At the AAL conference in Melbourne in July, one of the questions that I was asked was about medieval discussions of ex falso/impossibili/contradictione quidlibet sequiter (and it’s correlate, ad necessarium quodlibet sequiter).
The legitimacy of these inferences was significantly discussed during the 12th century, with the followers of Adam de Petit-Pont (called the Adamites or the Parvipontanians) accepting these principles, but others, in particular Abaelard and the so-called Nominales rejecting it. My point of contact with these discussions is via the Tractatus Emmeranus de Positio Impossibilis (1st half of the 13th C, edited by de Rijk in 1974), a treatise on the genre of obligationes which takes not a false statement but an impossible one as the positum. The author points out that the only way this genre can be sustained as a useful one is if the types of inferences used are restricted; in particular, the author singles out the “Adamite” thesis by name in order to identify it as unacceptable in the context of impossible positio:
And we should note that in this question everything does not follow from an impossible obligation. Thus, in this question one must not concede the consequence of the Adamites—namely that from the impossible anything follows [p. 218]
The principle is also attributed to Adam by John of Salisbury in the Metalogicon.
A couple of good modern discussions of this thesis can be found in Iwakuma 1993 (covering the 12th century), Spruyt 1993 (on the 13th century), and d’Ors 1993 (for the 14th century). In this post, I focus on Iwakuma’s paper and the 12th century.
The best-known 12th-century sources for the ex impossibili thesis are the Ars Meliduna and Alexander Neckam’s De Naturis Rerum (p. 125). The Ars Meliduna is written by a Melidunensis rather than a Parvipontanus, but the Parvipontani thesis is discussed in book IV, chapters 37-39. Neckam, on the other hand, accepts the thesis in a discussion in chapter 173, according to Iwakuma’s reinterpretation. In this section, Neckam starts off saying “Miror etiam quosdam damnare opinionem dicentium ex impossibili per se quodcumque sequi enuntiabile“, and gives six arguments. The first is a direct proof of ex impossibili, but Iwakuma says the remaining five “appear at first sight to have nothing to do with the thesis” (p. 126). The second argument is “one of the earliest presentations of the Liar’s paradox” (p. 126), and the other arguments “are very similar to the second in their structure” (p. 126). Iwakuma explains the relevance of the Liar paradox to questions of ex impossibili as follows:
Alexander’s second argument amounts to show that such an impossible antecedent is followed by a contradiction, namely Socrates says something both true and false. And once the contradiction holds, then it is easy to construct an argument to show that it is followed by any consequent, following the steps of the first argument…In just the same manner, one can interprete the other four arguments as proofs of the thesis. For in every argument of the four the antecedent of the first premise can be understood as impossible, and the conclusion is a contradiction (pp. 127-128).
But Iwakuma’s main aim in the paper is not discussion of Neckham and Ars Meliduna, but rather “to introduce two hitherto unknown texts and to comment on them, as well as on those already known, relevant to the thesis ex impossibili quidlibet sequiter” (p. 123). The two texts are MS Avranches 224, f. 3, from the late 12th century, and MS Munich clm 29520(2, also from the late 12th century.
The Avranches text discusses two counterarguments to the thesis, then gives two proofs of it (and a proof of the corollary ad necessarium) as well as responses to the purported counterarguments. Unfortunately, I neglected to scan the appendix in which this text is edited, and since Iwakuma doesn’t say anything further in the body of his article I can’t add anything more. (Guess I need to get that book via ILL again…or someday just purchase myself a copy!)
The Munich MS is incomplete, lacking the beginning so that it “leaves us uncertain as to what it discusses” (p. 130), though Iwakuma conjectures that it was probably written by a Parvipontanus on the ex impossibili thesis. As evidence, he points to a discussion of solutions to a problem set by Abelard wherein the solutions of the Nominales and the Melidunenses are rejected, but not the Parvipontani solution. In the same section, two more theses (ex inmodali sequitur modalis, et e converso and ex explicita sequitur inplicita, et e converso) are admitted; both of these are denied by the Porretani and Montani schools, but are consequences of ex impossibili, for:
a modal/immodal antecedent, if being impossible, is followed by any consequent, including immodal/modal ones. Similar considerations apply to inplicita/explicita propositions (p. 132).
If Iwakuma is right in his analysis and identification, then “this is one of the few valuable pieces of evidence on the Parvipontani‘s thesis from their own pens” (p. 133).
Anonymous. 2001. “The Emmeran Treatise on Impossible Positio“, in M. Yrjönsuuri, Medieval Formal Logic (Kluwer Academic Publishers): 217-223.
de Rijk, L. M. 1974. “Some Thirteenth Century Tracts on the Game of Obligation”, Vivarium 12: 94-123.
d’Ors, Angel. 1993. “Ex Impossibili Quodlibet Sequitur (John Buridan)”, in K. Jacobi, Argumentationstheorie (Brill): 195-212.
Iwakuma, Y. 1993. “Parvipontani‘s Thesis Ex Impossibili Quidlibet Sequitur: Comments on the Sources of the Thesis from the Twelfth Century”, in K. Jacobi, Argumentationstheorie (Brill): 123-133.
Spruyt, Joke. 1993. “Thirteenth-Century Positions on the Rule ‘Ex Impossibili Sequitur Quidlibet‘”, in K. Jacobi, Argumentationstheorie (Brill): 161-193.