, what is false (¬q in [5] and ¬p in [6]) does not appear. But, if this is so, it must also be admitted that, given a non-modulated inference such as that described above, the scenarios firstly considered are truly these ones:

[5]: Pat

[6]: Viv

This, clearly, is not a problem for Orenes and ^{Johnson-Laird’s (2012)} predictions, since, while [5] is not absolutely
inconsistent with the premise *Viv is here*, it is obvious that the
fact that Pat is here cannot be deduced from that premise alone. So the expected
response continues to be that the inference is not correct.

However, although it can seem to be the contrary, the situation does not radically change in the case of the modulated inference either. From what has been said, it is clear that the Mental Models of the example indicated are:

[5]: Bracelet

[6]: Jewelry

Obviously, it can be thought that, at first, modulation cannot have an influence on
[5]. If the information that the jewelry is not worn does not appear, it is
impossible to note that the scenario expresses a contradiction (the bracelet is worn
and jewelry is not) and has to be removed or modified. So, it could be stated that,
as in the previous case of the non-modulated inference, although it is not
incoherent with the premise *Lucia wore jewelry*, [5] cannot be
inferred from it (that Lucia wore jewelry does not lead to that she wore the
bracelet). In this way, it could also be claimed that really the inference should be
considered as incorrect, and that Orenes and ^{Johnson-Laird’s (2012)} account can only be accepted if it is supposed
that individuals always (or at least almost always) identify the Fully Explicit
Models in inferences with the structure of the tasks used by them. Nevertheless, in
their defense, ^{Orenes and Johnson-Laird
(2012)} could also argue that, given that individuals know the meanings of
both *bracelet* and *jewelry*, it is possible to
complete [5] (or, if preferred, to transform it into [1]) with the information that,
because the bracelet is worn, jewelry is worn as well. This is clearly possible in
the theory because it provides that modulation can help identify the Fully Explicit
Models as well (see, e.g., ^{Johnson-Laird & Byrne,
2002: 658}). Thus, the possibilities would be in this case [1] and [6],
that is, that Lucia wore both the bracelet and jewelry and that she only wore
jewelry. Such scenarios would be absolutely compatible with the premise, since what
the conclusion would provide would be that jewelry is always worn ([1] and [6]),
with the possibility that, in addition, the bracelet, which is, as said, jewelry
too, is worn as well [1]. Hence it can be held that, even when individuals only
identify the Mental Models, the prediction of MMT continues to be that they will
consider the modulated inference to be correct. After all, the premise is not denied
in any of the possible scenarios, and, as explained, the only additional element
(the bracelet) may be [1] or may not be [6].

Therefore, neither the fact that people tend to interpret disjunction as exclusive
nor the differentiation between Mental Models and Fully Explicit Models are a real
problem for Orenes and ^{Johnson-Laird’s (2012)}
account. Given that, based on the basic ideas of MMT, these are the two most
immediate objections that can be presented against their explanation, it can be
stated that their proposal is clearly coherent with such ideas.

Conclusion

Indeed, Orenes and ^{Johnson-Laird’s (2012)}
approach is incompatible neither essential theses of MMT (e.g., the distinction
between Mental Models and Fully Explicit Models) nor further developments of this
last theory (e.g., the finding that disjunction is mainly exclusive in the human
thought). The first point, the one related to the differentiation between types of
models, is especially important, since it have been said that a great weakness of
MMT is that it does not explain in detail neither the exact situations in which it
is more probable that the Fully Explicit Models are identified nor the exact
circumstances in which only the Mental Models are often taken into account (e.g.
^{López-Astorga, 2016}). Nonetheless, even
supposing that this weakness is real, as seen, it does not have an influence on the
arguments provided by ^{Orenes and Johnson-Laird
(2012)}.

On the other hand, as said above, other theories present alternative explanations to
the one of Orenes and ^{Johnson-Laird (2012)}.
However, such theories also have weaknesses. An example in this regard can be ^{López-Astorga’s (2015)} argumentation. In his
view, it is possible to continue to argue that a logic exists in the human mind. It
is only necessary to acknowledge, as, for instance, the mental logic theory (e.g.,
^{Braine & O’Brien, 1998}; ^{Gouveia, Roazzi, O’Brien et al. 2003}; ^{O’Brien, 2014}) does, that that logic is not
exactly classical logic. Thus, it can be proposed, as the mental logic theory does
too, that, while many rules of standard logic and the natural deduction calculus are
easily applied by human beings, not all of them are so, DIR being precisely one of
the schemata valid in the mentioned logic and the mentioned calculus that are
difficult for people. In this way, the results of the non-modulated inferences are
explained: if DIR is hard to individuals, it makes sense that that kind of inference
is considered as incorrect.

As far as the results achieved by Orenes and ^{Johnson-Laird (2012)} with the modulated inferences are concerned, the
solution given by ^{López-Astorga (2015)} is
simple. If a disjunction such as *Lucia wore the bracelet or she wore
jewelry* only refers to the scenarios [1] and [3], it is not a true
disjunction. If a formula such as p ∨ q is only true in the cases of p and q, and ¬p
and q, this fact reveals that its truth table in classical logic is not the one of
disjunction, but that of a formula such as (p ∨ ¬p) → q, where ‘→’ represents
conditional relationship, or just q. Thus, the real structure of the modulated
inference is not

q

-----------

So p ∨ q

But

q

--------------------

So (p ∨ ¬p) → q

or, still more clearly,

q

------

So q

It is evident that, regardless of the content of q, if this last formula is taken as
a premise and it is asked whether or not it follows from it that q is true, most
people will respond positively. And this shows that ^{López-Astorga’s (2015)} thesis that the actual logical form of the
conclusion in the modulated inference is not that of disjunction can be a good
explanation of why this last kind of inference is often considered as valid by
individuals.

However, a problem of ^{López-Astorga’s (2015)}
argumentation is that it must face results such as those presented by ^{Khemlani et al. (2014)} as well, according to
which individuals generally interpret dis-junctions as exclusive. This is an
important point because it reveals that the only real scenario of the conclusion in
the modulated inference can be, as indicated, just [3], and that what has to be
done, from this last account, is to think about a formula whose truth table in
standard logic shows that it is only true in the case of the combination ¬p and q.
Nevertheless, addressing this issue, which, in principle, does not seem very
difficult keeping ^{López-Astorga’s (2015)}
basic approach, would take us off the main aims of this paper, which are focused on
Orenes and ^{Johnson-Laird’s (2012)}
arguments.

In this way, what is truly interesting here of ^{López-Astorga’s (2015)} proposal is that, as it can be noticed, while it
can be an alternative explanation to that of Orenes and ^{Johnson-Laird (2012)}, it is also clear that the former depends
to certain extent on the latter. Indeed, as López-Astorga explicitly indicates
(^{López-Astorga, 2015: 148}), to recover
the real logical form of the conclusion in the modulated inference is absolutely
necessary to take into account the possibilities assigned to it by ^{Orenes and Johnson-Laird (2012)}, and to assume,
as MMT does, that modulation can remove some of these possibilities. And this means
that, ultimately, it is necessary to accept that the possibilities of disjunction
are usually [1], [2], and [3], and that, in the modulated inferences, [2] is
not.

So, in short, it appears that we can speak about an important strength of MMT.
Although an approach such as that of ^{López-Astorga
(2015)} is assumed and it is thought that, in the modulated versions of
the inference, p ∨ q is not the real logical form of the conclusion, the methodology
of MMT must be assumed too. And this is so because the possibilities of the
disjunctive sentences have to be considered to detect the true logical forms (as
said, ^{López-Astorga, 2015}, already comments
on this fact).

Furthermore, although the criticism raised by ^{López-Astorga (2016)} is correct and MMT needs to clarify the factors
that cause only the Mental Models to be identified in some cases and the Fully
Explicit Models in entirety to be detected in other instances, there is no doubt
that the distinction between these two kinds of models can be fruitful. In fact, as
shown by the literature on cognition, the distinction is very useful to account for
many intellectual phenomena. Hence, if MMT actually has problems in this way, it can
be expected that its proponents will work on a clear description of the exact
mechanisms that lead to one or other of these types of models in next studies.