Institute of Humanistic Studies “Juan Ignacio Molina”, University of Talca, Chile
Diodorus Cronus and Philo of Megara presented criteria for identifying true conditionals. Diodorus’ criterion has been said to be a version of that of Philo requiring that the conditional is always true. However, in this paper, based on the mental models theory and its analysis of possibilities, I try to show that those two interpretations are very different and that they do not refer to the same combinations of possibilities. In my view, Philo’s account can be linked to the material interpretation of the conditional. Nevertheless, Diodorus’ explanation can be related to that very interpretation and, in addition, to three different combinations of possibilities, none of them being that corresponding to the material interpretation.
Keywords: conditional; Diodorus Cronus; mental models; Philo of Megara; possibilities
There was an interesting discussion in the fourth century B.C. That discussion, which is described by Sextus Empiricus in Adversus Mathematicos and in Pyrrhoneae Hypotyposes, refers to the appropriate interpretation of a particular proposition (??????): the conditional. Actually, Sextus comments on four interpretations of the conditional proposition, the one of Diodorus Cronus, the one of Philo of Megara, the one that can be attributed to Chrysippus of Soli, and a fourth account that does not appear to be related to any author or ancient source. However, the interpretations that will be considered in this paper are only the first one and the second one.
Those two accounts are interesting because it has been said that Diodorus only adds a requirement to Philo’s criterion. That requirement is that the conditional can never be false (Mates, 1953, pp. 44-46). Nevertheless, if both interpretations are reviewed based on a contemporary theory on reasoning, the mental models theory (e.g., Johnson-Laird, 2010, 2012; Johnson-Laird & Byrne, 2002; Khemlani, Orenes, & Johnson-Laird, 2012, 2014; Orenes & Johnson-Laird, 2012), it can be noted that they are very different accounts and that hence that idea is not correct. In particular, the mental models theory can show us that, while Philo’s criterion is clearly linked to the material interpretation of the conditional, i.e., to the interpretation assumed by standard logic or by calculi such as that of Gentzen (1935), Diodorus’ interpretation does not only refer to that account, but also to other three combinations of possibilities that are different from that of the material interpretation. As explained below, two of those combinations are taken into account by the mental models theory, in particular, by Johnson-Laird and Byrne (2002), and the third one, which a priori seems to be impossible in practice, could be the result of the technical definition proposed by Diodorus.
To show all of this, I will start with a brief comment of the Sextus Empiricus’ description of the two indicated interpretations. Secondly, I will expose the general theses of the mental models theory that need to be considered in order to achieve my goals. Finally, I will explain why, based on that theory, it can be thought that Diodorus’ criterion is very different from Philo’s explanation and that Diodorus’ account is related in fact to other three combinations of possibilities as well, and not only to that material. I begin commenting the information offered by Sextus.
Sextus Empiricus and the criteria on the conditional
As said, Sextus Empiricus addresses four interpretations, but in this paper only that of Diodorus Cronus and that of Philo of Megara are relevant. Although, as it is known, the relationship between them was a teacher-student relationship, it seems opportune to start with Philo’s criterion, since the one of Diodorus appears to be a response to Philo’s thesis.
According to Philo of Megara, a conditional only is false when its antecedent (?????????) is true and its consequent (?????) is false. Really, as explained by O’Toole and Jennings (2004), it is hard to know for sure what Sextus truly wants to mean with the words ‘false’ (??????) and ‘true’ (??????) because in occasions he does not use ??????, but ?????, which should be translated as ‘sound’. In any case, regardless of whether or not Sextus thought that ?????? and ????? were synonymous, what is important here is that Philo’s criterion appears to be the material interpretation of the conditional. Indeed, if a conditional can only be false when its antecedent is true and its consequent is false, this fact means that it is necessarily true in the other remaining cases, i.e., when:
-The antecedent is true and the consequent is also true.
-The antecedent is false and the consequent is true.
-The antecedent is false and the consequent is false too.
So, it can be stated that undoubtedly Philo’s interpretation is the interpretation of classical logic. As mentioned by O’Toole and Jennings (2004), most of the logicians seem to agree on this point, which can be checked paying attention to, for example, the arguments offered by Mates (1953) or Boche?ski (1963).
On the other hand, following Pyrrhoneae Hypotyposes 2.110 and Adversus Mathematicos 8.115, it can be stated that Diodorus’ criterion establishes that “a sound conditional is one which neither was capable nor is capable of having a true antecedent and a false consequent” (O’Toole & Jennings, 2004, p. 479). Based on this, Mates (1953) appears to propose the idea that Diodorus’ conditional is only a Philo’s conditional that is never false.
But Sextus Empiricus also presents examples of these criteria. Regardless of whether they are actual examples taken from Philo and Diodorus or they are simply created by Sextus, it is interesting to think about them. The example assigned to Philo’s account is as follows (see Pyrrhoneae Hypotyposes 2.110):
“If it is day, I converse” (O’Toole & Jennings, 2004, p. 480).
Following Philo’s criterion, this conditional would be true when it is day and I converse, when it is not day and I converse, and when it is not day and I do not converse. On the contrary, it would be false only when it is day and I do not converse…Full Text PDF