University of St Andrews

Department of Philosophy

Arché Research Centre

Theories of Paradox in Fourteenth-Century Logic: Edition and Translation of Key Texts

St Andrews Institute of Mediaeval Studies

Leverhulme Trust

Medieval Logic Research Group

Principal Investigator: Stephen Read

Research Fellow: Barbara Bartocci

 1. The Logical Paradoxes

 2. Paul of Venice's Logica Magna

 3. Paul's Two Theories of Insolubles

 4. Segrave and Dumbleton

 5. Source Material

 6. References

Summary

The project, funded by a Leverhulme Research Project grant to Professor Read, began on 1 August 2017 and continues until 31 July 2021, having recently been extended to allow for disruption caused by COVID-19. It consists of preparing an edition of the Latin text, together with an English translation and commentary, of the late 14th-century treatise on Insolubles (logical paradoxes) by Paul of Venice from his Logica Magna; together with editions and English translations of two treatises on insolubles from earlier in the century, by Walter Segrave and John Dumbleton.

The main and most direct aim is scholarly and historical, to provide scholars and students with access to important and interesting texts from the 14th century on the logical paradoxes. The logical paradoxes have played a significant role in the development of philosophical ideas, not just in logic but also in philosophy of language, epistemology, metaphysics and even ethics and political philosophy, throughout the 20th and 21st centuries. They played a no less significant role in later medieval philosophy and were the subject of much debate and the spur to original ideas, arguably reaching their zenith in the 14th century. Much has been learned about the medieval debate in the past fifty years, in the writings of John Buridan, Thomas Bradwardine and others. But other interesting treatises remain unedited, many only surviving in contemporary manuscripts. Among these is the treatise on insolubles (logical paradoxes) by Paul of Venice, summarizing and developing theories and solutions from his predecessors in the 14th century, constituting the final treatise of his Logica Magna. Seven of the treatises from this huge work were edited and translated into English between 1978 and 1991. But the treatise on insolubles was not among them. The current project aims to edit and translate this treatise, which describes fifteen other theories which it rejects, then develops its own at length, together with a commentary; and to edit and translate two further treatises, those of Walter Segrave and John Dumbleton, writing in Oxford in the second quarter of the century, which Paul mentions and which remain unedited, containing rich ideas about alternative solutions, restrictio and cassatio respectively. Publication of these texts will allow a better overview of the development of solutions to the paradoxes through the 14th century, as well as giving further insight into the nature of the paradoxes and their possible solution.

In sum, the following publications are planned:

1. The Logical Paradoxes

Although the Liar paradox and similar puzzles were well known and much discussed in antiquity, the medieval interest in them seems to be quite independent and largely in ignorance of those discussions. (See Spade and Read 2018, §§1.1-1.2) On the one hand, their paradoxical nature seems not even to have been properly recognised until the end of the twelfth century, and the only reference from antiquity which is regularly cited is Aristotle’s discussion of the oath-breaker in his Sophistical Refutations, ch.25. The oath-breaker first says that he will break his oath and then proceeds to fulfil that oath by breaking a subsequent one. To be truly paradoxical, it would needs be one and the same oath which he both fulfils and breaks. This is what we find in the classic case of the Liar paradox, when someone says ‘I am lying’ (where this is all he says, or at least he means to refer to that utterance) or ‘This utterance is false’ (referring to that very utterance). For if it is true then it must be false (for that is what was said), so it is not true (since it can’t be both), and consequently by reductio ad absurdum it really is not true, and so is false (assuming it is either true or false, and so if not true, then false). But given, as we have just proved, that it is false, it is surely true (since that is what was said). Thus we have proved both that it is true and that it is false (indeed, that it is both true and not true), and that is paradoxical (literally, beyond belief). Something has surely gone wrong. But what is the mistake?

We can divide medieval discussions of the insolubles— logical paradoxes such as the Liar— into two main periods, before Bradwardine and after Bradwardine. Thomas Bradwardine wrote his treatise on Insolubles in Oxford in the early 1320s and it seems to mark a sea change in the solutions which were mainly favoured. Up until the 1320s two types of solution were the focus of attention, restrictio and cassatio (though only two treatises are known which favoured the latter). Supporters of restrictio (restrictivism, aka restrictionism) proposed an outright ban on self-reference— in some authors, a blanket ban, in most a limited ban under which self-reference may not be combined with a negative (or privative) expression. Walter Burley, a prime advocate of the latter form of restrictivism, wrote:

“According to earlier writers, there are three sources of insolubles. The first source comes from the combination of an intentional verb with the expression ‘false’ or anything convertible with it or with the denial of ‘true’... Concerning the first source it should be recognised that whenever the same act has reflection on itself with a privative determination, namely, with the determination ‘false’ or anything convertible with it, then the act is restricted ... For a term is restricted when it does not imply its superior [that is, any term it falls under] ... It should be recognised that a part can never supposit for a whole of which it is part when, putting the whole in place of the part, there results reflection of the same on itself with a privative determination. So if one says ‘I say a falsehood’, the term ‘falsehood’ does not supposit for ‘I say a falsehood’ because, if the whole is put in place of the term ‘falsehood’, there results reflection on itself with a privative determination.” (Burley §§3.01-3.03 in Roure 1970, pp.271-2, correcting the text against De Rijk 1966, p.90)

Bradwardine attacked several versions of restrictivism mercilessly in chs.3-4 of his Insolubles. While this was the pre-eminent proposal for solving the insolubles before his attack, it had few proponents thereafter (a rare exception is found in Walter Segrave’s Insolubles: Spade 1975, p.113). An anonymous later author wrote:

“But because there are both many authorities and strong arguments against this opinion, we can probably say that whereas in propositions where an act has reflection on itself or where a part supposits for the whole of which it is part there is a tortuous and unusual way of speaking or thinking; nonetheless, it is possible where what supposits for the whole is a general or universal term, e.g., ‘Every proposition is false’, but never when the term is singular, e.g., ‘This is false’, referring to itself, or ‘You are an ass or this disjunction is false’, or ‘This inference is valid, so you are an ass’, referring to that very inference itself, and so on.” (Pironet 2008, p.324)

Bradwardine also dismissed cassationism (cassatio) as nonsense, just as its supporters dismissed insolubles as nonsense. The verb ‘to cass’ (archaic, and derived from the Latin ‘cassare’) means to render null and void. Cassationism is regularly rejected in thirteenth-century treatises (indeed, as mentioned, it is defended in only two that survive). Bradwardine concluded after only a brief discussion: “the view of the nullifiers is sufficiently nullified.” Cassationism received its most extensive and well-argued defence in John Dumbleton’s Summa Logicae, written in the late 1330s and 1340s (see Spade 1975, pp.63-65).

Other theories were proposed and defended before Bradwardine’s revolution. He considers another couple in the fifth chapter of his treatise. But it was Bradwardine’s own proposal, along with his detailed and extensive criticism of restrictivism, which initiated a whole new era in discussions of the insolubles. His aim was to develop a solution to the insolubles which placed no restriction on self-reference or the theory of truth. His first and third postulates were:

(1) “Every proposition is true or false” (and, implicitly, not both)
(3) “The part can supposit for the whole of which it is part and its opposite and for what is convertible with them.” (Bradwardine 2010, ¶6.3)

Along with his definition of a true proposition as “an utterance signifying only as it is”, a false one as “signifying other than it is”, and some basic logical principles, he was able to establish his second conclusion, that “any proposition that signifies itself not to be true or to be false, also signifies itself to be true and is false.” The basis of this claim was his novel proposal, encapsulated in his second postulate:

(2) “Every proposition signifies or means as a matter of fact or absolutely everything which follows from it as a matter of fact or absolutely.”

The ingenious use of this postulate in proving his second conclusion is well worth studying, as is his application of it in analysing a succession of insolubles.

However, although most if not all subsequent writers on insolubles owe a debt to Bradwardine, few followed him completely, and in particular, it seems to have been his powerful second postulate that was not popular. Two alternative proposals presented in Oxford in the 1330s, both responding to Bradwardine’s idea but in different ways, dominated subsequent discussion of the insolubles. They were due to William Heytesbury and Roger Swyneshed.

Discussion of the insolubles was shot through, as noted in §3, by the language of obligations. But in Heytesbury’s case, it was not just the language (of granting, denying etc) but the whole structure of obligational disputations which frames the discussion. Most insolubles need a background scenario—minimal as in the Liar, which assumes that the Liar utterance is self-referential (and not just apologising for a lie just told, for example), or more extensive in such cases as the common medieval scenario where a stock character called ‘Socrates’ says one and only one thing, namely, ‘Socrates says a falsehood’, or elaborate examples where, say, a landowner, troubled by vagabonds, has set up a gallows by a bridge over a river dividing his lands, decreeing that everyone who wishes to cross the bridge must declare their business and where they are going, to be let across if they speak truly but hung on the gallows if they lie: only to be confronted by Socrates (yet again) saying that his sole business is to be hung on the gallows.

Heytesbury takes each insoluble to be the positum in an obligation. He first distinguishes an insoluble scenario from an insoluble proposition:

“[A] scenario of an insoluble is one in which mention is made of some proposition such that if in the same scenario it signifies only as its words commonly suggest, from its being true it follows that it is false and vice versa ... [A]n insoluble proposition is one of which mention is made in some insoluble scenario such that if in the same scenario it signifies only as its words commonly suggest, from its being true it follows that it is false and vice versa.” (Heytesbury 1987, p.236; Pironet 2008, p.284. For an alternative translation, see Heytesbury 1979, p.46)

Heytesbury’s solution turns on the specification of the precise signification of the insoluble. The scenario may specify that the insoluble signifies only as its terms commonly suggest (precise sicut termini communiter pretendunt) or it may leave it open, signifying as the terms commonly suggest but not necessarily only in that way (non sic precise). (See Heytesbury 1987, p.238; Pironet 2008, pp.284-5) If it were left completely open, Hunter and others realise that the respondent could then do no more than doubt the positum, since he would not know how it signified. (See, e.g., Hunter in Pironet 2008, p.303)If the opponent adds ‘precise’, then the scenario should not be accepted, for we noted in §3 that the first rule of obligations is that no intrinsically impossible obligation should be accepted, and with this restriction the usual contradiction, that the positum is both true and false, or should be both granted and denied, is immediately forthcoming. Heytesbury’s third Rule applies when the Opponent does not add the ‘precise’ restriction. In that case, he says, the obligation should be accepted, the insoluble should be granted as following, but that it is true should be denied.

Recall the earlier proof that ‘I am lying’ is both true and false. What is common to Bradwardine’s and Heytesbury’s solutions (and most others) is that they accept the first leg of the argument, using reductio ad absurdum to infer that the insoluble is false, but they find some way to block the second leg, arguing from its falsity, already granted, to its truth. Bradwardine, for example, blocks this move by reminding his reader that the truth of a proposition requires that the proposition signify only as it is, and since it signifies both that it is false (by the meaning of ‘lie’ or ‘false’) and that it is true (by his second conclusion), it cannot signify only as it is (since it cannot be both true and false), so it is false. Heytesbury is more discreet. He writes:

“But if someone asks what the proposition uttered by Socrates signifies in this scenario other than that Socrates says a falsehood, I say that the Respondent will not have to respond to that question, because from the scenario it follows that the proposition will signify other than that Socrates says a falsehood, but the scenario does not specify what that is and so the Respondent does not have to respond any further to what was asked.” (Heytesbury 1987, p.240. See also Pironet 2008, p.286, and for an alternative translation, Heytesbury 1979, pp.49-50)

The respondent is only required to say ‘I grant it’, ‘I deny it’, ‘I doubt it’ and so is not obliged to enter into discussion of what the positum may or may not signify—that is for the Opponent to do. Later authors came to distinguish the exact or primary signification of a proposition (what the words commonly suggest, or “according to the common institution of the idiom”, or “the common institution of grammar”) from its secondary or consequential signification, the two combining to make up its principal or total signification. But caution is needed, for these terms are often used slightly differently by different authors (and are often translated differently by different translators).

A later author, identified by Spade as ‘Robert Fland’, but arguably properly called ‘Robert Eland’ (see Read and Thakkar 2016), presented Bradwardine’s and Heytesbury’s solution and invited the reader to choose between them:

“So ‹these› two responses [sc. Bradwardine’s and Heytesbury’s] are better than the others for solving insolubles. Therefore the respondent should choose one of them for his solution to the insolubles.” (Spade 1978, p.65)

What seems to have happened, however, is that the popular solution was to combine them. We find such a solution in a number of treatises, several anonymous (including that of pseudo-Heytesbury, so called by Spade (1975, pp.35-36) because his treatise is so closely modelled on that of Heytesbury, and in treatises ascribed to John of Holland and to John Hunter (Johannes Venator). (Those of pseudo-Heytesbury, John Hunter and another anonymous treatise are edited in Pironet 2008; that of John of Holland in Bos 1985, pp.123-46; and another anonymous treatise in Spade 1971.) It is the solution commonly found in the teaching manuals at Oxford now known as the Logica Oxoniensis (see De Rijk 1977), and is the basis of Paul of Venice’s solution in his Logica Parva (see §5 below). When it comes to the third Rule, instead of refusing to specify what the additional signification is, pseudo-Heytesbury writes: “It must be said that ... ‘A falsehood exists’ signifies conjunctively, namely, that a falsehood exists and that that very proposition is true.” (Pironet 2008, p.292)

Hunter spells it out more clearly. Suppose Socrates says ‘Socrates says a falsehood’ and there is no other proposition, call it A. The claim is that A is false. Then we might be tempted to argue as follows:

“A is false, Socrates says A, so Socrates says a falsehood. The inference is valid, the premises are true, so the conclusion is true too. But Socrates says the conclusion. So Socrates says the truth.”

Not so, Hunter replies:

“It should be denied that Socrates says the conclusion. Rather, he says another proposition similar in sound but not in signification, because what was said by Socrates signifies that Socrates says a falsehood (and thus that it itself is false, since Socrates said nothing else) and that it itself is true. But any other similar proposition which is not said by Socrates signifies only that Socrates says a falsehood.” (Pironet 2008, p.305)

But none of them offer any argument, as Bradwardine had done, to show that this additional signification is that the proposition itself is true. We might call this the modified Heytesbury solution. That John of Holland subscribes to the modified view is not immediately obvious, but clear enough when his text is examined carefully. In response to a counter-argument, he writes:

“I deny the first inference, namely ‘A is a falsehood, and Socrates says A, therefore Socrates says a falsehood’, because it is a fallacy of the restricted and unrestricted. For the conclusion signifies many things conjunctively, namely that Socrates says a falsehood and something else (according to some people, viz that ‘Socrates says a falsehood’ is true)." (John of Holland 1985, 130:10-14)

That parenthetical clause contains the modified Heytesbury solution. At this point, John has mentioned it but hasn't yet committed himself to it. But in his response to the fourth counter-argument, he writes:

“I reply that ‘Socrates does not say a falsehood’ is not the contradictory of the insoluble, because the insoluble is not a singular proposition. Hence it is not necessary that this proposition contradict it. But if one asks, what then is the contradictory of this insoluble, I say that it is ‘Not-(Socrates says a falsehood)’ signifying disjunctively that either Socrates does not say a falsehood or that ‘Socrates says a falsehood’ is not true. Then this disjunction is true if one of its parts is, namely: ‘Socrates says a falsehood’ is not true. And the reason is that from the fact that the insoluble signifies conjunctively that Socrates says a falsehood and that ‘Socrates says a falsehood’ is true (at least as many people say), it is necessary that its contradictory is a proposition in which the negation is preposed to the whole." (ibid., 130:28 - 131:7)

That clearly commits John of Holland to the modified view, and stands (to my mind) significantly in contrast to Heytesbury's view, which refuses to specify what the additional meaning is (and is why other thinkers reject Heytesbury's view, e.g. Paul of Venice in his Logica Magna §1.12.5). But none of those who hold the modified view seems to offer any argument, as Bradwardine had done, to show that this additional signification is that the proposition itself is true.
Ralph Strode clearly distinguishes the modified Heytesbury solution from Heytesbury’s own solution, about which he writes:

“Regarding this third opinion, namely, that of Heytesbury, in so far as it agrees with Thomas Bradwardine's opinion, I consider it to be true, namely, in that it claims that it is impossible for an insoluble proposition to signify only as the words commonly suggest. For example, supposing that the proposition ‘A falsehood exists’ is the only proposition, it is impossible that it only signifies that a falsehood exists. But in so far as it is claimed that, in the given scenario, it is not decided or stated by the Respondent what else that proposition signifies, or in what other way that proposition signifies, I do not consider it to be true.” (Ralph Strode, Tractatus de Insolubilibus, ms Erfurt Amploniana Q 255, f.10va)

Strode proceeds in the Third Part of his treatise to apply his preferred solution to a range of insolubles. His response to the widely discussed scenario in which Socrates says only ‘Socrates says a falsehood’, labelled ‘A’, he writes:

“Regarding the solution to this insoluble it should be realised that close attention be given whether in the presentation of the scenario it is supposed that the insoluble proposition signifies only as the words prima facie suggest, or it is supposed that they signify in that way but not with the addition of the adverb ‘only’. If it was given in the first way, the scenario should in no way be accepted, because the scenario is impossible, as was clearly stated above. If it was given in the second way, then the scenario should be accepted, and generally so in every insoluble scenario. Furthermore, one should deny that A is true and grant that A is false and also that the proposition uttered by Socrates is false.” (loc.cit.)
He spells out the reason for those verdicts about the truth and falsehood of the insoluble in response to the next insoluble he considers, namely, where all and only those who speak the truth will receive a penny, and Socrates pipes up, ‘Socrates will not receive a penny’:
“And so, just as ⟨in the case of⟩ the proposition ‘Socrates says a falsehood’, supposing that he says only that, the proposition is insoluble, the proposition ‘Socrates will not receive a penny’ is an insoluble proposition in the scenario described, and consequently in line with what was established earlier, it signifies itself to be false and itself to be true.” (ibid., f.10vb)


Roger Swyneshed’s solution was at root very different. His aim was to provide a solution without the suggestion of hidden meanings, but taking the expressions at face value, so that the principal signification is just what it is commonly taken to be, what the words commonly suggest (though this phrase, which seems to originate with Heytesbury, is not used by Roger). Rather, he focussed on the fact that all the insolubles entail their own falsehood (though, of course, not only insolubles do that). Where Bradwardine and Heytesbury demanded for truth that everything that a proposition signified, including any hidden secondary or additional signification, should obtain, Swyneshed proposed that truth should require that a proposition not entail its own falsehood:

“There are four definitions ... The second is this: a true proposition is a proposition not falsifying itself, signifying principally as it is either naturally or by an imposition by which it was last imposed to signify.
Third definition: a false proposition is an utterance falsifying itself or an utterance not falsifying itself signifying principally other than it is either naturally or by an imposition by which it was last imposed to signify.”

Then Swyneshed defines an insoluble:

“[A]n insoluble as put forward is a proposition signifying principally as it is or other than it is ‹which is› relevant to inferring itself to be false or unknown or not believed, and so on.” (Swyneshed 1979, pp.185-6; 1987, p.182)

Swyneshed derives three famous iconoclastic conclusions from his theory:

(1) There is a false proposition that signifies principally as it is
(2) There is a formally valid inference where the false follows from the true
(3) There are two mutual contradictories that are both false. (Swyneshed 1979, pp.188-9; 1987, pp.186-8)

The Liar serves to illustrate and support all three claims. For ‘This very proposition is false’ does signify as it is, for it signifies that it is false (and, for Swyneshed, that is all it signifies) and by Swyneshed’s account of truth and falsehood, it is false, for it falsifies itself, that is, it entails its own falsehood. It also serves to establish the second claim: for consider the inference:

The conclusion of this inference is false
So the conclusion of this inference is false.

The conclusion is a version of the Liar, asserting of itself that it is false. Swyneshed claims that the inference is valid, for the premise signifies exactly the same as the conclusion, namely, it predicates the same property (being false) of the same thing (the conclusion). Swyneshed, like several fourteenth-century authors, takes preservation of signifying as it is, rather than truth-preservation, as the criterion of validity. Nonetheless, the premise is true, since it correctly (by his lights) says that the conclusion is false, and the conclusion is false, because it falsifies itself.

The third claim is in some ways the most puzzling and surprising. How can contradictories both be false? Did Aristotle not introduce the notion of contradictories as pairs of propositions that cannot both be true and cannot both be false? Not so, according to Whitaker (1996), who reminds us that what Aristotle actually wrote was:

“As men can affirm and deny the presence of that which is present and the presence of that which is absent and this they can do with reference to times that lie outside the present: whatever a man may affirm, it is possible as well to deny, and whatever a man may deny, it is possible as well to affirm. Thus, it follows, each affirmative statement will have its own opposite negative, just as each negative statement will have its affirmative opposite. Every such pair of propositions we, therefore, shall call contradictories, always assuming the predicates and subjects are really the same and the terms used without ambiguity.” (Aristotle 1938, pp.123-5: De Interpretatione ch. 6, 17a27-33)

Whitaker claims that Aristotle proceeds in the subsequent chapters of De Interpretatione to argue against what Whitaker (p.79) calls the Rule of Contradictory Pairs (RCP: that, of any pair of contradictories, one is true and the other false), giving examples in ch.7 of pairs both of which are true, in ch.8 pairs each of which is false, and in ch.9 (regarding the future sea-battle) that (RCP) fails for future contingents.

Swyneshed argues for his third claim by again taking ‘This proposition is false’ and its pair ‘This proposition is not false’, each referring to the former. The latter, he says, denies of the former what the former affirms of itself. So by Aristotle’s account, they are a contradictory pair. But the former is false because it falsifies itself, and the latter is false because it says, falsely, that the former is not false. So we have a pair of contradictories both of which are false.

Swyneshed adds “or unknown or not believed and so on” at the end of his definition of insolubles in order to include what are now called epistemic paradoxes, which the medievals included under the title ‘insoluble’. The most famous example is perhaps the Knower paradox, in the forms ‘This proposition is not known’, or ‘You do not know this proposition’. Suppose it were known. Then it would be true, and so not known. Hence by reductio ad absurdum, it is unknown. That is, we have just proved that it is unknown, which is what it signifies. So it is true, and moreover, since we have proved it, we know that it is true and so it is known. Swyneshed’s response is to question the second leg of the argument. Let A be the proposition ‘A is unknown’:

“‘A is unknown’ should be granted, and it should be granted that I know A to signify principally in this way. And the inference, “therefore, I know A” should be denied. But it is necessary to add that A is not relevant to inferring itself not to be known. And if that is added, it should be denied. For it follows directly, ‘A is unknown, therefore, A is unknown’.” (Swyneshed 1979, p.209)

So far we have looked only at developments at Oxford, following Bradwardine’s proposal. There were remarkably similar developments at Paris, remarkable not least for the fact that their differences suggest that there may not have been any direct influence. We can perhaps divide them again into two branches, the first stemming from John Buridan’s ideas, the second from Gregory of Rimini’s.

Buridan’s solution to the insolubles is nowadays perhaps the most famous of all medieval solutions, having been discussed extensively over the past fifty years. In fact, Pironet 1993 showed that Buridan’s ideas developed over the course of three or four decades in some five works. His early suggestion was that every proposition signifies its own truth. This is strongly in contrast with the Oxford tradition, both with Bradwardine and with pseudo-Heytesbury and his group, who only claimed that insolubles signify their own truth. Like Bradwardine, but unlike pseudo-Heytesbury and the others, Buridan offered a proof that every proposition signifies its own truth:

“For every proposition is affirmative or negative. But each of them signifies itself to be true or at least from each it follows that it is true. This is clear first concerning affirmatives, for every affirmative proposition signifies that its subject and predicate supposit for the same, and this is for it to be true ... Secondly, it is clear concerning negatives, for a negative does not signify that the subject and predicate supposit for the same, and this is for the negative proposition to be true.” (Buridan 1994, p.92. Although Buridan claimed that a proposition is true if it signifies as it is, he thought this phrase seriously misleading and cashed it out in terms of supposition: an affirmative proposition is true if subject and predicate supposit for the same, and a negative proposition is true if subject and predicate do not supposit for the same. See, e.g., Buridan, Sophismata, ch.2: ‘On the Causes of the Truth and Falsity of Propositions’ (2001, pp.845-62), especially the Fourteenth Conclusion, pp.858-9)

Then propositions such as the Liar are self-contradictory, signifying both that they are true and that they are false, and so are simply false:

“Regarding this proposition, ‘I say a falsehood’, I grant that it is false ... because it not only signifies that it itself is false, but also, from the general condition of a proposition, it signifies that it itself is true, and is not true.” (Buridan, Quaestiones in primum librum Analyticorum Posteriorum, Q.10, cited in Pironet 1993, p.295 n.4)

However, in later works, Buridan had doubts about this claim. In his Sophismata, composed some twenty or more years later, in the 1350s, he rehearsed his earlier view before criticising and revising it:

“For some people have said, and so it seemed to me elsewhere, that although this proposition [‘Every proposition is false’] does not signify or assert anything according to the signification of its terms other than that every proposition is false, nevertheless, every proposition by its form signifies or asserts itself to be true … But this response does not seem to me to be valid, in the strict sense ... rather, I [am going to] show that it is not true that every proposition signifies or asserts itself to be true.” (Buridan 2001, pp.967-8)

His objection was that this claim either implies that every proposition is metalinguistic, in always talking about the truth of some proposition (which he believes is not so), or commits one to the postulation of significates, some real correlate of the proposition (which to Buridan was anathema: these correlates are the notorious complexe significabilia, whose existence Buridan strongly contested; see, e.g., Klima 2009, §10.2). Buridan’s revised view was that every proposition implies its own truth, or at least would do so if it existed:

“Therefore, we put this otherwise, in a manner closer to the truth, namely, that every proposition virtually implies another proposition in which the predicate ‘true’ [would be] affirmed of the subject that supposits for [the original proposition]; and I say ‘virtually implies’ in the sense in which the antecedent implies that which follows from it.” (Buridan 2001, p.969)

Buridan’s account is one of the well-known solutions to the insolubles which Paul of Venice does not include in his survey of fifteen alternative solutions in the treatise on ‘Insolubles’ in his Logica Magna. But he does include Albert of Saxony’s account, which is very similar to Buridan’s early view. (See, e.g., Spade and Read 2018, §3.6)

Paul also omits Gregory of Rimini’s solution from his survey of previous opinions, which is again surprising, since Gregory is one of the few authors to whom Paul refers by name in the Logica Magna, and indeed, Gregory had been Prior General of the Order of Augustinians in the 1350s, Paul’s own order. But Gregory’s solution was taken over and adapted by Peter of Ailly, and Peter’s view is discussed, commented and criticised at length by Paul.

To understand Gregory’s approach, we need to recall that the medievals, following Aristotle’s lead, divided language into three levels, written, spoken and mental. (See Aristotle 1938: De Interpretatione ch.1, and Read 2014) Boethius (writing in the sixth century CE) cited Aristotle as saying: “[S]poken [words] are signs (notae) of impressions (passionum) in the soul, and the written ones are those of the spoken ones.” By the fourteenth century, it was common to speak of a complete parallelism of mental, spoken and written propositions. Spoken and written propositions signify by human imposition (ad placitum), mental propositions signify naturally. Gregory seems to have inferred that only spoken and written propositions can be insolubles—the paradoxical situation cannot infect mental language. Consequently, (spoken and written) insolubles correspond to non-insoluble mental propositions, in fact, to a conjunction of two such mental propositions, the first of which captures the primary or customary signification of the insoluble, and the second of which says that the first conjunct is false. For example, taking the Liar again, the spoken proposition ‘This proposition is false’, referring to itself, call it A, corresponds to the conjunctive mental proposition whose first conjunct says that (the spoken proposition) A is false, and whose second conjunct says that the first conjunct is false. Neither conjunct of the mental proposition is self-referential, nor is either insoluble or contradictory. In fact, the first conjunct is true (A is false) and the second conjunct is false (since it says falsely that the first conjunct is false), so the whole mental conjunction is false, and so the corresponding spoken proposition A is false too.(No text on insolubles by Gregory survives, and so this is a reconstruction of Gregory’s view by Spade and others. See Peter of Ailly 1980, pp.6-7)

In Peter of Ailly (1980, p.6), Spade suggests that Gregory’s solution was a development of Bradwardine’s, and that Marsilius of Inghen’s was also. However, Marsilius’ solution (probably developed in the 1360s) has strong similarities to that of pseudo-Heytesbury and others discussed above. Spade quotes Marsilius (in translation), discussing the common example where there is only one Socrates, who only says ‘Socrates says a falsehood’:

“The reply is that the sophism is false. For it amounts to the conjunction ‘Socrates says a falsehood and it is false that Socrates says a falsehood’. But that is false on account of its second part. Therefore, although it is always as is signified by its first conjunct, nonetheless it is not as is signified by its second conjunct.” (Peter of Ailly 1980, p.98 n.56)

So in general, an insoluble is expounded as a conjunction whose first conjunct expresses what the terms commonly suggest and whose second conjunct contradicts this and says that is false. But since insolubles falsify themselves, that second conjunct says that it is false that it is false, that is, that it is true, as Bradwardine, Heytesbury and their successors proposed.

Gregory’s solution, probably dating from the 1340s, was taken up and adapted by Peter of Ailly in his treatise on Concepts and Insolubles, written in 1372. Where Gregory claimed that insolubles were false, corresponding to false mental conjunctions, Peter argued that the phenomena are better explained by realising that insolubles are equivocal, both true and false, corresponding to two different mental proposition, the two conjuncts from Gregory’s theory, now to be seen not as conjoined but as each separately corresponding to the insoluble. For example, A, in saying of itself that A is false, is true (in corresponding to the true mental proposition that says that A is false), but in saying that what it says (i.e., the mental proposition that says that A is false) is false, it is false. Now you see it, now you don’t; Peter tries to capture the flip-flop behaviour that insolubles exhibit. (See Spade’s comment in Peter of Ailly 1980, pp.12-13)

In the treatise on insolubles in the Logica Magna, after presenting and discussing fifteen alternative solutions, Paul of Venice presents his own, a development of Roger Swyneshed’s account. Paul adds to Swyneshed's view that a proposition is true if things are as it signifies, unless it falsifies itself, his theory of the exact (or core) signification of a proposition (its significatum adaequatum): a proposition is true if its exact significate is true, unless that is incompatible with its truth.

2. Paul of Venice's Logica Magna

The Logica Magna was the subject of a British Academy project, begun over forty years ago and involving a number of scholars, to edit and translate into English the whole work in twenty volumes. Eventually, seven volumes appeared, but the final treatise, on Insolubles, was not among them, and the project was abandoned in the 1990s. Whether that was the right outcome for the other projected volumes is moot, but the treatise on insolubles is an important record for possible approaches to the paradoxes (still a live debate today, as noted above), and it is important that it receive a modern edition. Although Paul’s compendium was printed in 1499, most modern readers need an apparatus explaining the discussion, and many need a translation. Moreover, although the 1499 text is fairly reliable, it needs some correction against the sole surviving manuscript.

Here is an analysis of the whole of the Logica Magna, showing the place of the treatise on Insolubles in it (where E is the 1499 incunable, M the manuscript, and 'Fasc' refers to the divisions proposed in the incomplete British Academy edition). Note also the different position of the treatise on Syllogisms in M and in E: in M it is placed straight after the discussion of subject-predicate (or categorical) propositions; in E it comes much later, after the discussion of molecular propositions and the account of truth, falsehood, signification and modality, even though Paul concentrates only on assertoric, that is, non-modal, categorical syllogisms:

Folios in E

Folios in M

Sections

Chapters

Fasc.

Treatises and chapters in E

Part I

2ra

3ra

1: On Terms

i: Categorematic and Syncategorematic Terms

I:1

1: de terminis

3vb

9vb

ii: Naturally Significant and Arbitrarily Significant Terms

c.ii: secunda divisio terminorum

8ra

10vb

iii: Common and Discrete Terms

c.iii: tertia divisio terminorum

12va

16rb

iv: Immediate and Mediate Terms

c.iv: quarta divisio terminorum

16ra

20rb

2: On Supposition

i: Material Supposition

I:2

2: de suppositionibus terminorum

18ra

23ra

ii: Simple Supposition

20ra

25vb

iii: Personal Supposition

22ra

28va

iv: Supposition in respect of ampliative words

26vb

34va

v: Supposition of Relatives

31va

40va

3: On Terms that Render Supposition Confused

i: On the Term ‘differt’

I:3

3: de terminis confundentibus

34ra

43vb

ii: Exclusive Expressions

4: de dictionibus exclusivis: “in hoc capitulo”

35vb

46ra

5: de regulis exclusivarum

38ra

48va

iii: Exceptive Expressions

6: de dictionibus exceptivis

39ra

50ra

7: de regulis exceptivarum

41ra

52ra

iv: Reduplicatives

I:4

8: de reduplicativis

42ra

53rb

v: On the ‘just as’ construction

9: de dictione sicut

43vb

55vb

vi: On Comparatives

10: de comparativis

47va

60va

vii: On Superlatives

11: de superlativis

49rb

62vb

viii: On ‘Maximum’ and ‘Minimum’

12: de maximo et minimo

52rb

68ra

13: de obiectionibus et solutionibus argumentorum

56ra

71rb

4: On Terms that can be taken Categorematically or Syncategorematically

i: On ‘Whole’ taken Categorematically

I:5

14: de toto cathegorematice tento

57vb

73ra

ii: On ‘Always’ and ‘Eternal’

15: de semper et aeternum: “Secundus terminus qui”

58vb

74va

iii: On ‘Infinite’

16: de isto termino infinitum

61ra

77va

iv: On ‘Immediate’ and ‘Mediate’

17: de isto termino immediate

63rb

80rb

v: On ‘Begins’ and ‘Ceases’

18: de incipit et desinit

70va

89vb

5: On Intensionality

i: On Hyper-intensional Terms

I:6

de terminis officiabilibus

71ra

90va

ii: On Modal Propositions

19: de propositione exponibili

73ra

93ra

iii: On Hyper-intensional Propositions

20: de propositione officiabili

76rb

97va

iv: On Compounded and Divided Senses

21: de sensu composito et diviso

82ra

104vb

6: On Knowledge, Doubt, Necessity and Contingency

i: On Knowledge and Doubt

I:7

22: de scire et dubitare

91ra

115rb

ii: On Future Contingents

I:8

23: de necessitate et contingentia futurorum

Part II

101ra

128rb

1: On Subject-Predicate Propositions

i: On the Definition of ‘Proposition’

II:1

1: de propositione

101vb

129va

ii: On Subject-Predicate Propositions

2: de propositione cathegorica

109ra

139rb

iii: On the Quantity of Propositions

3: de propositione in genere

111ra

141vb

4: de quantitate propositionum

113rb

144rb

iv: On the Square of Opposition

II:2

5: de figuris propositionum (capitulum de figura)

114rb

145rb

v: On the Equivalence of Propositions

6: capitulum de equipollentiis

115rb

146ra

vi: More on the Square of Opposition

7: capitulum de natura situatorum in figura (de lege positorum in figura)

122vb

155va

vii: On the Conversion of Propositions

8: de conversione propositionum

172ra

158rb

2: On Syllogisms

i: On the First Figure

II:7

13: de syllogismis capitulum

174rb

161ra

ii: On the Second Figure

174vb

161vb

iii: On the Third Figure

176rb

163rb

iv: On Irregular Syllogisms

124va

164rb

3: On Molecular Propositions

i: On ‘When’-Propositions

II:3

9: de hypotheticis propositionibus

125vb

165vb

ii: On ‘Where’-Propositions

126ra

166ra

iii: On ‘Why’-Propositions

127ra

167ra

iv: On ‘Although’-Propositions

127vb

168ra

v: On Properly Molecular Propositions

129va

170rb

vi: On Conjunctive Propositions

131rb

172va

vii: On Disjunctive Propositions

134rb

175va

viii: On Conditional Propositions

II:4

139rb

180va

ix: On Entailment Propositions

147rb

189rb

x: Rules of Inference

II:5

158rb

201va

4. On Truth, Signification and Modality

i: On Truth and Falsehood

II:6

10: de veritate et falsitate propositionum

162ra

205va

ii: On the Significate of the Proposition

11: de significato propositionis capitulum

167vb

211vb

iii: On the Modality of Propositions

II:7

12: de necessitate contingentia possibilitate et impossibilitate propositionum

177ra

216va

5. On Obligations

i: On Obligations in General

II:8

14: de obligationibus

181ra

221va

ii: On Positio

191va

234vb

iii: On Depositio

192rb

236ra

6. On Insolubles

i: On Previous Opinions

II:9

15: de insolubilibus

194rb

238va

ii: The Author’s Opinion

195vb

240rb

iii: Objections and Replies

197rb

241rb

iv: On the Famous Insoluble ‘Socrates says a falsehood’

197vb

243ra

v: On Covert Insolubles

198rb

243va

vi: On Quantified Insolubles

198vb

244rb

vii: On Non-Quantified Insolubles

199ra

244va

viii: On Merely Apparent Insolubles

 

3. Paul's Two Theories of Insolubles

There are few summaries and presentations of Paul’s solution(s). A very brief account of that in the Logica Magna is given in Spade 1975, pp.83-4, and a slightly fuller, but somewhat confused one in Bottin 1976, pp.148-51, who conflates the very different solutions in the Logica Magna and the Logica Parva. That in Bochenski 1970, pp.247-51 is also muddled and misleading: after correctly reproducing a selection of Paul’s divisions and assumptions, he writes: “Paul’s own solution is very like that of the eleventh [viz Albert of Saxony’s] and twelfth [viz Heytesbury’s] opinions, and so we do not reproduce his long and difficult text”, giving instead a one-page summary. This summary bears no relation whatever to what Paul writes in the Logica Magna, not even to the passages Bochenski has cited from it, nor to Heytesbury, but is similar in many ways to Albert’s solution (see Albert of Saxony 1988, pp.346-7).

Paul of Venice wrote at least four treatises, of varying lengths, on the insolubles: a short treatise (about 6500 words) in his Logica Parva (Paul of Venice 1984, 2002), a longer treatise (a little over 20,000 words, the main object of the current project) in his Logica Magna, a brief discussion in sophism 50 of his Sophismata Aurea (Paul of Venice 1483), and further discussion in his Quadratura (Paul of Venice 1493). It is by no means certain that these four works have a single author. In particular, Alan Perreiah (who edited and translated the Logica Parva) has repeatedly argued that the Logica Magna is not by Paul, on account of inconsistencies between this treatise and the others (Paul of Venice 1984, pp.327-43, 2002, p.xviii, Perreiah 1978 §1, 1986, pp.73-127). The author of the Logica Parva and the Quadratura adopts pseudo-Heytesbury's solution, whereas the author of the Logica Magna and the Sophismata Aurea defends Swyneshed's solution. We reject Perreiah's arguments, in brief arguing that the Logica Parva and the Quadratura are teaching manuals, where Paul is simply presenting the standard curriculum for elementary students, whereas in the Logica Magna and the Sophismata Aurea he is presenting his own solution. At the end of the Logica Parva he writes: "Notice that not everything I have said here, or in other treatises, I have said according to my own view, but partly according to the view of others, in order to enable young beginners to progress more easily." We noted above that two of the leading solutions to the insolubles in the fourteenth century were those of pseudo-Heytesbury and his followers (adapting Heytesbury’s distinctive solution) and of Roger Swyneshed. Paul follows both of these solutions in different works: pseudo-Heytesbury’s in the Logica Parva and in the Quadratura, and that of Swyneshed in the Logica Magna and the Sophismata.

Recall that Heytesbury, and his followers such as Hunter and Holland, distinguish an insoluble scenario in which it is specified that the insoluble signifies only as the terms suggest, in which case the scenario (or obligatio) should not be accepted, from a scenario where that exclusion clause is omitted, in which case the insoluble and its scenario can be accepted and the insoluble denied. Paul follows this division at the start of the chapter on insolubles in his Logica Parva, marking the distinction as that between what is simply, or unrestrictedly, an insoluble (insolubile simpliciter) and what is an insoluble restrictedly (insolubile secundum quid—note that Perreiah translates this as ‘according to a condition’). Then he presents two conclusions: the first is that no simple or unrestricted insoluble should be accepted:

"No scenario from which what is unrestrictedly an insoluble arises should be accepted. E.g., if anyone proposes that ‘Every proposition is false’, signifying only in that way, is the only proposition, the scenario should not be accepted because a contradiction follows." (Paul of Venice 2002, p.132)

On the other hand, any scenario from which there arises what is an insoluble restrictedly, that is, without the exclusion clause, should be admitted:

"Every scenario from which what is restrictedly an insoluble arises should be accepted; and one grants the proposed insoluble by saying it is false. E.g., suppose that ‘This is false’ is a proposition referring to itself which signifies as the terms suggest—call it A. Then the scenario is accepted, and A is granted, and it is said that it is false. If one argues like this: ‘A is false, therefore it signifies other than is the case’, I grant it. ‘But A only signifies that it is false, therefore that it is false is not so’. I deny the minor premise, and if it is asked what else it signifies, I say that it signifies that A is true, and that is the reason why A is false. So it should be said that every proposition which is an insoluble restrictedly signifies conjunctively, namely, as it terms suggest and that it is true." (Paul of Venice 2002, pp.132-3)

This is the solution we noted is found in pseudo-Heytesbury, Holland, Hunter and others, following Heytesbury in accepting an insoluble scenario only if it is left open that the insoluble proposition has a secondary or additional signification, and (unlike Heytesbury) specifying that additional signification as asserting its own truth. Paul spells it out towards the end of ch.6:

"It should be noted that an insoluble has two significates, one exact (adequatum) and one principal. The exact significate is a subject-predicate significate similar to the insoluble utterance. E.g., the exact significate of ‘Socrates says a falsehood’ is Socrates saying a falsehood or that Socrates says a falsehood. But the principal significate is a compound significate, e.g., that Socrates says a falsehood and that the proposition is true." (Paul of Venice 2002, p.149)

We find the same approach to insolubles in Paul’s Quadratura. This work is not about squaring the circle or quadrature, but is a highly formal and artificial series of two hundred sophisms arranged in four parts of fifty chapters each. The reason for the strange title ‘Quadratura’ is that each chapter “is fortified with four Conclusions and as many corollaries or more”. Each of the four main parts focuses on a particular question:

"First, whether the same inference can be both valid and invalid; secondly, whether the same proposition can be both true and false; thirdly, whether disparate things are verifiable of the same thing; fourthly, whether two incompatibles can be both true or both false."

The fifteenth chapter of part 1 is explicitly concerned with insolubles, specifically with the following argument:

"This inference ⟨call it B⟩ is valid: A will signify only that everything ⟨that is or will be⟩ true will be false, so A will be false; and this inference ⟨B⟩ is invalid. So the question is true."

Paul shows that the exclusion clause in the scenario (‘A will signify only that …’) must be amended. This follows from his second Conclusion:

"There is some proposition [namely, ‘Every proposition is false’] signifying principally purely predicatively which at some time will signify principally in a compound way. Nonetheless, there will be no change in it, nor will any new imposition be added to it."

For “when it will be the only proposition it will signify principally that every proposition is false and that it is true, just like other insolubles, whose significations reflect wholly on themselves.” He continues, in discussing the third Conclusion, to remark that “this conjunctive significate is called the principal significate of A, although it is not the exact ⟨significate⟩ but only the first part ⟨is⟩”. Without the exclusive phrase (that is, that A signifies only that everything true will be false), argument B is invalid, but if it is retained, Paul does not accept the scenario, “because it implies a contradiction”.

In response to a later sophism, the second in part 2, Paul spells out his use of ‘exact signficate’ and ‘principal significate’ in greater detail, and links them to the notions of truth and falsity:

"Finally, then, it should be said that it is because it immediately signifies a truth that any proposition is true, and it is because it immediately signifies a falsehood that any proposition is false, where outside the case of insolubles ‘immediately’ means the same as ‘exactly’. But in the case of insolubles it means the same as ‘principally’. Hence ‘A man is an animal’ is true because it immediately signifies a truth, that is, it exactly ⟨signifies⟩ the truth that a man is an animal; but ‘This is false’, referring to itself, is false because it immediately signifies a falsehood, that is, it principally ⟨signifies⟩ a falsehood, namely, that this is false and that this is not false."

However, whereas in the Logica Parva and the Quadratura Paul subscribes to the pseudo-Heytesbury solution to the insolubles, in the Logica Magna he defends a version of Swyneshed’s solution. Surprisingly, pseudo-Heytesbury’s solution does not appear among the fifteen solutions that Paul considers in his first chapter (in varying detail) and rejects, whereas Heytesbury’s own solution is considered, being the first solution to which Paul devotes more than a few lines. One of Paul’s objections to the solution turns, in fact, on Heytesbury’s reluctance to specify what the additional signification is which renders an insoluble false. This, and many of the other objections which Paul levels against Heytesbury’s view, are drawn from Peter of Mantua’s Insolubles (or possibly from a third text on which they both draw). Suppose ‘A falsehood is said’ signifies principally (that is, wholly and exactly) that God exists. Heytesbury had claimed that a proposition could have a further signification in addition to what it standardly signifies, so presumably it could signify something completely different, such as, that God exists. If so, it would be necessarily true. But Heytesbury’s view is that ‘A falsehood is said’ is false as uttered in the proposed scenario. Yet the conclusion of an inference which is clearly valid and whose premise is in doubt should not be denied—for if one denies the conclusion of an inference one recognises to be valid, one is committed to denying the premise. So Heytesbury’s solution proposing such hidden and unspecified significations is unacceptable.

Paul devotes the whole of the first chapter of his treatise on insolubles in the Logica Magna to the rejection of these other proposed solutions. The treatment can be seen as falling into four groups, the first three groups corresponding to three sources on which Paul draws: first, he runs rapidly through seven of the eight alternative solutions considered by Bradwardine (2010, chs.3-5), for the most part summarising almost verbatim Bradwardine’s own criticism. He then turns to Heytesbury’s criticism of alternative solutions, starting with the second of the four solutions considered by Heytesbury (the first one Heytesbury rejects is Swyneshed’s, which Paul will himself accept), that of John Dumbleton. The next (third on Heytesbury’s list) is Kilvington’s, and then Paul comes to Bradwardine’s own solution (Heytesbury’s fourth). Thus the first ten solutions considered are all from Oxford, or at least, those discussed at Oxford in the two decades from the early 1320s to the early 1340s. With the eleventh solution, Paul turns to his third source, namely, Peter of Mantua, the eleventh being Albert of Saxony’s solution, presented at Paris in the early 1350s, and possibly the same as John Buridan’s own early solution, the first view discussed by Mantua; and next to Heytesbury’s (Mantua’s second), as noted above. (See Strobino 2012, p.484.) Before proceeding to the third view discussed by Mantua, Paul considers Peter of Ailly’s solution at some length, seemingly drawing directly on Ailly’s own treatise, to the discussion of which Paul appends (without distinguishing it by number) a criticism of Mantua’s solution. Finally, Paul turns to a rejection of restrictivism, the first solution rejected by Bradwardine and the third by Mantua. But Paul deals with the specific form given to it by Walter Segrave in Oxford in the 1320s or early 1330s, who in fact defends restrictivism in the face of Bradwardine’s objections (see below), attributing the solution to the fallacy of accident, a suggestion not dealt with by either Bradwardine or Mantua.

After this extended discussion of alternatives (occupying a quarter of his treatise), Paul sets out to develop his own solution, based firmly on Roger Swyneshed’s proposals from the 1330s. In his second chapter, he systematically lays out his distinctions (divisiones) and assumptions (suppositiones), then draws seven Conclusions and seven Corollaries. The basic idea is Swyneshed’s, to provide a solution which does not depend on postulating tacit or hidden or consequential significates for insoluble propositions beyond what is clearly shown—what they standardly suggest or indicate by the straightforward combination of their parts (in Heytesbury’s phrase, ‘sicut termini communiter pretendunt’). Instead, as Swyneshed had proposed, Paul tightens the criterion for truth, to exclude those that falsify themselves, weakening the criterion for falsehood to admit those examples that do falsify themselves even if otherwise impeccable. Roger’s second and third notorious Conclusions reappear as Paul’s fifth and second respectively. Paul will later describe the second Conclusion as a fundamental principle, perhaps the fundamental principle (two other fundamental principles are mentioned in ch.8, on merely apparent insolubles). Consequently, Paul defines an insoluble as a self-falsifying proposition, that is, “a proposition having reflection on itself wholly or partially implying its own falsity or that it is not itself true”.

Paul’s adoption of Swyneshed’s solution to the insolubles, and in particular, his acceptance of Roger’s second Conclusion as his fifth, overturns several claims Paul had made in the earlier chapter on consequence (De Rationali) in the Logica Magna, in particular, his third Rule, that valid inference is always truth-preserving. As we saw when considering Roger’s solution in §4, the simple Liar, deemed by both of them to be false, follows immediately from the (for them, true) statement that it is false. Both premise and conclusion state of the same thing (the conclusion) that it is false. Nonetheless, their accounts of validity are different: Roger’s is the requirement that ‘signifying as it is’ be preserved, a definition he shares with Buridan. (See Spade 1983, pp.105, 113 and Buridan 2014, I 3, p.67.) Paul, however, identifies validity with the opposite of the conclusion’s being incompatible with the premise:

"A valid inference which signifies in accordance with the composition of its elements may be defined as one in which the contradictory of its conclusion would be incompatible with the premise of that inference, given that these signify as they do; and by ‘as they do’ I refer to what they customarily signify." (Paul of Venice 1990b, p.80)

There is a further inconsistency within the Logica Magna between the treatise on insolubles and the chapter on consequence in Paul’s treatment of what might be called the inferential Knower paradox (see, e.g., Anderson 1983):

This is unknown to you, so this is unknown to you,

where both occurrences of ‘this’ refer to the conclusion. In Paul of Venice (1990b, p.197) he presents this as a counterexample to his ninth Rule, that knowledge is closed under known consequence. For the conclusion is an example of the Knower paradox and cannot be known, on pain of contradiction (knowledge entails truth and so if it were known it would be unknown). So the premise is not only true but known to be true (by the argument just given). Paul considers four possible responses (including restrictivism and cassationism as the second and third), suggesting in his preferred response, the fourth, that the premise is also unknown, like the conclusion, but promising that there will be “more about this when we come to deal with the insolubles” (p.200).

When he comes to discuss this example in the fourth Conclusion in the second chapter of the treatise on insolubles, Paul agrees that it is an insoluble, but claims that the premise is true:

"For it is evident that this inference is formally valid, because one cannot see how the opposite of the conclusion can be compatible with the premise. But the premise is known by you, because you know that the conclusion is not known, since it is an insoluble that asserts that it itself is unknown. But the conclusion is not known by you."

Thus it is indeed a counterexample to the ninth Rule of the chapter on consequence: knowledge is not closed under known consequence. He also seems here to be extending the notion of insoluble. In the passage cited above, insolubles are defined as self-falsifying propositions—and indeed he remarks later that it follows that many propositions called insolubles by others are not really insoluble insofar as they do not fit his definition. Swyneshed (1979, §81), in fact, had already extended the conception of insoluble to include propositions that are relevant to inferring themselves not to be known, and Paul seems to be implicitly following Swyneshed here. However, in ch.8 of the Insolubles, Paul includes ‘This is not known by you’ among the merely apparent insolubles: “although they are not insoluble since they do not have reflection of falsity on themselves”.

In the rest of the treatise, Paul considers the objections which Heytesbury had directed at Swyneshed’s solution, many of which had already been addressed in discussing his own Conclusions and Corollaries; shows at some length how his solution deals with the much-discussed example where Socrates only says ‘Socrates says a falsehood’; extends the account to deal with other examples, such as that where Socrates says that his sole business is to be hung on the gallows, which are not obviously insolubles until the background scenario is added, to examples like ‘A falsehood exists’ which become insoluble in a suitable scenario, and to examples involving exclusive and exceptive propositions, such as ‘Only a false proposition is exclusive’ (assuming it is the only exclusive proposition) and ‘No proposition except A is false’ (where this is A and is the only exceptive proposition).

Turning to the discussion of insolubles in the Sophismata Aurea, it turns out to be very derivative from that in the Logica Magna. The work consists of a collection of fifty sophisms, many very familiar from other collections of sophisms, many turning on an equivocation between compounded and divided senses (what would nowadays be called a scope ambiguity), e.g., ‘Every proposition or its contradictory is true’ and ‘Everything false if it is impossible is not true’. The final sophism, no.50, is ‘Socrates says a falsehood’, given that there is only one Socrates and that is all he says. The sophismatic arguments leading to contradiction are mostly drawn, essentially verbatim, from the treatise on insolubles in the Logica Magna. Paul then presents four Conclusions and associated corollaries in order to defuse the sophismatic arguments. The first of these Conclusions is in effect a statement of the Swyneshed programme, to solve the insolubles without resort to any hidden signification, claiming that “every subject-predicate insoluble signifies exactly according to the composition of its terms”, and its first corollary consequently rejects all those solutions which turn on there being such a secondary signification, describing the standard opinion as specifying that second significate as “that it is true”, that is, that of pseudo-Heytesbury and the Logica Oxoniensis. The subsequent Conclusions repeat six of the seven Conclusions in the Insolubles treatise in the Logica Magna. The work concludes with a discussion of the paradox of signification, ‘This proposition signifies other than it is’, drawn again from Logica Magna.

4. Segrave and Dumbleton

Walter Segrave was writing in Oxford in the 1320s or ‘30s, defending a restrictivist theory explicitly in response to Bradwardine’s criticisms. His treatise is preserved in three mss, one incomplete (Spade 1975, pp.113-15). It constitutes an extensive and detailed response to Bradwardine, defending restrictivism by presenting a well-thought out reason for the restriction of supposition required to avoid contradiction. Where Burley, and Bradwardine, both attributed the fallacy in insolubles to what Aristotle described as the fallacy of the conditional and the unconditional (simpliciter et secundum quid), Segrave attributed it to the fallacy of accident, turning on a variation in the supposition of the middle term and the extremes in what might otherwise appear to be a sound syllogism.

John Dumbleton was, like Bradwardine, one of the famous Oxford Calculators, whose main interest was in mathematical physics. His discussion of insolubles occurs as the second chapter of Part I (Summa Logicae) of his magnum opus, Summa Logicae et Philosophiae Naturalis, a huge work running to some 400,000 words (Spade 1975, pp.63-65). The whole work was transcribed by James Weisheipl from a single ms (Vat.lat. 6750) in the early 1950s when preparing his Oxford D.Phil. thesis on Dumbleton’s natural philosophy, but that transcription was never published, and exists, it seems, in a single copy in the Library of the Pontifical Institute of Medieval Studies in Toronto. Useful as it is, it is in a very preliminary state, with many insecure, and arguably mistaken, readings, and needs comparison with the texts of the other extant mss of Dumbleton’s Summa which also contain this early section on insolubles. (Two mss are incomplete in lacking Part I; all are incomplete in lacking Part X, which Dumbleton refers to but arguably never completed before he succumbed to the Black Death in 1348 or 1349.) The 19 chapters on insolubles are preceded by an extended discussion of signification in 5 chapters, which is important for understanding Dumbleton’s solution to the insolubles and so needs to be included in the edition. The chapters on insolubles are followed by two chapters on knowledge and doubt, the whole comprising the first article of Part I, the Summa Logicae. Thus it makes sense to the first article as a whole. In his theory of insolubles, Dumbleton revives a solution much criticised by Bradwardine and others, cassationism, otherwise advocated only in a single treatise from the early 13th century (De Rijk 1966), which claims that insolubles are not in fact propositions at all.

In addition, five of the mss contain five further chapters, one on Insolubles, the others making up a short introduction to supposition theory, obligations and other logical issues, a Summulae as it is often known. (The treatise on Obligations was edited by Kretzman and E.Stump from one manuscript, in ‘The anonymous De Arte Obligatoria in Merton College Ms.306’, in E. P. Bos (ed.), Mediaeval Semantics and Metaphysics, Studies Dedicated to L. M. de Rijk, Ph.D. on the Occasion of his 60th Birthday, Ingenium, Nijmegen, 239–80.) The additional chapters are arguably by Dumbleton himself, or by a follower of his, for the doctrine is consistent with the Summa Logicae itself.

Paul Spade argues (Heytesbury 1979, p.73) that Cajetan’s identification (in his 15th-century commentary on Heytesbury’s Insolubles) of the second view criticised by Heytesbury, and consequently the eighth discussed by Paul of Venice, as Dumbleton’s cannot be right, since Dumbleton’s treatise itself argues against Heytesbury’s view. But this is a weak argument, for Heytesbury, Swyneshed, Dumbleton and others were all working together in Oxford in the 1330s and would have been aware of each others’ ideas and so could easily end up criticising each other.

5. The Source Material

The known manuscripts and early printed texts to be used are as follows:

6. References

  1.    Albert of Saxony. ‘Insolubles’. In The Cambridge Translations of Medieval Philosophical Texts, vol. I: Logic and the Philosophy of Language, trans. N. Kretzmann and E. Stump. Cambridge: Cambridge University Press, 1988, 338-68.
  2.    Albert of Saxony. Logik: Lateinisch-Deutsch (Perutilis Logica), ed. and tr. H. Berger. Hamburg: Meiner 2010.
  3.    Anderson, C.A. 1983. 'The Paradox of the Knower', The Journal of Philosophy 80, 338–355.
  4.    Aristotle, 1938. Categories, On Interpretation, Prior Analytics. (The Loeb Classical Library: Heinemann). Categories and On Interpretation ed. and tr. Harold P. Cooke, Prior Analytics ed. and tr. Hugh Tredennick.
  5.    Aristotle. De Sophisticis Elenchis, Translatio Boethii, Fragmenta Translationis Iacobi et Recensio Guillelmi de Moerbeke (Aristoteles Latinus VI 1-3), ed. B. Dod. Leiden: Brill 1975. 
  6.    Ashworth, E. Jennifer. ‘Paul of Venice on Obligations:  The Sources for both the Logica Magna and the Logica Parva Versions’, in Knowledge and the Sciences in Medieval Philosophy, Vol. 2, ed. Simo Knuuttila et al. Publications of Luther-Agricola Society 1990, 407-415. 
  7.    Bochenski, 1970. History of Formal Logic, translated by Ivo Thomas (Chelsea Pub.Co.), second edition.
  8.    Bos, E.P., 1985. John of Holland: Four Tracts on Logic (Suppositiones, Fallacie, Obligationes, Insolubilia). Artistarium 5 (Ingenium).
  9.    Bottin, F, 1976. Le Antinomie Semantiche nella Logica Medievale (Editrice Antenore).
  10.    Bradwardine, Thomas. Insolubilia. Edition, English translation and Introduction by Stephen Read. (Dallas Medieval Texts and Translation 10.) Leuven: Peeters 2010.
  11.    Buridan, John. 1994. Quaestiones Elencorum, ed. R. van der Lecq and H.A.G. Braakhuis (Ingenium).
  12.    Buridan, John. 2001. Summulae de Dialectica, tr. G. Klima (Yale UP).
  13.    Buridan, John. 2004. Summulae de Practica Sophismatum, ed. F. Pironet (Brepols).
  14.    Conti, Alessandro, ‘Paul of Venice’, The Stanford Encyclopedia of Philosophy. Ed. E. N. Zalta (Summer 2017 Edition)
  15.    De Rijk, Lambertus M. ‘Some Notes on the Mediaeval Tract De insolubilibus, with the Edition of a Tract Dating from the End of the Twelfth Century.’ Vivarium 4 (1966), 83-115.
  16.    De Rijk, Lambertus M. Logica Modernorum: A Contribution to the History of Early Terminist Logic. Vol. 1: On the Twelfth Century Theories of Fallacy. Assen: Van Gorcum 1962.
  17.    De Rijk, Lambertus Marie, 1977. ‘Logica Oxoniensis: an attempt to reconstruct a fifteenth-century Oxford manual of logic’, Medioevo 3, 121-64.
  18.    Field, Hartry. Saving Truth from Paradox. Oxford: Oxford UP 2008.
  19.    Hauser, Kai. ‘Gödel’s program revisited: Part I: The turn to phenomenology’, Bulletin of Symbolic Logic 12 (2006), 529-90.
  20.    Heytesbury, William. On “Insoluble” Sentences: Chapter One of His Rules for Solving Sophisms. Tr. Paul Vincent Spade. “Mediaeval Sources in Translation,” vol. 21. Toronto: Pontifical Institute of Mediaeval Studies, 1979.
  21.    Heytesbury, William, 1987. Insolubilia, in Il Mentitore e il Medioevo, ed. L.Pozzi (Edizioni Zara), 201-57.
  22.    Klima, Gyula, 2009. John Buridan (OUP).   
  23.    Martin, Christopher J. ‘Obligations and Liars.’ In Sophisms in Medieval Logic and Grammar. Ed. Stephen Read. Dordrecht: Kluwer 1993, 357-81; reprinted in Medieval Formal Logic. Ed. M. Yrjönsuuri, Kluwer 2001, 63-94.
  24.    Paul of Venice. Logica Magna. Venice 1499.
  25.    Paul of Venice. Quadratura. Venice 1493. 
  26.    Paul of Venice, 1483. Sophismata Aurea (Pavia, Nicolaus Girardengus, de Novis). [repr. Venice: Bonetus Locatellus, for Octavianus Scotus, 1493] 
  27.    Paul of Venice. Logica Parva. Tr. A.R. Perreiah. Munich/Vienna: Philosophia Verlag 1984.
  28.    Paul of Venice. Logica Parva. Ed. A.R. Perreiah. Leiden: Brill 2002.
  29.    Paul of Venice. Logica Magna, Secunda Pars: Tractatus de Veritate et Falsitate Propositionis et Tractatus de Significato Propositionis. Ed. F. del Punta and tr. M.M. Adams. Oxford UP for the British Academy, 1978.
  30.    Paul of Venice. Logica Magna, Prima Pars: Tractatus de Terminis. Ed. N. Kretzmann. Oxford UP for the British Academy, 1979.
  31.    Paul of Venice. Logica Magna, Prima Pars: Tractatus de Scire et Dubitare. Ed. P. Clarke. Oxford UP for the British Academy, 1981.
  32.    Paul of Venice. Logica Magna, Secunda Pars: Tractatus de Obligationibus. Ed. E.J. Ashworth. Oxford UP for the British Academy, 1988.
  33.    Paul of Venice. Logica Magna, Secunda Pars: Capitula de Conditionali et de Rationali. Ed. G. Hughes. Oxford UP for the British Academy, 1990.
  34.    Paul of Venice. Logica Magna, Secunda Pars: Tractatus de Hypotheticis. Ed. A. Broadie. Oxford UP for the British Academy, 1990.
  35.    Paul of Venice. Logica Magna, Prima Pars: Tractatus de Necessitate et Contingentia Futurorum. Ed. C.J.F. Williams. Oxford UP for the British Academy, 1991. 
  36.    Perreiah, A.R. ‘Insolubilia in the Logica Parva of Paul of Venice.’ Medioevo 4 (1978), 145-71.
  37.    Perreiah, A.R. Paul of Venice: a Bibliographical Guide. Philosophy Documentation Center 1986.
  38.    Peter of Ailly. Concepts and Insolubles: An Annotated Translation. Tr. Paul Vincent Spade. “Synthese Historical Library” vol. 19. Dordrecht: Reidel 1980.
  39.    Peter of Mantua. Logica. Padua 1477.
  40.    Pironet, Fabienne, 1993. ‘John Buridan on the Liar paradox: study of an opinion and chronology of the texts’, in Argumentationstheorie, ed. K. Jacobi (Brill), 293-300.
  41.    Pironet, Fabienne. ‘William Heytesbury and the treatment of Insolubilia in 14th-century England.’ In Unity, Truth and the Liar: The Modern Relevance of Medieval Solutions to the Liar Paradox. Ed. Shahid Rahman et al. Berlin: Springer-Verlag 2008, 255-333.
  42.    Pozzi, Lorenzo. Il Mentitore e il Medioevio. Edizioni Zara 1987. 
  43.    Read, Stephen. ‘The Liar paradox from John Buridan back to Thomas Bradwardine.’ Vivarium 40 (2002), 189-218.
  44.    Read, Stephen, 2014. ‘Concepts and meaning in medieval philosophy’, in Intentionality, edited by Gyula Klima, Fordham University Press, 9-28.
  45.    Read, Stephen and Thakkar, Mark, 2016. ‘Robert Fland, or Elandus Dialecticus?’, Mediaeval Studies 78, 167-80.
  46.    Roure, M.-L., 1970. ‘La problématique des propositions insolubles au XIIIe siècle et au début du XIVe, suivie de l’édition des traités de W.Shyreswood, W. Burleigh et Th. Bradwardine’, Archives d'histoire doctrinale et littéraire du moyen âge 36-37, 205-326.   
  47.    Spade, Paul Vincent, 1971. ‘An anonymous tract on Insolubilia from Ms Vat.Lat.674. An edition and analysis of the text’, Vivarium 9, 1-18.
  48.    Spade, Paul Vincent. The Medieval Liar. Toronto: Pontifical Institute of Medieval Studies 1975.
  49.    Spade, Paul Vincent, 1978. ‘Robert Fland’s Insolubilia: an edition, with comments on the dating of Fland’s works’, Mediaeval Studies 40, 56–80.
  50.    Spade, Paul Vincent. ‘Roger Swyneshed's Insolubilia: Edition and Comments.’ Archives d'histoire doctrinale et littéraire du moyen âge, 46 (1979), 177-220.
  51.    Spade, Paul Vincent, 1983. ‘Roger Swyneshed’s theory of insolubilia: a study of some of his preliminary semantic notions’, in History of Semiotics, ed. A. Eschbach and J. Trabant (John Benjamins), 105-13.
  52.    Spade, Paul Vincent. ‘The manuscripts of William Heytesbury’s Regulae solvendi sophismata: conclusions, notes and descriptions’, Medioevo 15 (1989), 271-314.
  53.    Spade, Paul Vincent and Read, Stephen. ‘Insolubles’, The Stanford Encyclopedia of Philosophy. Ed. E. N. Zalta. (Fall 2018 Edition).
  54.    Strobino, Riccardo. ‘Truth and Paradox in Late XIVth Century Logic: Peter of Mantua's Treatise on Insoluble Propositions.’ Documenti e studi sulla tradizione filosofica medievale, 23 (2012), 475-519.
  55.    Swyneshed, Roger, 1979. Insolubilia, in Spade (1979).
  56.    Swyneshed, Roger, 1987. Insolubilia, in Il Mentitore e il Medioevo, ed. L.Pozzi (Edizioni Zara), 173-99. 
  57.    Zupko, Jack, 2018. ‘John Buridan’, The Stanford Encyclopedia of Philosophy Edward N. Zalta (ed.). (Fall 2018 Edition).

Steering Committee

[Updated 20 May 2020]