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DOI: 10.24234/wisdom.v21i1.741 Miguel LOPEZ-ASTORGA
STOIC LOGIC FROM THE THEORY OF MENTAL MODELS
Abstract
An essential point about Stoic philosophy is why certain arguments and rules are basic in their logic. That is the case of the indemonstrables and the themata. It has been proposed that assuming the theory of mental models, one can think that the five indemonstrables and two of the themata are easy to understand for the human mind. This can explain why those arguments and rules are essential components in Stoic logic. In addition, it is relevant because, given that the theory of mental models tries to capture the real way people reason, it can show that Stoic logic is closer to the manner individuals naturally make inferences than modern propositional calculus. The present paper is intended to move forward in this direction. It has two aims: one of them is to give an account from the theory of mental models of all of the themata. The other one is to argue that a simple schema that is correct in modern propositional calculus, and which, however, is not deemed as a true syllogism in Stoic logic, is difficult for people according to the theory of mental models. Those are further pieces of evidence that Stoic logic describes the way human beings think to a greater extent than modern logic.
Keywords: indemonstrables, Stoic logic, syllogism, themata, theory of mental models.
Introduction
The reason why the indemonstrables (avoOToSeiKioi) in Stoic logic are deemed as indemonstrable is hard to understand from modern propositional calculus. This is because only one of them (and they are five) is actually indemonstrable following modern logic. However, this problem can be resolved if modern propositional calculus is ignored and another framework is taken into account (see, e.g., Lopez-Astorga, 2017). Furthermore, it has already been said that modern logic is not the best approach to interpret the Stoic system (Bobzien, 1996).
In this way, it has been proposed that, while Stoic logic does not seem to be consistent with modern propositional calculus, it appears to be coherent with some contemporary cognitive theories, for example, the mental logic theory (e.g., Lopez-Astorga, 2015) and the theory of mental models (e.g., Lopez-Astorga, 2017). This paper will focus on the theory of mental models. Its
purpose will be to show that Stoic logic is even more compatible with that theory than what the literature has revealed. To do that, the paper will move forward in two senses. On the one hand, it will deal with the themata (Qe^axa). On the other hand, it will analyze an argument that, following Bobzien (1996), cannot be admitted as a real syllogism (ouAXoyio^cx;) in Stoic logic, although the argument is correct in modern logic.
The themata are important rules in Stoic logic. They help determine whether or not an argument is a syllogism. If the argument is an indemonstrable or can be transformed, employing an analysis (avaluoi^) process, into an indemonstrable, it is a syllogism. The themata are essential because they are rules allowing making those analysis processes. It seems that the themata are four (De Lay, 1984, Galeni De Placitis Hippoc-ratis et Platonis 114, 1-10) or five (there are two versions of the third one; see, e.g., Bobzien, 1996). However, only the first thema and the third one (its two versions) are preserved. Bob-
zien (1996) describes the themata kept and, in addition, presents a rebuilding of the missing ones. That reconstruction will be assumed here. Thus, the first point of this paper will be to review whether all of the themata, in the way Bob-zien (1996) considers them, can be easily understood from the theory of mental models. The literature has already shown that with regard to the indemonstrables (Lopez-Astorga, 2017) and the first thema and one of the versions of the third thema (Lopez-Astorga, 2016), which is not the version Bobzien (1996) takes into account. So, Bobzien's reconstructions of the second and the fourth themata, as well as the version of the third one she accepts, will be addressed below.
The second point of this paper will have to do with an argument Bobzien (1996) also indicates. It is an argument sound in modern calculus. However, it is not admissible in Stoic logic. The reason for that is simple: it is not an indemonstrable, and it cannot be transformed into an indemonstrable by means of themata. Thereby, what the present paper will try to explain is that the argument is difficult for people within the theory of mental models. Accordingly, the accounts below can be further evidence in favour of the idea that Stoic logic is closer to the theory of mental models than to modern logic. This can mean that, if the theory of mental models explains how the human mind works, Stoic logic is also closer to the real way human beings reason than modern propositional calculus.
To achieve these goals, first, the paper will present the Stoic indemonstrables and the four themata rebuilt by Bobzien (1996). Then, it will explain some important theses of the theory of mental models. The third section will briefly comment on the accounts based on the theory of mental models in the literature about the reasons why the indemonstrables, the first thema, and one of the versions of the third thema were deemed as basic inferences in Stoic logic. Fourth, the other version of the third thema and the reconstructions of the second and fourth ones by Bobzien (1996) will be considered in order to
show that they are coherent with the way people make inferences according to the theory of mental models. The last section will deal with the mentioned argument Bobzien identifies and argues that it is hard to accept following that very theory.
The Stoic Indemonstrables and Four Thematha
It seems that Chrysippus of Soli was the Stoic philosopher that first presented the indemonstra-bles (e.g., Sextus Empiricus, AdversusMathema-ticos 8, 223) (Mau, 2011; O'Toole & Jennings, 2004). As said, only one of them is indemonstrable in modern propositional calculus (see also, e.g., Lopez-Astorga, 2017). It is MPP (Modus Ponendo Ponens) (Marcovich, 1999, Diogenes Laertius, Vitae Philosophorum 7, 80).
MPP:
First premise: If p, then q
Second premise: p
Conclusion: q
An important point about MPP is that its first premise (Xrj^a) is a conditional (ouvrm^evov). This is relevant because there is a difference from modern logic here. Modern logic follows the view of the conditional Philo of Megara gave (e.g., Sextus Empiricus, Adversus Mathematicos 8, 113) (Mau, 2011; Mates, 1953). That view is known as 'the material interpretation of the conditional'. It provides that a conditional such as the first premise of MPP is true ". .whenever is not the case that the antecedent is true and the consequent false" (O'Toole & Jennings, 2004, p. 479). However, Chrysippus of Soli, and, hence, Stoic logic, claims one more requirement: to be true, a conditional needs " ..a certain relationship between the clauses. In particular, the contrary (avuKei^evov) of the second clause (Xrjyov) must be inconsistent with (^axexoa) the first clause (^you^evov)" (Lopez-Astorga, 2017, p. 311). Therefore, the first premise in MPP is true if and only if the contrary of q fights (^dxexai) p. Undoubtedly, this makes Stoic logic very different
from frameworks such as that of Gentzen (1934, 1935). Likewise, it also removes paradoxes such as those of the implication (for a description, see, e.g., Orenes & Johnson-Laird, 2012). As it is well known, the material interpretation of the conditional allows inferring a conditional with any antecedent from any formula if this last formula is taken as the consequent of the new conditional. If, as in Stoic logic, a relation between the two clauses is necessary, a paradox of that kind is not possible.
The second indemonstrable is MTT (Modus Tollendo Tollens) (see Diogenes Laertius, Vitae Philosophorum 7, 80) (Marcovich, 1999). MTT:
First premise: If p, then q Second premise: Not-q Conclusion: Not-p
Of course, in MTT, as far as the first premise is concerned, not-q must also be inconsistent with p.
MPT I (Modus Ponendo Tollens I) is the third
indemonstrable (see Diogenes Laertius, Vitae Philosophorum 7, 80) (Marcovich, 1999). MPT I:
First premise: Not-(p and q) Second premise: p Conclusion: Not-q
Another indemonstrable is MPT II (Modus ponendo Tollens II) (see Diogenes Laertius, Vi-tae Philosophorum 7, 81) (Marcovich, 1999). MPT II:
First premise: Either p or q, but not both of them
Second premise: p Conclusion: Not-q
The first premise in MPT II includes 'but not both of them' because disjunction (Sie^euy^evov) is exclusive in Stoic logic (e.g., Cicero, Topica 14, 56-57) (Reinhardt, 2003; Bochenski, 1963). This is another difference from modern logic, in which disjunction is inclusive.
And MTP (Modus Tollendo Ponens) is the last indemonstrable (see Diogenes Laertius, Vi-tae Philosophorum 7, 81) (Marcovich, 1999).
MTP:
First premise: Either p or q, but not both of them
Second premise: Not-p Conclusion: q
Obviously, disjunction keeps being exclusive in MTP.
Regarding the themata, only the descriptions of the first one (Pseudo-Apuleius, De Interpreta-tione 209, 12-14) (Thomas, 1970) and two versions of the third one (respectively, Alexander of Aphrodisias, Aristotelis Analyticorum Priorum 278, 12-14) (Wallies, 1883); Simplicius, In Aristotelis De Caelo 273, 2-4 (Heiberg, 1894)) are known. There are discussions about the problem of the themata (e.g., Mignucci, 1993). Nevertheless, the rebuilding Bobzien (1996) made will be assumed here. This is because that reconstruction can suffice to make a point of this paper. Bob-zien's (1996, pp. 152-153) rebuilding is as follows ('IT, 'T2', 'T3', and T4' refer, respectively, to the first, second, third, and fourth themata; they are not axioms, and the reason for that is explained below):
T1: If [if (p and q) then r] then [if (p and not-r) then not-q]
T2: If {[if (p and q) then r] and [if (r and p) then s]} then [if (p and q) then s] T3: If {[if (p and q) then r] and [if (r and s) then t]} then [if (p and q and s) then t] T4: If {[if (p and q) then r] and [if (r and p and s) then t]} then [if (p and q and s) then t] As said, Bobzien (1996) only assumes one of the two versions of the third thema (that in Simplicius, In Aristotelis De Caelo 273, 2-4). On the other hand, T1, T2, T3, and T4 are expressed above in a simplified way. For example, Bobzien (1996) also includes an 'expanded version' of T1. In that version, the first conditional, that is, (1), does not have two conjuncts in its first clause. It can have more conjuncts.
(1) If (p and q) then r
In addition, it is obvious that in T2, conditional (2) can have other forms such as (3) and (4).
(2) If (r and p), then s
(3) If (r and q) then s
(4) If (r and p and q) then s
Likewise, s can be replaced in T3 with a set of propositions such as (5).
(5) s:,...,sn
And these two circumstances can happen in T4 too. On the one hand, (6) could be substituted with (7) or (8).
(6) If (r and p and s) then t
(7) If (r and q and s) then t
(8) If (r and p and q and s) then t
On the other hand, s could be replaced with (5) in T4 as well.
Furthermore, although Bobzien (1996) does not express the themata as conditionals, but resorting to premises and conclusions, it is justified to express them as above. It seems that the Stoics admitted conditionalization, that is, the process by means of which the premises of an argument can be deemed as the first clause of a conditional, and the conclusion as the second clause of that very conditional (e.g., Sextus Empiricus, Pyrrhoneae Hypotyposes 2, 137) (Mau, 2011; O'Toole & Jennings, 2004). This does not make T1, T2, T3, and T4 axioms. The reason is that, as indicated, Stoic logic is not modern logic. So, the conditionals suppose relations between their antecedents and consequents which are not required in modern logic.
Thus, the way the themata worked was akin to this one:
Given an argument such as the following (which is taken from Bobzien, 1996, p. 153):
First premise: p
Second premise: Not-q
Conclusion: Not-(if p then q)
That argument can be transformed into MPP by virtue of T1.
The theory of mental models shows that all these components of Stoic logic appear to be natural for the human mind and easy to understand. This can explain the basic character of those components in Stoic philosophy. However, before describing the account of the indemon-strables and the themata that can be given from
the theory of mental models, it is necessary to comment on some theses of this last theory.
The Theory of Mental Models as a Dual Process Theory
The theory of mental models proposes several explanations of human reasoning (e.g., Khemla-ni, Byrne, & Johnson-Laird, 2018). It is important to note that many of those theses move the theory away from modern logic (see also, e.g., Johnson-Laird, 2010). Nevertheless, the central thesis of the theory of mental models to make a point of this paper is that their proponents deem it a dual-process theory (see also, e.g., Johnson-Laird, Khemlani, & Goodwin, 2015). A dual-process theory (e.g., Evans, 2008) is a theory distinguishing two systems in the human mind. Those systems are usually named 'System 1' and 'System 2'. System 1 refers to intuitive processes. When using System 1, people do not spend much time drawing conclusions. On the other hand, when mental processes are more reflexive, individuals resort to System 2. In that case, they spend time and think in a more logical way. There are several dual-process theories (see also, e.g., Evans, 2009), but the manner this applies to the theory of mental models is explained below.
The theory of mental models claims that 'sentential connectives' lead people to consider the possibilities representing the situations that can be true for those connectives and their propositions (see also, e.g., Johnson-Laird & Ragni,
2019). If Stoic logic is addressed, the relevant sentential connectives are the conditional, exclusive disjunction, and conjunction. Given a conditional such as (9),
(9) If p, then q
Its possibilities are in (10).
(10) 0(p & q) & 0(not-p & q) & 0(not-p & not-q)
(10) expresses a 'conjunction of possibilities' (see also, e.g., Espino, Byrne, & Johnson-Laird,
2020). '0' stands for possibility. Nonetheless, it
does not work as the operator of the possibility in normal modal logics. In normal modal logics, (11) does not follow from (9).
(11) Op
This is because normal modal logics understand the conditional as Philo of Megara does. Thereby, (9) is false only when, as indicated, 'the antecedent is true and the consequent false'. Accordingly, if p is false, (9) is true even if p is, in addition, impossible. This is different in the theory of mental models. As (10) shows, (12) can be deduced from (9) (see, e.g., Espino et al., 2020).
(12) 0(p & q)
As far as a sentence such as (13) is concerned,
(13) Either p or q, but not both of them
It also has a conjunction of possibilities. It is (14) (see also, e.g., Khemlani, Hinterecker, & Johnson-Laird, 2017).
(14) 0(p & not-q) & 0(not-p & q)
The theory of mental models is different from logic in this case too. In normal modal logics, the inference of (11) from (13) is not allowed either. The reason is obvious. (13) can be true even if (11) is false, that is, even if p is not possible. If the other disjunct, that is, q, is true, that already makes (13) true. However, as (14) reveals, one of the possibilities that (13) enables to deduce in the theory of mental models is (15) (see, e.g., Khemlani et al., 2017).
(15) 0(p & not-q)
Lastly, conjunction is easy to capture in the theory of mental models. Conjunction such as (16) only expresses one possibility: (12).
(16) p and q
In this case, (12) cannot be really denominated 'possibility'. Given that there is only one possibility, (12) is a fact (see, e.g., Khemlani et al., 2017).
However, the most interesting point of the theory for this paper is that, as said, it is a dual-process theory. This means that the possibilities indicated for the conditional and exclusive disjunction are not always identified. To note all of them, it is necessary to use System 2. If people only resort to System 1, they can consider just
what is true, ignoring what is false in the conjunctions of possibilities (see also, e.g., Byrne & Johnson-Laird, 2020).
Thus, System 1 only allows being aware of (12) in the case of the conditional. The other two possibilities in (10) are not taken into account because they refer to situations with p being false. So, they cannot be represented with just System 1 (see also, e.g., Johnson-Laird, 2012).
Regarding exclusive disjunction, what System 1 does not consider is the false disjuncts. Thereby, the resulting conjunction of possibilities is (17) (see also, e.g., Quelhas, Rasga, & Johnson-Laird, 2019).
(17) Op & Oq
In the first possibility in (17), q is missing. The reason is that it is false, and System 1 does not pay attention to falsity. The same can be said with regard to p in the second possibility.
This framework makes it possible to account for why the Stoics deemed the indemonstrables and the themata as essential components in their logic. In fact, that account has already been partly given in the literature. The next section shows this.
The Theory of Mental Models, the Indemonstrables, the First Thema, and the Version of the Third Thema by Alexander of Aphrodisias
Actually, there are two versions of the theory of mental models. The description in the previous section corresponds to the second one, that is, the most updated version (for the first one, see, e.g., Oakhill & Garnham, 1996). This is important because, in the literature analyses of the indemonstrables, the first thema, and Alexander's version of the third thema based on the theory are to be found (for the indemonstrables, see Lopez-Astorga, 2017; for the first thema and Alexander's version of the third one, see Lopez-Astorga, 2016). Nevertheless, those analyses have been made from the initial version.
But, as far as the aims of the present paper are
concerned, this does not have significant relevance. The main difference between the two versions is just in the relation between the possibilities. The original version proposes that the possibilities are linked by means of disjunctions. In the second version, the links are by means of conjunctions since, as indicated, the proponents use the expression 'conjunction of possibilities' (e.g., Khemlani et al., 2018). This has no influence on the accounts of Stoic logic given from the theory of mental models. To remove any doubt in this way, this section will describe the explanations in the literature based on the initial version of the theory again, but using the machinery and terminology of the second version.
Regarding the indemonstrables, it is easy to note why they were deemed as basic schemata in Stoicism. The explanation resorting to the initial version of the theory of mental models for the five indemonstrables can be found in Lopez-Astorga (2017). The following accounts are based on that explanation. However, as said, the present paper uses the updated version of the theory.
MPP is not hard because it only requires System 1. If the only possibility for (9) is (12), it is evident that in a scenario with p, q needs to be as well.
One might think that MTT is different. Given that its second premise is not-q, one might suppose that all of the possibilities corresponding to the first premise, that is, to a sentence such as (9), have to be taken into account. As pointed out, those possibilities are (10). Nevertheless, in particular, the necessary possibility is the third one in (10), that is, (18).
(18) 0(not-p & not-q)
Possibility (18) is the necessary one because not-q appears in it. Thus, it reveals that not-q can be the case only if not-p is also the case. In fact, this is the usual argumentation from the theory of mental models to explain why MTT is often more difficult than MPP for people: individuals need to use System 2 and detect (18) (see, e.g., Byrne & Johnson-Laird, 2009).
Nonetheless, it is enough to remember Chry-sippus' thesis with regard to the case in which a conditional can be true: the denial of the consequent has to fight the antecedent. This means that when it is known that (9) is true, it is known that
(19) is true too.
(19) If not-q then not-p
Accordingly, given (19) and only resorting to System 1, the possibility identified is, again, (18). And, as indicated, (18) shows that the only circumstance in which not-q can happen is when not-p occurs as well.
The account for MPT I is also easy. The first premise presents a negated conjunction. The only possibility for conjunction is (12). However, in MPT I, the conjunction is negated. So, if the second premise is p, q needs to be false. This can be underlain to a greater extent with the possibilities that the theory of mental models assigns to sentential connectives when negated. When a proposition is negated, its possibilities are those missing in the set of that very proposition when affirmed. In other words, when a sentential connective is negated, its conjunction of possibilities matches the complement of the possibilities of that connective when it is not negated (e.g., Khemlani, Orenes, & Johnson-Laird, 2014). Therefore, the possibilities of a sentence such as
(20) are (21).
(20) Not-(p & q)
(21) 0(p & not-q) & 0(not-p & q) & 0(not-p & not-q)
In (21), p, the second premise, is true just in the first possibility, that is, in (15). Furthermore, in that possibility, not-q is the case.
The first premise of MPT II is an exclusive disjunction. Hence, if only System 1 is used, the conjunction of possibilities is (17). Given that the second premise is p, the possibility of q is eliminated. Thus, because the disjunction is exclusive, the second premise also reveals that q cannot happen. This leads to not-q, that is, to what corresponds to the first possibility of the conjunction of possibilities of exclusive disjunction when System 2 works. In other words, that leads to
(15).
Finally, MTP does not require a lot of effort either. It needs to use System 2, as the first premise is again a sentence such as (13). As explained, in the case of exclusive disjunction, System 1 only allows identifying (17). Nevertheless, the second premise of MTP is the negation of one of the disjuncts, which implies considering the possibilities System 2 can deploy, that is, (14). In this way, from the first possibility in (14), that is, (15), it is possible to conclude that, if not-q is true, p must happen.
However, although System 2 leads this process, it is not necessarily hard. The first premise is an exclusive disjunction. Accordingly, it is evident that one of its two disjuncts has to be true. If the information the second premise provides is that one of the disjuncts is false, the only scenario that can be thought is that the other disjunct is the case. Thereby, the possibilities of System 2 are displayed without a great cognitive analysis.
As far as the themata are concerned, as mentioned, the first one and a version of the third one (that of Alexander of Aphrodisias, Aristotelis Analyticorum Priorum 278, 12-14) (Wallies, 1883) have already been dealt with from the theory of mental models. That has been done in Lopez-Astorga (2016). As in the case of the in-demonstrables, the analysis was based on the old version of the theory. That analysis is reproduced again then. Nonetheless, as also in the previous case, the accounts here follow the updated version of the theory of mental models.
The analysis in the literature also resorted to the Stoic idea of conditionalization. Therefore, it presented the first thema and Alexander's version of the third thema as conditionals too (Lopez-Astorga, 2016). Starting with T1, its first conditional is (1). Compound conditionals of this kind, with a conjunction in the antecedent, were not a problem in Stoic logic (e.g., Bobzien, 1996). Hence, they can be deemed as any other conditional. That allows making two points. First, System 1 leads to the possibility in which both clauses (the if-clause and the then-clause)
are true, that is, to (22).
(22) 0[(p & q) & r]
Second, (1) is a Stoic conditional. So, it also enables to detect the third possibility for conditionals in System 2 without difficulties. In this case, that possibility would be (23).
(23) 0[not-(p & q) & not-r]
In this way, individuals know that not-r can be the case only if (20) is. Nonetheless, not-r is not the only component of the antecedent of the second conditional in T1, which is (24).
(24) If (p and not-r), then not-q
The antecedent of (24) includes p too. Accordingly, as explained for MPT I, (20), which is in (23), and p allow drawing not-q (remember that conjunction is truth-functional in Stoic logic; see, e.g., Bobzien, 1996). Therefore, as indicated in T1, given p and not-r, the conclusion is not-q.
As also mentioned, the version of the third thema that has been reviewed under the theory of mental models is not that Bobzien (1996) assumed, but the one coming from Alexander of Aphrodisias. If it is conditionalized, that version can be expressed as T3A.
T3A: If {[if (p and q) then r] and (if s then q)} then [if (p and s) then r]
Only using System 1, the possibility of the first conditional in T3A is (22). On the other hand, one of the second conditional, that is, (25), is (26).
(25) If s, then q
(26) 0(s & q)
Possibility (26) reveals that, if s happens, q needs to occur as well. So, if the situation is that both p and s happen, which is what the antecedent of the last conditional, that is, (27), provides,
(27) If (p and s) then r
There is no doubt that q is also present. But, as (22) points out, if both p and q occur, r is the case too, which leads to the consequent of (27).
Nonetheless, the theory of mental models not only can explain why T1 and T3A were essential rules in the Stoic system. The theory can also show that T2, T3, and T4 have structures that are easy to capture. The following section is devoted
to this issue.
T2, T3, and T4 from the Theory of Mental Models
System 1 also allows understanding T2 without difficulties. The possibility of its first conditional, which matches (1), is again (22). Likewise, following System 1 as well, the possibility of the second conditional, that is, (2), is (28).
(28) 0[(r & p) & s]
This last possibility reveals that whenever r and p are true, s is also true. Thus, by virtue of (22), if p and q are the case, r is the case too. However, by virtue of (28), if p and r happen, s happens too. Hence, what the last conditional in T2 expresses, that is, (29), is correct.
(29) If (p and q) then s
The case of T3 is very similar: it can also be understood by considering only System 1. The first conditional is once again (1), which leads to possibility (22). The second conditional, that is, (30), is linked, if only System 1 is taken into account, to (31).
(30) If (r and s) then t
(31) 0[(r & s) & t]
This means that it is not possible the conjunction of r and s without t. Accordingly, the result is evident. Given a scenario in which, as in the antecedent of the third conditional, that is, (32), propositions p, q, and s occur,
(32) If (p and q and s) then t
If, as (22) shows, p and q are not possible without r, then p, q, and s necessarily imply t. This is because t must happen when r and s occur, and p and q cause r to be true.
Lastly, the first conditional in T4 continues to be (1), which means that, following System 1, its possibility is in this case (22) as well. On the other hand, if the only system working is System 1, the second conditional, that is, (6), is linked to possibility (33).
(33) 0[(r & p & s) & t]
Possibility (33) expresses that when r, p, and s occur, t must also happen. Hence, when, as in the
antecedent of the last conditional, that is, (32), p, q and s are true, t is true too. As said in previous cases, p and q lead to r. If, in addition, s happens, t happens as well. The reason is that p along with r and s lead to t.
Therefore, T2, T3, and T4 are also, according to the theory of mental models, easy rules for the human mind. That can explain their role in Stoic philosophy. Nevertheless, the links between Stoic logic and the theory of mental models can be seen even clearer if another argument is reviewed. It is an argument that is sound in modern propositional calculus. However, it is not a syllogism in Stoic logic.
A Sound Argument in Modern Logic that is not a Syllogism
The argument is the following: First premise: If not-(p and q), then r Second premise: Not-p Conclusion: r
This argument (presented in Bobzien, 1996, p. 174) is correct in modern prepositional logic: the second premise makes the antecedent of the first premise true. If p is not true, (20) is true. Therefore, r also has to be true. Otherwise, the situation would match the only case in which, following the material view, that is, Philo's view, the conditional does not hold. As indicated, that is the case in which 'the antecedent is true and the consequent false'.
Nevertheless, according to Bobzien (1996), the argument is not a syllogism in Stoic logic. It cannot be transformed into an indemonstrable by means of the themata. Nevertheless, this point brings Stoic logic even closer to the theory of mental models. This is because the argument is difficult under this last theory. If only System 1 is considered, the possibility of the first premise is (34).
(34) 0[not-(p & q) & r] Possibility (34) establishes that whenever (20) is true, r happens. Nonetheless, the problem with the argument is that unlike MPT I, its second
premise denies one of the conjuncts of the negated conjunction. Hence, the second premise does not make the truth value of the other conjunct evident. In this way, to note that the argument is correct, it is necessary to deploy all of the possibilities the theory of mental models attributes to negated conjunctions; that is, it is necessary to use System 2 and transform (34) into (35).
(35) 0[(not-p & q) & r] & 0[(p & not-q) & r] & 0 [(not-p & not-q) & r]
Furthermore, the additional difficulty is that even (35) is not the complete result of processing the first premise of the argument by means of System 2. The combinations corresponding to the situation in which the antecedent does not occur would have to be displayed too. Thus, all the possibilities related to the first premise would be those in (36).
(36) 0[(not-p & q) & r] & 0[(p & not-q) & r] & 0 [(not-p & not-q) & r] & 0[(p & q) & r] & 0[(p & q) & not-r]
Actually, (35) would suffice to note that, if not-p is the case, r is the case as well, whether or not q is the case. In (35), q is the case in the first possibility, that is, in (37).
(37) 0[(not-p & q) & r]
And it is not in the third one, that is, in (38).
(38) 0 [(not-p & not-q) & r]
However, both in (37) and in (38) r also happens. In spite of this, to come to (35) already requires a detailed and extensive analysis, that is, already requires to resort to system 2. So, it is difficult to infer the conclusion.
Conclusion
The theory of mental models not only can, as the literature shows, offer an account of why the indemonstrables, T1, and T3A were considered as basic components in Stoic philosophy. The theory can also give similar explanations for T3, T2, and T4. Likewise, its framework allows understanding why certain arguments, as in the previous section, were not accepted as syllogisms in Stoicism.
The fact that the theory of mental models is a dual process theory, and that, therefore, distinguishes the processes linked to System 1 from the processes corresponding to System 2, seems to be the key in this way. To assume the inde-monstrables and the themata, only System 1 is necessary in many cases. In the cases in which System 2 should be used, individuals can come to this last system in a direct and easy way. Furthermore, regarding arguments such as the last one addressed in the present paper, which are not syllogisms, the circumstances are different: the analysis of possibilities needs to be exhaustive, and System 2 is absolutely required. Thus, the inferences are deemed as difficult arguments.
All of this leads to several conclusions compatible with those of other works relating to Stoic logic and the theory of mental models (e.g., Lopez-Astorga, 2016, 2017). First, the Stoic framework seems to be more similar to the theory of mental models than to modern proposition-al calculus. This is important since the theory of mental models is a theory about the manner people reason. So, the conclusion appears to be obvious: if the theory of mental models is a correct proposal, Stoic logic is closer to the way individuals reason than modern propositional logic.
This last point is interesting, as it leads to research to what extent Stoic logic is also able to explain experimental results that the theory of mental models accounts for. An example in this regard can be the results in different reasoning tasks that are compatible with the predictions of the theory of mental models (see, e.g., any of the works supporting the theory cited here). There are already studies in this direction (e.g., Lopez-Astorga, 2021). However, given that the results published in cognitive science and psychology are significant, maybe there is much work to do.
Acknowledgements
PIA Cognitive Sciences, Center for Research on Cognitive Sciences, Institute of Humanistic Studies. Fondecyt Continuity Fund for Senior
Researchers, code FCSEN2102, University of Talca (PIA Ciencias Cognitivas, Centro de Investigación en Ciencias Cognitivas, Instituto de Estudios Humanísticos, Universidad de Talca. Fondo Fondecyt de Continuidad para Investigadores Senior, código FCSEN2102, Universidad de Talca).
References
Bobzien, S. (1996). Stoic syllogistic. In C. C. W. Taylor (Ed.), Oxford Studies in Ancient Philosophy (pp. 133-192). Oxford, UK: Clarendon Press. Bochenski, I. M. (1963). Ancient formal logic. Amsterdam, The Netherlands: North-Holland.
Byrne, R. M. J., & Johnson-Laird, P. N. (2009). "If' and the problems of conditional reasoning. Trends in Cognitive Science, 13(7), 282-287. https://doi.org/10.1016-/j.tics.2009.04.003 Byrne, R. M. J., & Johnson-Laird, P. N. (2020). If and or: Real and counterfactual possibilities in their truth and probability. Journal of Experimental Psychology: Learning, Memory, and Cognition, 46(4), 760-780. https://doi.org/10.1037-/xlm0000756 De Lay, P. (1984). Galeni De Placitis Hippocra-tis et Platonis (On the Doctrines of Hippocrates and Plato by Galen, in Latin). Berlin, Germany: Akademie-Verlag.
Espino, O., Byrne, R. M. J., & Johnson-Laird, P. N. (2020). Possibilities and the parallel meanings of factual and counterfactual conditionals. Memory & Cognition, 48, 1263-1280. https://doi.org/10.3758/s-13421-020-01040-6 Evans, J. St. B. T. (2008). Dual-processing accounts of reasoning, judgment, and social cognition. Annual Review of Psychology, 59(1), 255-278. https://doi.-
org/10.1146/annurev.psych.59.103 006. 093629
Evans, J. St. B. T. (2009). How many dual-process theories do we need? One, two or many? In J. St. B. T. Evans, & K. Frankish (Eds.), In two minds: Dual processes and beyond (pp. 33-54). Oxford, UK: Oxford University Press. http://doi.org/10.1093/acprof:oso/9780 199230167.003.0002 Gentzen, G. (1934). Untersuchungen über das logische Schließen I (Studies on logical reasoning I., in German). Mathematische Zeitschrift (Mathematical Journal, in German), 39(2), 176-210. https://-doi.org/10.1007/BF01201353 Gentzen, G. (1935). Untersuchungen über das logische Schließen II (Studies on logical reasoning II, in German). Mathematische Zeitschrift (Mathematical Journal, in German), 39(3), 405-431. https://doi.org/10.1007/BF01201363 Heiberg, J. L. (1894). Simplicii in Aristotelis De Caelo Commentaria (Simplicius in Aristotle's Commentaries on Heaven, in Latin). Berlin, Germany: Reimer. Johnson-Laird, P. N. & Ragni, M. (2019). Possibilities as the foundation of reasoning. Cognition, 193. https://doi.org/10.1016-/j.cognition.2019.04.019 Johnson-Laird, P. N. (2010). Against logical form. Psychologica Belgica, 50(3/4), 193-221. http://doi.org/10.5334/pb-50-3-4-193
Johnson-Laird, P. N. (2012). Inference with mental models. In K. J. Holyoak, & R. G. Morrison (Eds.), The oxford handbook of thinking and reasoning (pp. 134-145). New York, NY: Oxford University Press. https://doi.org/10.10-93/oxfordhb/9780199734689.013.0009 Johnson-Laird, P. N., Khemlani, S., & Goodwin, G. P. (2015). Logic, probability, and human reasoning. Trends in Cognitive
Sciences, 19(4), 201-214. https://doi.-org/10.1016/j.tics.2015.02.006 Khemlani, S., Byrne, R. M. J., & Johnson-Laird, P. N. (2018). Facts and possibilities: A model-based theory of sentential reasoning. Cognitive Science, 42(6), 18871924. https://doi.org/10.1111/cogs. 12634
Khemlani, S., Hinterecker, T., & Johnson-Laird, P. N. (2017). The provenance of modal inference. In G. Gunzelmann, A. Howes, T. Tenbrink, & E. J. Davelaar (Eds.), Computational foundations of cognition (pp. 663-668). Austin, TX: Cognitive Science Society. Khemlani, S., Orenes, I., & Johnson-Laird, P. N. (2014). The negation of conjunctions, conditionals, and disjunctions. Acta Psychologica, 151, 1-7. https://doi.org/-10.1016/j.actpsy.2014.05.004 López-Astorga, M. (2015). Chrysippus' inde-monstrables and mental logic. Croatian Journal of Philosophy, 15(43), 1-15. López-Astorga, M. (2016). The first rule of Stoic logic and its relationship with the inde-monstrables. Tópicos. Revista de Filosofia, 50, 9-23. https://doi.org/10.215-55/top.v0i50.724 López-Astorga, M. (2017). Chrysippus' inde-monstrables and the semantic mental models. Eidos, 26, 302-325. https://-doi.org/10.14482/eidos.26.8437 López-Astorga, M. (2021). Execution in selection task depends on Chrysippus' criterion for the conditional. Schole, 15(2), 501-512. https://doi.org/10.25205/19-95-4328-2021-15-2-501-512 Marcovich, M. (1999). Diogenes Laertius. Vitae Philosophorum (Diogenes Laertius. Philosophers' Lives, in Latin). Stuttgart & Leipzig, Germany: B. G. Teubner. Mates, B. (1953). Stoic logic. Berkeley and Los Angeles, CA: University of California Press.
Mau, J. (2011). Sexti Empirici Opera (Works by Sextus Empiricus, in Latin). Berlin, Germany and New York, NY: De Gruyter.
Mignucci, M. (1993). The Stoic themata. In K. Doering, & T. Ebert (Eds.), Dialektiker und Stoiker: Zur Logik der Stoa und ihrer Vorläufer (Dialectics and Stoics: On the logic of the Stoa and their predecessors, in German) (pp. 217-238). Stuttgart, Germany: Franz Steiner.
O'Toole, R. R., & Jennings, R. E. (2004). The Megarians and the Stoics. In D. M. Gabbay, & J. Woods (Eds.), Handbook of the history of logic, Volume 1. Greek, Indian and Arabic logic (pp. 397-522). Amsterdam, The Netherlands: Elsevier. https://doi.org/10.1016/S1874-5857-(04)80008-6
Oakhill, J., & Garnham, A. (Eds.) (1996). Mental models in cognitive science. Essays in honour of Phil Johnson-Laird. Hove, UK: Psychology Press.
Orenes, I., & Johnson-Laird, P. N. (2012). Logic, models, and paradoxical inferences. Mind & Language, 27(4), 357-377. https://doi.org/10.1111/j.1468-0017.-2012.01448.x
Quelhas, A. C., Rasga, C., & Johnson-Laird, P. N. (2019). The analytic truth and falsity of disjunctions. Cognitive Science, 43(9). https://doi.org/10.1111/cogs. 12739
Reinhardt, T. (2003). Marcus Tullius Cicero. Topica. New York, NY: Oxford University Press.
Thomas, P. (1970). Apulei Platonici Madauren-sis Opera quae supersunt (Works by Apuleius of Madaur, in Latin). Stuttgart, Germany: B. G. Teubner.
Wallies, M. (1883). Alexander of Aphrodisias. Aristotelis Analyticorum Priorum (Prior Analytics by Aristotle, in Latin). Berlin, Germany: Reimer.
