Oh, look, it’s Thursday again! Time to write another medieval logic post. We’re still doing Aristotle in my intro class, so I haven’t any new interesting medieval tidbits from class prep to share. So I guess I’ll just return to the question of ex impossibili sequitur quidlibet in the 13th century; you can find part 1 of the discussion there. In this part we return to Spruyt 1993 and continue our discussion of views opposing ex impossibili, taking up the positions in the Sophistaria wrongly attributed to Walter Burley, Peter of Spain’s Syncategoremata, and Henry of Ghent’s Syncategoremata.
First, Spruyt notes that it isn’t actually clear whether the author of the Sophistaria was an opponent or proponent of the thesis, but she takes him to be an opponent since “he starts off his treatise with arguments in favor of the rule, and quite a number of Medieval authors who wish to defend a certain position often begin by presenting arguments to the contrary” (p. 168). These arguments include:
(1) The necessary follows from the impossible. But nothing is more contrary to the impossible than the necessary. So if even the necessary follows from the impossible, anything which is less contrary to the impossible will also follow (and everything is less contrary to the impossible than the necessary).
(2) An argument from the definition of valid argument (which Spruyt doesn’t further discuss).
(3) If you are a man and not a man, then you are not a man. If you are a man, then either you are a man or anything is true. Thus, if you are a man and not a man by disjunctive syllogism, anything is true.
The arguments against the position are especially interesting to me because they involve impossible positio. First, assume that nothing that follows from something is contrary to it (contra (1)). If anything followed from an impossibility, then in impossible positio, then “one can never give inadequate answers to an impossible positio” (p. 168). It follows then that nothing would be contrary to the impossible, “there would be no such thing as positing the impossible, or an impossible position, and this is false” (p. 168). That is, if there is nothing contrary to the impossible, then the impossible isn’t really impossible after all.
The next argument against “has even more to say about the notion of ‘impossible'” (p. 168). First, a distinction is made between things which are impossible per se and things which are impossible “in virtue of a natural situation”. Then, a conditional sentence is paired up with a corresponding syllogism in such a way that if the syllogism is not valid, then the conditional isn’t true. In a syllogism, one cannot get from one and the same assumption to the same conclusion both affirmed and denied, so in the corresponding conditional, this can’t happen either:
from a man being an ass follows a man being an animal and not the denial of a man being an animal at the same time. Thus, from the impossible man being an ass does not follow just anything (p. 169).
One consequence of this argument is that syllogisms with impossible premises must be unsound.
The third argument against is that if the rule were sound, it would have to be based on some topic, but it is not clear which topic would apply.
The distinction between things which are impossible per se and which are only naturally impossible comes up again when the author discusses his own position (which Spruyt says is not clear); it appears that he accepts the rule when the impossibilities involved are “impossible containing contradictory opposite statements”, i.e., ones that are impossible in themselves (p. 169). (This notion of gradations of impossibilities is not uncommon in 13th-century texts; William of Sherwood also makes similar distinctions.)
Next up: Peter of Spain’s views as in the Syncategoremata (dating c1230). His position is based on the fact that in his view, “if we wish to infer consequent from antecedent, there must also be some topical relationship involved” (p. 170): There needs to be more than just the relationship of consequence between the antecedent and the consequent. As Spruyt puts it, with this view in hand, “small wonder that he does not adhere to it”, that is, ex impossibili, for there will, in general, be no topical relation underpinning the consequence from an impossibility to an arbitrary sentence.
Henry of Ghent’s view in his Syncategoremata (written about 30 years after Peter’s) is similar to Peter’s. He takes the consequence relation between antecedent and consequent as being one of cause and effect, which limits the types of things that can follow from an impossibility, because, in general, an impossibility will not be the cause of an arbitrary other thing. In order to block the validity of ex impossibili following immediately from the standard definition of a true conditional, he modifies the definition so that the “conditional is true when the conditionalized truth of the antecedent posits the truth of the consequent” (p. 173) — something that cannot be done with an impossibility.
In part three, we’ll look at Spruyt’s discussion of 13th-century arguments in favor of the rule.
Medieval Logic & Semantics 