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ABSTRACT
Impossible antecedents entered the scene of medieval logic around the 1120s and soon started to dominate this scene, becoming one of the most debated issues from the second half of the twelfth century onwards. This article focuses on theories of counterpossibles from this period and aims to offer an overview of the different responses offered by twelfth-century logicians on whether everything, something, or nothing follows from an impossible statement. Rather than trying to historically reconstruct the positions of the different authors – an operation that may be premature given the insufficient notions we still have of the authorship and dating of sources – I aim to provide a provisional map of the main arguments that were advanced in favour of or against the ex impossibili quodlibet principle. The article also aims to show that the strategies developed by twelfth-century logical schools in their approach to counterpossibles survive in logical discussions of the same topic during the thirteenth and the early fourteenth century, especially in the syncategoremata and sophistaria traditions.
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Notes
1 For the former of these examples, see Williamson Citation2017, 202; an example similar to the second one is found in an anonymous treatise from the end of the 12th century, ms. Paris BNF lat. 3713, f. 87v.
2 Or at least, this reading was orthodox until the ‘hyperintensional revolution’ taking place in logic and metaphysics during the last two decades; cf. e.g. Nolan Citation2014, 149ff.; Berto and Jago Citation2019, 1–2.
3 See e.g. Williamson Citation2017; see also Lewis Citation1973 and Stalnaker Citation1968 for a canonical defence of vacuism.
4 Cf. in particular Berto et al. Citation2018 and the bibliography quoted there. For an overview of contemporary ideas on counterpossibles and a defence of non-vacuism, see Berto and Jago Citation2019. For recent studies on non-vacuist approaches in scientific reasoning, see e.g. Wilson Citation2021; McLoone, Grützner, and Stuart Citation2023.
5 See Boethius Citation1969, 250 (my translation and emphasis): ‘The necessity of a hypothetical proposition, and the nature of those propositions in which the relations of connection are linked to one another, requires a consequence, as when I say: “If Socrates sits, then [Socrates] is alive” it is neither necessary that he sits nor that he lives, but if he sits, it is necessary that he is alive. Similarly, when we say: “If the sun moves, necessarily it will reach the West” it analogously means that, if the sun moves, it will arrive to the West. The necessity of the proposition consists in the immutability of the consequence. Again, when we say: “If it is possible that a certain book is read, then it is possible that the reading arrives to its third line,” the necessity of the consequence is once again preserved, since if it is possible that a book is read, then necessarily it is also possible that the reading arrives up to the book’s third line. Hypothetical propositions are negated only by those propositions that destroy their entire substance. Now, the substance of hypothetical propositions lies in this, in the fact that their necessity of consequence has the capacity to always remain intact.’
6 See e.g. Peter of Spain Citation2014, 115. For another 13th-century mention of the principle, see Roger Bacon Citation2009, 82: ‘One should know, moreover, that a conditional proposition is false whose antecedent cannot be true without the consequent being true, e.g., “If it is a man, it is an animal”; a conditional proposition is false whose antecedent can be true without the consequent being true, as in “If it is an animal, it is a man”. Every conditional proposition that is true is necessary; every one that is false is impossible.’
7 See e.g. Burley Citation1980, 128–129; Buridan Citation2015, 67.
8 See for instance Paul of Venice Citation2002, 52:‘A sound consecution is one in which the opposite of the conclusion is incompatible with the premises.’
9 Alighieri (Citation1979), Monarchia II, v, 22. For the use of logical theories of consequences in Dante’s works, see Pelizzari (Citation2020).
10 See Abelard Citation1970, 285. On Abelard’s ‘discovery’ of the paradoxes of strict implication see Martin Citation2004, 183; Martin Citation2018, 344.
11 See De Rijk Citation1988, 169–212. The treatise is found in ms. Biblioteca Vaticana, lat. 7678, 73ra–82ra.
12 This treatise is included in ms. Paris, Bibliothèque Nationale de France, lat. 3713, ff. 86r–88v. I am very grateful to Yukio Iwakuma for sharing his preliminary transcription of the text with me. In the following folio of the same manuscript, another treatise is found on similar topics, which Iwakuma called Tractatus Parvipontanus II de enuntiabilibus. I will only briefly mention this second treatise in this article, at page 14. Apart from this one, all references are to the first of the two treatises. On the topic of counterpossibles we also have two other fragmented and poorly preserved logical texts, one included in ms. Paris, Bibliothèque Nationale de France, Lat. 2904, p. IVb and the other in ms. Leipzig, Universitätsbibliothek, Fragm. lat. 32, ff. 1–3. I would like to thank Chris Martin for bringing this last text to my attention.
13 This text is currently unedited, and preserved in ms. London, British Library, Royal 2D XXX, ff. 95ra–102vb.
14 The Ars is preserved in ms. Oxford, Bodleian Library, Digby 174, ff. 211ra–241rb. An analysis and some excerpts of it are edited in De Rijk Citation1967, 264–390 and in Iwakuma Citation1993.
15 For an edition of this short treatise, see Iwakuma Citation2013. On the Albricani’s views on the ex impossibili principle, see also the Introductiones Montane Maiores, where the anonymous author admits that in many cases true propositions correctly follow from false ones, and possible or impossible propositions follow from the impossible – while the opposite is never the case (Bos and Spruyt Citation2017, 100).
16 As Courtenay (Citation1993) pointed out, we have some textual evidence that associates the school with the view that everything follows from an impossible, which is referred to as the ‘opinio Nominalium’. This evidence comes from the aforementioned Tractatus Vaticanus, whose author identifies instead as one of the ‘Reales’ and a supporter of the idea that nothing follows from a falsehood. See De Rijk Citation1988, 206. I will return to this passage from Tractatus Vaticanus at the end of Section 4.
17 De Rijk Citation1988, 204 (my translation): ‘That conditional proposition is true in which the antecedent cannot be true without the consequent, while it is false when the antecedent can be true without the consequent. But in a conditional proposition in which an impossible antecedes and something else follows, [the antecedent] cannot be true without the consequent, because it cannot be true, and thus cannot be true without the consequent. Therefore, a conditional in which an impossible antecedes is said to be true. And thus, this is true: “If Socrates is a donkey, Socrates is a human or Socrates is a goat.” ’
18 See Spruyt Citation1993, 182: ‘Every conditional is true whose antecedent is not true without the consequent. But of any conditional in which the antecedent is impossible, the antecedent cannot be true without the consequent, because the antecedent cannot be true. Therefore, that conditional in which an impossible antecedes is true.’
19 De Rijk Citation1988, 204 (my translation): ‘Consequently it is asked about the rule that says that from an impossible everything follows. From which this follows: “If a human is a donkey, a human is a goat.” And that this rule is true can be demonstrated in the following way. This conditional is true: “If Socrates is a donkey, Socrates is a goat,” in which an impossible antecedes. Which is proved in this way: “If Socrates is a donkey, Socrates is a donkey or he is a goat.” And this argument (argumentatio) follows by means of the topic from a subjective part, or the topic from a disjunctive part, which is the same. Then this is deduced, “but Socrates is not a donkey,” which is true simpliciter. Thus, from the beginning: “If Socrates is a donkey, Socrates is a goat.” In a similar way, the same can be shown about anything else [de quodlibet alio], in this way: “If Socrates is a donkey, Socrates is a human”; because if Socrates is Socrates, Socrates is a donkey or a human; but Socrates is not a donkey; thus, he is a human. Therefore, from the beginning: “If Socrates is a donkey, Socrates is a human.” ’
20 Disjunction introduction (also called Addition) is the inferential rule which allows one to infer, from the truth of ‘P’, the truth of ‘P or Q’, for any Q. The author of Tractatus Vaticanus does not employ this as an independent inferential rule based on the meaning of the connective, but rather provides a topical warrant to ground its validity: the inferential move from (I.1) to (I.2) is presented as an instance of the locus ‘from a subjective part’. The same topical warrant for the rule may be seen in another late 12th-century source, the Tractatus Parvipontanus de enuntiabilibus, and also in works from later authors, such as Peter of Spain’s Syncategoreumata.
21 Disjunctive syllogism is an inferential rule according to which, from the pair of premises ‘P or Q’ and ‘not-P’, one is allowed to conclude that ‘Q’.
22 Martin further elaborated on the connection between proofs of ex impossibili quodlibet and the development of extensional disjunction in Martin Citation2022b, still unpublished.
23 Cf. ms. Paris 3713, f. 86v (my translation): ‘Having seen how from “Socrates is a donkey” it follows that “Socrates is a human,” it must be seen how from the same [antecedent] any enuntiabile follows. If Socrates is a donkey, Socrates is a human; thus, if Socrates is a donkey, Socrates is a human or any enuntiabile is true. But if Socrates is a donkey, Socrates is not a human; therefore, if Socrates is a donkey, any enuntiabile is true. To give confirmation to an argument of this sort, the following rule is given: any time that two [statements] follow from the same one under disjunction (sub disiunctione), if the contradictory of one [disjunct] follows from the antecedent, then the other [disjunct] follows from the same [antecedent].’
24 See ms. Paris 3713, f. 87r (my translation): ‘Having said how from that particular kind of enuntiabilia [i.e. statements that are per se impossible] all enuntiabilia follow, it remains now to say which is the kind of enuntiabilia that is entailed by all others. This is the necessary per se, according to this rule: every hypothetical conditional in which the consequent is necessary per se is true, just as every hypothetical conditional is true in which the antecedent is a per se impossible.’ This statement is followed by proofs of the principle necessarium ex quolibet sequitur.
25 Neckam Citation1863, 288–289 (translated in Martin Citation2018, 348): ‘I am amazed that some condemn the opinion of those saying that from what is per se impossible there follows any proposition (enuntiabile). This may be confirmed with many arguments or made clear with a few. For is it not the case that (1) if Socrates is a human being and Socrates is not a human being, then Socrates is a human being. But (2) if Socrates is a human being, then Socrates is a human being or a stone; therefore (3) if Socrates is a human being and Socrates is not a human being, then Socrates is a human being or a stone, but (4) if Socrates is a human being and Socrates is not a human being, then Socrates is not a human being; therefore (5) if Socrates is a human being and Socrates is not a human being, then Socrates is a stone.’
26 Under the assumption that, for Neckam, the two sentences ‘Socrates is not a human’ and ‘Not: Socrates is a human’ are equivalent.
27 These impossibilities are ‘natural’ or ‘metaphysical’ inasmuch as, according to the essentialist views shared by logicians of this time, the nature of, say, humans is incompatible with being a donkey, or with irrationality. There are thus some metaphysical laws (grounded on the natures of things) that make statements like ‘A human is a donkey’ impossible.
28 Neckam seems to also consider paradoxical statements like ‘Socrates says that he is lying’ as instances of per se impossibility, and he offers another set of arguments showing that also from propositions of this sort everything would follow. See the arguments in Neckam Citation1863, 289 ff.
29 As examples of 13th-century sources, see for instance the discussion on the ex impossibili quodlibet principle discussed in the Sophistaria attributed to Burley, analysed and edited in Spruyt Citation1993. As Spruyt mentions here (Citation1993, 169), the important limitation on the validity of the ex impossibili principle is also used in Peter of Spain’s discussion of counterpossibles.
30 See ms. Paris 3713, f. 86r. The distinction between per se and per accidens impossibilities is extensively discussed in the anonymous Tractatus Parvipontanus I de enuntiabilibus. Regretfully, the text is not edited and has not received any critical analysis so far, but it is particularly valuable for scholars interested in the earliest medieval discussions on the nature of impossibility. The Tractatus distinguishes between different sub-kinds of statements that are per se impossible, which include metaphysical impossibilities (‘A human is a donkey’; ‘Socrates is Plato’), logical or syntactical impossibilities (‘Socrates exists and Socrates does not exist’), and impossible statements of other sorts.
31 See Iwakuma Citation1993, 141–142.
32 See p. IVb: ‘Quinti concedunt ultimum illatum, dicentes ex quodlibet impossibili per se sequi quidlibet.’
33 See Stephen Langton Citation2014, 402, 407. While discussing the truth of the inference ‘If a virgin gives birth, she has slept with a man,’ Langton mentions the view of some according to whom ‘from something that is impossible per se, everything follows’. Langton provides the idea of ‘per se impossibility’ with a theological twist: he describes it as the impossible ‘which cannot be true’ and that thus ‘cannot be made true even by God’. Since a virgin giving birth is, indeed, impossible, but not in the per se sense, because God could make it true, the inference in question is not vacuously true according to him, because not everything follows from an antecedent which is impossible but not per se. Langton thus employs the notion of per se impossibilities to safeguard the use of certain impossible statements as premises in non-vacuous inferences, provided that these impossibilities are within the domain of what God could bring about.
34 De Rijk Citation1988, 204 (my translation): ‘The following is argued: If Socrates is a donkey, then Socrates exists; and this follows in virtue of the topic from a subjunctive part. And if Socrates exists, Socrates is Socrates; in virtue of the topic from the same. And if Socrates is Socrates, Socrates is a human, in virtue of the topic from a subjective part. And if Socrates is a human, Socrates is not a donkey, by means of the topic from opposites. Therefore, from first to last: if Socrates is a donkey, Socrates is not a donkey. Because, then, that Socrates is not a donkey follows from the impossible [statement] which is its contradictory opposite, there is all the more reason why any other [statement] must follow from the same impossible, since all these other statements would be less incompatible [minus repugnent] to that impossible [statement] than its contradictory opposite. And so it is evident that from an impossible everything follows.’
35 Cf. Iwakuma Citation1993, 135. The Avranches text, in which we find the exact same argument, also continues with a second part of the proof:
If Socrates is a donkey, then Socrates is a donkey or any proposition whatsoever is true
If Socrates is a donkey, then (Socrates is not a donkey and Socrates is a donkey) or anything is true;
If Socrates is a donkey, then anything is true.
36 See ms. Paris BnF, lat. 3713, f. 89r.
37 Burley Citation2000, 146: ‘For example, “If a man is an ass, you are sitting.” This inference is a good one, and holds through the rule “Anything follows from the impossible.” The rule relies on the topic “from the less”. For the impossible seems to be less true than anything else. Therefore, if the impossible is true, it follows through the topic “from the less” that anything else will be true.’
38 As said earlier, the Avranches text is edited in Iwakuma Citation1993, 134–138. The theory of counterpossibles presented here is briefly mentioned in Martin Citation2018, 348, f24. As Martin states, the text discusses a number of arguments against and in favour of the principle, and eventually concludes with the acceptance of vacuism. For a more detailed analysis of the position on counterpossibles offered in the Avranches text see also Lenzen Citation2021. I agree with Lenzen’s general analysis of the text, but not in some of its elements. See for instance my vs. his interpretation of the notion of ‘(im)pertinens’ proposition used in the text, as briefly explained in the following paragraph and footnote.
39 For a similar terminology in the discussion of counterpossibles, see also ms. Paris BnF lat. 2904, p. IVB, where the anonymous author says that, if we can prove that from an impossible something that is entirely irrelevant to it (nihil pertinet ad ipsum) follows, then everything else follows from it. The use of the term ‘(im)pertinens’ to signify the relevance or irrelevance of a certain proposition to another in a relation of consecution is also used in later medieval logic, and particularly in the tradition of obligations. For a different interpretation of ‘(im)pertinens’ in the Avranches text, see Lenzen Citation2021, 10–13 (but such an interpretation is said to be highly speculative by Lenzen himself).
40 See Iwakuma Citation1993, 134. Interestingly, at the end of the treatise the author will refute this argument by saying that the antecedent ‘Socrates is a donkey’ is indeed relevant to the consequent ‘God exists,’ because the former is impossible only in virtue of the composition of its parts, but the parts themselves (that is, Socrates and donkeys), in order to exist, require the existence of God. What I think is interesting here is the analysis of the antecedent’s impossibility as depending on the composition of the antecedent’s parts. Similar ways of analysis will return in the 13th century. See for instance Peter of Spain’s discussion of impossible antecedents in the Syncategoreumata, where he distinguishes the notion of impossible as impossibility in itself from impossibility depending on the composition/division of parts. This analysis is at the basis of Peter’s answer to vacuism (for Peter’s divisio impossibilium see Peter of Spain Citation1992, 234–236; Spruyt Citation1993, 171–172).
41 This text is preserved in ms. Vienna VPL 2237, f. 31r and edited in Iwakuma Citation2013, 30–31.
42 Preserved in ms. London, British Library, Royal 2D XXX, ff. 95ra–102vb.
43 Cf. on this Martin Citation2022a, 122–123, where Martin also associated the Meludinenses’ views on this matter with the idea supported by Frege (Citation1980, 182) that nothing can be concluded on the basis of false premises.
44 See also Spruyt Citation1993, 162–163 on this issue as a general point in 13th-century debates on the ex impossibili principle.
45 This view is also supported in other parts of the Ars Meliduna, where the author discusses true statements. On this, see also Martin Citation2022a; 121–122.
46 See Iwakuma Citation1993, 145. In support of this view, the author appeals to Aristotle’s Prior Analytics and Boethius’s De Hypotheticis Syllogismis.
47 In Peter of Spain’s argument against the ex impossibili principle, for instance, we find a similar reasoning. Peter states that in every argument the ratio inferendi depends on a certain relationship holding between the premises and the conclusion, which produces faith in something doubtful, but impossible premises have no such relationship to other propositions. See Peter of Spain Citation1992, 233: ‘Now it is argued to the contrary [i.e. that not everything follows from an impossible] as follows. Since a conclusion is a proposition proved by an argument or arguments, and every argument provides the rationale for inferring it according to some logical relationship or relationships (for an argument forms the rationale which produces faith in something doubtful, and faith in something that is doubtful can only be produced by means of a logical relationship or logical relationships), therefore it is necessary that wherever some thing is concluded from something else and something follows from something else, that there be some logical relationship, or relationships, involved in virtue of which the one can follow from the other. Now the impossible “that a man is an ass” has no such logical relationships, whether directly or indirectly, to the impossible “that whiteness is blackness” or “that justice is injustice.” Therefore from the impossible “that a man is an ass” the impossibles mentioned will not follow. Therefore it is not the case that from the impossible anything follows.’
48 In John of Salisbury’s Metalogicon, this thesis is attributed to Adam of the Little Bridge, the founder of the Parvipontani. See Iwakuma Citation1993, 124.
49 Iwakuma Citation1993, 142–143 (my translation): ‘Assuming the opposite view [i.e. that something does follow from a falsehood], it can be proved that (1) from a certain proposition its contradictory follows; (2) that two contradictory propositions follow from the same [antecedent]; and (3) from a certain proposition follows another one which cannot be true with it. It seems that all these three are against the art [of logic – contra artem]. For indeed, just as no thing can simultaneously be and not be, in the same way two [contradictories] cannot [follow] from the same proposition.’
50 See Courtenay Citation1993; cf. footnote 16 above.
51 Cf. De Rijk Citation1988, 206–207 (my translation): ‘It should be said that truly according to the opinion of some – namely, the nominalists – from the impossible everything follows. However, according to the truth (secundum veritatem), nothing follows from an impossible, and this is the opinion of the realists. That this opinion is correct can be made manifest by considering things as they are in nature, since things that are in art – that is, in logic – must always be seen in analogy (proportio) and resemblance (similitudo) [to the way things are in nature]. And indeed, in nature things are in such a way that it cannot happen that two opposite species are included in the same subject. As if in a certain subject heat is found, the opposite species (that is, coldness) cannot be found in the same subject. Similarly, also in logic it will be the case that opposite species will not be included in the same subject. Therefore, from the fact that donkey-ness inheres in Socrates this consequent must not follow, that goat-ness is also in Socrates, since donkey and goat are of different species. Thus, from positing one it does not follow that the other is posited in the same subject, but rather the privation of the other should be posited. And so, with these conclusive reasons, we correctly agree that from the impossible nothing follows.’
52 Cf De Rijk Citation1988, 204–205. See for comparison a similar argumentation in the Avranches text, in Iwakuma Citation1993, 134–135.
53 The list of arguments in question is rather long, so I only quote the ending remarks, found in De Rijk Citation1988, 206 (my translation): ‘Antecedent and consequent (or, anteceding and consecution) are relative forms, like paternity and filiation. And indeed, the antecedent is connected to the consequent and vice versa, like father to son. And yet, relative forms are within existing substances, like paternity and filiation. Therefore, since impossibles are non-things, there could be no consecution or anteceding among impossibles. Thus, the impossible could not be the antecedent or the consequent in a conditional. And if this is the case, from an impossible nothing follows.’