‘Everything True Will Be False’: Paul of Venice and a Medieval Yablo Paradox

In his Quadratura, Paul of Venice considers a sophism involving time and tense which appears to show that there is a valid inference which is also invalid. Consider this inference concerning some proposition A : A will signify only that everything true will be false, so A will be false. Call this inference B . A and B are the basis of an insoluble-that is, a Liar-like paradox. Like the sequence of statements in Yablo's paradox, B looks ahead to a moment when A will be false, yet that moment may never come. In the Quadratura, Paul follows the solution to insolubles found in the collection of elementary treatises known as the Logica Oxoniensis, which posits an implicit assertion of its own truth in insolubles like B . However, in the treatise on insolubles in his Logica Magna, Paul develops and endorses a different solution that takes insolubles at face value. We consider how both types of solution apply to A and B : on both, B is valid. But on one, B has true premises and false conclusion, and contradictories can be false together; on the other (following the Logica Oxoniensis), the counterexample is rejected.


A Temporal Paradox
Paul of Venice's Quadratura is not a treatise on squaring the circle or on determining area in any way. Rather, it is an introductory logic text framed around four doubtful questions (dubia), each of which prompts fifty sophisms in a wide variety of philosophical areas, whose resolution allows Paul to present his students with a whole gamut of theories and arguments. The quadrature is a pun, since each of the fifty sophisms provoked by the four doubtful questions is resolved by four conclusions (or theses) and at least as many corollaries. 1 The first question asks whether the same inference can be both valid and invalid. The fifteenth chapter, invoking an insoluble -a Liar-like paradox -as its particular sophismatic puzzle (Capitulum de insolubilibus), argues as follows: CONTACT Stephen Read slr@st-andrews.ac.uk This needs careful analysis. At its heart is a self-referential proposition A, where we assume that A will signify only that everything true will be false. This is the premise of inference (consequentia) B, whose conclusion is that A will be false. Paul argues first that B  Rijk 1982, p.177. The puzzle was not original to Paul: it also appears about 40 years earlier in Richard Ferrybridge's Consequentiae: see Pozzi 1978, 262-71. is invalid and then that B is valid. It follows, he says, that our overall question is true: the same inference (e.g. B) can be both valid and invalid.
Let us first consider Paul's argument that B is valid. He claims that the opposite of its conclusion is incompatible with its premise. The reason is that Paul takes proposition A to be equivalent to 'Everything that is or will be true is false'. This was standard practice according to the medieval doctrine of ampliation, whereby a future-tense copula ampliates its subject from the present to the future. That is, any proposition of the form Every S will be P is equivalent to Everything that is or will be S will be P, and similarly for 'Some S will be P', 'No S will be P' and 'Not every S will be P'. For example, 'Everything white will be black' is true if everything which is now white will be black and everything which will at any future time be white, will (at some future time, before, at or after the other) be black. If anything that is now or at some future time will be white will not also at some future time be black, the proposition 'Everything white will be black' was taken to be false. 3 When A is understood in this way, the proof that B is valid is straightforward. Take the contradictory opposite of its conclusion, viz 'A will not be false'. So assuming A exists, A will be true. 4 But according to the premise, everything that will be true will be false, so A will be false, contradicting the assumption that A will not be false. So as long as A will exist, and as part of B it does exist, the premise of B is incompatible with the contradictory of its conclusion, so B is valid. This is not exactly how Paul presents the argument, but formulating the reasoning in this way brings out how remarkably similar it is to the reasoning in Yablo's paradox. Yablo's paradox explicitly invokes a sequence of propositions, each referring to all subsequent propositions in sequence: 5 S 1 : for all k > 1, S k is untrue S 2 : for all k > 2, S k is untrue · · · Suppose that for some n, S n is true. Then for all k > n, S k is untrue, in particular, S n+1 is untrue and for all k > n + 1, S k is untrue. But the second conjunct (i.e. 'for all k > n + 1, S k is untrue') is what S n+1 says, so S n+1 is (also) true. Accordingly, there can be no n such that S n is true, that is, for all n, S n is untrue. But Yablo points out that if every S n is untrue, then 'the sentences subsequent to any given S n are all untrue, whence S n is true after all ' (p. 252) Contradiction.
Yablo's intermediate conclusion that every S n is untrue matches Paul's intermediate conclusion above that given that everything true will be false, A will be false and so B is valid. So far, so good. But then Paul, like Yablo, realises that this conclusion will lead to paradox. For there is a counterexample to the validity of B: suppose that as long as A exists, it will continue to signify that everything true will be false, and that in fact it will continue to be the case that everything true will be false. Then things will be as A signifies, so A will continue to be true, and consequently B's conclusion will be false even though its premise (stating how A signifies) will at the same time be true. So B is invalid. Paradox.
Paul envisages A itself continuing to exist, whereas Yablo describes a succession of distinct propositions S n . The similarity between the two paradoxes can be made greater by considering successive utterances of A in Paul's sophism, each assumed to signify that everything true will be false, especially since, as noted above, the medievals considered propositions to be distinct tokens. Moreover, note that Paul described the first syllogism he mentions as an instance of Baroco, so he thinks of 'A will not be false' as an O-proposition, that is, a particular negative, and so he takes A to be a general term. Then each utterance, or instance, of A closely matches S n for some n.

Theories of Insolubles From Bradwardine to Paul of Venice
Paul's Quadratura is preserved in three manuscripts and an incunabulum of 1493 (Venetus 1493). The Vatican manuscripts (Vat.Lat.2133 and 2134) have a colophon to the fourth doubt, which reads: Here end the sophistical Determinations with their tables composed by me, brother Paul of Venice of the Order of the Brother Hermits of St Augustine while I was teacher in the Convent at Padua and Bachelor of the same most Holy Order. 6 The Determinations were an exercise which Paul completed for his Magister Artium at Padua between October 1399 and July 1400, after which he enrolled as a Bachelor of Theology. 7 In the Quadratura, he draws on the solution to the insolubles which he had presented in his Logica Parva, composed shortly after his return to Italy from three years' study at the Augustinian Convent in Oxford, a solution derived from one he describes in his Logica Magna as that 'which is now generally maintained by everyone '. 8 But it is different from his favoured solution in the Logica Magna. We need to look back to Thomas Bradwardine's iconoclastic solution presented in his Insolubilia in the early 1320s to trace the origins of these three solutions.
Bradwardine's revolutionary idea was that insolubles, indeed, all propositions, might mean (denotare) or signify (significare) more than is immediate at first glance, or as Heytesbury would put it ten years later, more than the words commonly suggest (verba communiter pretendunt). Bradwardine proposed a principle to govern this multiplicity of overt and hidden meanings, his second postulate: every proposition signifies or means anything which follows from anything it signifies or means. 9 Then he was able to provide a rather clever and subtle proof that any proposition which signifies its own falsity also signifies its own truth. Since what seems to be characteristic of many insolubles is that they signify their own falsity, it follows that they are implicitly contradictory in signifying both (overtly) that they are false and (covertly) that they are true. So if truth requires, as seems most plausible if one thinks of multiple meanings as being essentially conjunctive, that everything a proposition signifies must obtain, it is impossible for everything these insolubles signify to obtain, and so something they signify fails to obtain and they are all false. Moreover, although they are false and they signify that they are false, it does not follow (as the standard argument goes) that they are true, for their being false is only part of what they signify, so although one might say they are partly true (true secundum quid), they are also partly false, and so are as a whole (simpliciter) false.
Most of Bradwardine's successors, in a flurry of treatments of the insolubles in the fourteenth century, took up the idea that there might be hidden, additional, meanings to propositions, but few were willing to endorse Bradwardine's second postulate and the proof using it to show that insolubles also signify their own truth. One of those rare followers was Ralph Strode, who wrote in his Insolubilia (see Spade 1975, item LIII, pp.87-91): [. . . ] the earlier generation [. . . ] correctly understood little or nothing about insolubles. After them arose the prince of modern natural philosophers, namely, master Thomas Bradwardine, who was the first to come across something of value about insolubles, for which reason his opinion deserves to be quoted more extensively for the use of the young. So this reverend doctor precedes the plainer description of Aristotle's opinion with first some divisions, definitions, assumptions and conclusions, from which everything which follows shines more clearly. 10 But Bradwardine's fellow Calculator, John Dumbleton, observed that Bradwardine's postulate entails the absurdity that every impossible proposition signifies everything and that any proposition whatever signifies everything necessary, by the principles that from an impossibility anything follows and that every necessary truth is entailed by anything: When in certain treatises it is said that every proposition signifies whatever follows from it given how things are or no matter how they are, this should not be maintained wholly or universally without qualification, since there are some necessary inferences whose conclusions do not signify as their premises do . And there are other formal inferences whose conclusions do not mean the same as their premises [. . . ] For the first case : take this inference, ' Some man is an ass, therefore some man is a goat', which is said to be necessary since it cannot be that some man is an ass unless he is a goat, so the inference is necessary, although there is no necessary relation between the premise and the conclusion. For the second case : take this inference in the mind, 'Whiteness exists, therefore a first cause exists'. This is formally valid , understanding only whiteness by the subject of the premise, not relating it with any cause of it, because its premise does not signify in any way about any thing other than what is signified by its subject, and its subject does not signify a first cause in any way, since then 9 Bradwardine 2010, § 6.3: Secunda est ista: quelibet propositio significat sive denotat ut nunc vel simpliciter omne quod sequitur ad istam ut nunc vel simpliciter. On the justification for interpreting this as a closure condition, see Bradwardine 2010, 'Introduction' § 5, p.17 any intention would signify a first cause. For this reason, it is clear that the premise signifies in a way in which the conclusion does not signify. 11 Another of Bradwardine's fellow Calculators, William Heytesbury, 12 notoriously suggested that if someone presented an apparent insoluble adding that what it appeared to signify is all it signifies, one should reject it outright, whereas if it was presented without that stipulation, it should be accepted but that it is true should be denied, on the grounds that it must have some hidden meaning whch failed to obtain. 13 Heytesbury was able to act in this seemingly cavalier way because he framed his solution in the language of obligations, whereby the Respondent, to whom he was offering this advice, was only allowed to accept or reject the initial obligation (or positum) presented to him by his Opponent, and (once accepted) to grant, deny or doubt subsequent propositions which the Opponent proposed. 14 In particular, when challenged as to what this hidden meaning might be, on which the whole success of the solution turned, Heytesbury could invoke the framework of obligations theory to say that the Respondent was under no obligation to specify what it might be, but only to respond by granting or denying. 15 He was thus able to tread the narrow but consistent line of granting the insoluble but denying that it was true.
Unsurprisingly, many subsequent writers were frustrated by Heytesbury's caution, though they were happy to adopt his framework of obligations theory. They adapted his solution to specify that the hidden meaning was in fact an assertion of the truth of the insoluble. That claim is inconsistent with its overt meaning, and accordingly the insoluble is granted but its truth is denied and its falsehood granted. This solution became very popular, at least in Oxford, and was incorporated in most of the Oxford logic textbooks of the late fourteenth century, which were collectively dubbed the Logica Oxoniensis in De Rijk 1977. Among later proponents were John of Holland and John Hunter (aka Venator). 16 We might call it the 'modified Heytesbury solution'. A possible link between Heytesbury's and these later treatises is that of Ralph Strode, whose solution is explicitly based on combining Bradwardine's and Heytesbury's. His treatise was composed in Oxford, probably in the late 1350s, and so before Holland's and Hunter's treatises. 17 Strode writes (f.10va): 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, 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 'There is a falsehood' is the only proposition, it is impossible that it only signifies that there is a falsehood. But in so far as it is claimed that, in the given scenario, it need not be specified 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. 18 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' (Sortes dicit falsum), labelled 'A', he writes: Regarding the solution to this insoluble it should be realised that close attention should 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. 19 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. 20 It is this modified Heytesbury solution which Paul presents in the chapter on insolubles in his Logica Parva. It is explicitly directed at students, and does not necessarily represent his own view. He writes at the end of the chapter: Similar solutions attributing an additional signification, or something similar, to insolubles were offered by John Buridan, Albert of Saxony, Gregory of Rimini, Peter of Ailly, Marsilius of Inghen and others. But some were unpersuaded. 22 Notable among them was Roger Swyneshed, another Calculator, writing in Oxford in the 1330s. His aim was to find a viable solution to the insolubles by taking them at face value, and his big idea was that insolubles falsify themselves -in an intuitive sense which he set out to make formal and precise. That is, for Swyneshed, the interesting characteristic of insolubles is that they imply their own falsehood. Indeed, that's usually the first leg of a proof of contradiction from them: first, we show that they are false, then feel forced to infer that they must also be true, since that's what they say. Swyneshed avoids this second leg of the paradox argument by broadening the definition of 'false': a proposition is false (he says) if either things are not as it signifies (in the normal communiter pretendunt sense of 'signifies') or they falsify themselves (in the sense that they imply their own falsehood). 23 For example, 'This proposition is false' falsifies itself because from 'This proposition is false' we can immediately infer that it is false; 'Every proposition is false' falsifies itself in the sense that it implies that it itself, being a proposition, is false; 'What Socrates says is false' falsifies itself if it is the only proposition uttered by Socrates, since we can then infer that it is itself false. In general, a proposition is true if and only if things are as it signifies and it does not falsify itself. 24 So, given that 'This proposition is false' is false, since it falsifies itself, we cannot infer that it is true (on the grounds that things are as it signifies) since it does not meet the extra condition of not falsifying itself.

Paul's Two Solutions to the Temporal Paradox
As we noted in Section 2, Paul offers different solutions to the insolubles in different works. In the Logica Parva and the Quadratura, the solution he favours is the modified Heytesbury solution; in the Logica Magna and the Sophismata Aurea, it is Swyneshed's.
That Paul applies the modified Heytesbury solution to the insolubles in the Quadratura is clear from the second and third Conclusions which he sets out in preparing his response to the temporal insoluble we considered in Section 1. The second Conclusion states: There is some proposition 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. 25 As proof, Paul claims that 'Every proposition is false' satisfies this claim, assuming that at some time it will be the only proposition, for its principal (that is, total) signification is (now) purely predicatively that every proposition is false. But, he says, [. . . ] 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. 26 The point is reiterated and elaborated in discussing his third Conclusion, namely: It is possible for every proposition to be false and for 'Every proposition is false' to signify exactly that every proposition is false. 27 Again, the relevant scenario is one where 'Every proposition is false' (call it A) is the only proposition. Then, he says, I claim that in this scenario A signifies that every proposition is false and that A is true. This conjunctive significate is called the principal significate of A, although it is not the exact significate , which is only the first part. 28 That is, when A is the only proposition, it signifies conjunctively and principally that every proposition is false and that A is true, but its exact significate is that every proposition is false, as the third Conclusion claims. This response is clearly very different from Swyneshed's solution and belongs to the tradition started by Bradwardine and continued by Heytesbury where an insoluble has a further implicit signification. But unlike Heytesbury himself, Paul commits himself squarely to the claim that the additional signification is that the insoluble itself is true, in the way we have seen that the modified Heytesbury solution does. Paul's response to the temporal insoluble is to accept that B is valid, and to deny that there is any scenario in which its premise is true and conclusion false, as was claimed in the sophism. In particular, the scenario described in the sophism itself is impossible and so fails to show that B is invalid. Recall that the argument was that we could 'assume that as long as A will exist, A will signify only that everything true will be false, and that it will be the case that everything true will be false as long as A will exist '. It follows, it was claimed, that 'on this assumption, the premise of B is true according to the scenario '. Not so. For if A is indeed an insoluble, it will not signify only that everything true will be false, that is, what it standardly signifies, but it will also signify that it itself is true, from which it will follow that A is false.
So is A an insoluble? In his Logica Parva (where, recall, he also endorses the modified Heytesbury solution), Paul defines an insoluble as 'a proposition signifying consequentially (assertive significans) its own falsehood ', 29 later distinguishing insolubles unconditionally (insolubile simpliciter) from insolubles conditionally (secundum quid): An insoluble unconditionally is one to which a scenario is attached which implies a contradiction if admitted [. . . ] An insoluble conditionally is one to which a scenario is attached which does not imply a contradiction if admitted. 30 He gives us an example, in which it is assumed that the proposition 'No proposition is true' is the only proposition and signifies only as the terms suggest. A contradiction follows, so in this scenario, 'No proposition is true' is an insoluble and signifies both as its terms suggest (and hence that it is not true) and that it is true, and so the scenario is impossible and 'No proposition is true' is false assuming it is the only proposition. Similarly, in the supposed counter-example in the Quadratura: if A signifies only that everything true will be false, and everything true will in fact be false (so things are as A signifies, so A is true), a contradiction will follow in that B will be both valid and invalid. So things cannot be as A signifies, whence A is false and A signifies consequentially its own falsehood, and is thus an insoluble. Accordingly, A must have a further implicit signification, namely of its own truth, and its falsehood is explained. Indeed, we can read this from B itself, that is, 'A will signify only that everything true will be false, therefore A will be false'. For if A is ever true, it will follow by contraposition, given that B is valid, that A does not signify only that everything true will be false, but must have some further signification, namely, that it itself is true. So the proposed scenario is impossible. Paul concludes this chapter of his Quadratura with the words: From this it is clear how to respond to the original argument, namely by granting this inference: 'A will signify only that everything true will be false, therefore A will be false', and as for the counter-instance, I do not accept the scenario, because it implies a contradiction, as has been clearly seen. Hence etc. 31 leaving his readers to put the pieces together. The upshot is that B is valid, and the counterexample is rejected, for if A is true and everything true will be false, A will be an insoluble, and so will not signify only that everything true will be false, and thus the premise of B will be false.

Paul's Preferred Solution
In his Logica Magna, however, Paul rejects Heytesbury's solution, and passes over the modified version in silence. That is odd, since his main criticism there of Heytesbury's solution is its reluctance to specify what the implicit signification of insolubles is. 32 In any case, having rejected Heytesbury's, together with fourteen other putative solutions, Paul adopts and adapts Swyneshed's solution and applies it at length to a range of insolubles. The temporal paradox is, however, not among them, so it is an interesting exercise to see how Swyneshed's, and Paul's, solution § 1.12.3.1.2-1.12.3.2.3. 33 Swyneshed's type of solution was applied to the temporal paradox a century later in Lax 1512, ff.147r-148r and Jean de Celaya's Insolubilia (Roure 1962, pp.277-8).
To start with, the definition of insoluble in the Logica Magna is a little different, and indeed, comes in two forms, a narrower and a broader one. Paul gives the narrower one in the second chapter of the treatise on insolubles in the Logica Magna: An insoluble proposition is a proposition having reflection on itself wholly or partially implying its own falsity or that it is not itself true. 34 Paul comments that his definition excludes many propositions counted as insolubles by others, such as 'Socrates will not cross the bridge' and 'Plato will not have a penny', for he says, they do not have reflection on themselves. But he is not consistent here, for in the fifth chapter he includes them under what he calls 'insolubles that don't appear at first glance to be insolubles' (insolubilia que prima facie insolubilia non apparent). It is in the eighth chapter that he comes to further cases that he believes only appear to be insolubles, such as 'This proposition is not known to you' and 'This is in doubt for you', which others would include as epistemic insolubles. 35 Swyneshed had himself given a broader definition which included these epistemic paradoxes: An insoluble as put forward is a proposition signifying principally as things are or other than things are which is relevant to inferring itself to be false or unknown or not believed, and so on. 36 Paul himself is tempted to broaden his definition to include the epistemic insolubles, when, for example, he presents the fourth Conclusion in the Logica Magna: There is a formally valid inference, known by you to be so, signifying exactly by the composition of its parts, where the premise is known by you, yet the conclusion is not known by you. 37 The example he gives is what may be called the Inferential Knower Paradox: 38 This is unknown to you, therefore this is unknown to you where each occurrence of 'this' refers to the conclusion. For, he says, 'the premise is known by you, because you know that the conclusion is not known, since it is an insoluble that implies that it itself is unknown. ' 39 Thus the idea in the broadening of the definition is to say that just as propositions which imply their own falsehood are self-falsifying and so are false, so too propositions which imply they are not known are not known and those which imply they are not believed are not believed, and so on.
The fourth Conclusion, in its denial of logical omniscience, may seem attractive. It does indeed seem true that we can know the axioms of some theory, and its rules of inference, but not know all the consequences of those axioms. But that is more a matter of psychology 34 Venetus 2022, § 2.1.8: Propositio insolubilis est propositio habens supra se reflexionem sue falsitatis aut se non esse veram, totaliter vel partialiter illativa. 35 See, e.g. Bradwardine 2010, ch.9 and Swyneshed 1979, §IV. 36 Swyneshed 1979 than what lies behind the fourth Conclusion, which is a matter of logic. And that Conclusion is nothing like as dramatic as Paul's fifth Conclusion in the Logica Magna, which seems to undermine the whole idea of proof: There are some formally valid inferences which signify exactly by the composition of their principal parts, where the premise is true and the conclusion false. 40 This was Swyneshed's second iconoclastic Conclusion, and the example is the same: 41 (*) This is false, therefore this is false, where each occurrence of 'this' refers to the conclusion. For the conclusion falsifies itself, and the premise truly records this fact. When, in the chapter on 'Consequence' in the Logica Magna, Paul mentioned the Inferential Knower paradox, he anticipated the fourth Conclusion about insolubles, by including an important caveat in his Ninth Rule: Suppose that a certain inference is valid, is known by you to be valid, is understood by you, and signifies primarily in accordance with the composition of its elements; suppose too that its premise is known by you, and that you know that what is false does not follow from anything that is true; then its conclusion is also known by you. 42 That caveat ('you know that what is false does not follow from anything that is true') may have seemed anodyne at the time, but it means he can alert us to expect 'more about this when we come to deal with the insolubles '. 43 But he was not so careful when he stated his Third Rule in the chapter on 'Consequence': If the premise of a valid inference which signifies primarily in accordance with the composition of its parts is true, then the conclusion is also true. 44 (*) is a counterexample. As we have seen, the premise is true, the conclusion false. Moreover, the inference is valid, since the opposite of the conclusion is incompatible with the premise. Indeed, for Paul, it is formally valid, as he stated in the fifth Conclusion, for, he says, who would claim that these are compatible, 'This is false' and 'This is not false', referring to the same thing? Surely, no-one who wishes to avoid a greater absurdity. 45 If proof consists in validly inferring a conclusion successively from premises already proved, proof now fails, for if Paul is right, it allows us validly to infer falsehoods from truths.
In the case of our temporal paradox, proposition A does not imply its own present falsehood, as we have seen, but it does imply its own future falsehood. For suppose everything true will be false, and suppose that A will not be false. Since everything that is or will be true will be false, A is not true and will not be true, and so is and will be false. Contradiction. Hence A will be false. Thus, assuming that everything true will be false, A implies its own future falsehood. That is, if things are as A signifies, it will be false. Taking 'insoluble' in the narrow sense, A is not strictly an insoluble, but following Paul's practice in the chapter of Logica Magna on merely apparent insolubles (that is, which appear to be insolubles but are not), Paul's response would be that, just as 'each proposition asserting that it itself is unknown is not known', 46 and 'a proposition asserting that it itself should be denied should be denied ', 47 so too A, asserting that it will itself be false, will be false.
Alternatively, Paul could follow Swyneshed, as he does in chapter 5 ('On Propositions which are not Obviously Insolubles'), and include propositions like A which assert of themselves that they will be false as insolubles. For he there includes the example where all those who speak truly will receive a penny and Socrates says 'I will not receive a penny' as insolubles. He writes: I grant that Socrates will not receive a penny and consequently that he says a falsehood. And then in reply to the argument [therefore it is not true that he will not receive a penny, and consequently, he will receive a penny], I deny the inference, since one should add [ Paul's claim is that Socrates' statement that he will not receive a penny is false not because he will receive a penny, but because it falsifies itself. And the proof that it falsifies itself is in the paradox. Assuming that Socrates will not receive a penny (that is, what Socrates said), it follows that he will, and so what he said was false.
Similarly, 'Everything true will be false' is false not necessarily because something true will not be false (though indeed, that would falsify it) but even when everything true will be false it will be false because it falsifies itself. Thus, whether we class the temporal paradox as an insoluble or not, it will be false if things are as it signifies; while if things are not as it signifies, then it is false. So either it is false, or it will be false. Thus Paul offers two diagnoses of the sophism: in the Quadratura, following the modified Heytesbury solution, his answer is that B is valid and the purported counterexample fails since if B's conclusion were false, A would be an insoluble and so it would not signify only that everything true will be false, and accordingly B's premise would also be false; while, though Paul does not discuss the temporal sophism in the Logica Magna or the Sophismata Aurea, his own solution, following Swyneshed's, lets us choose whether to include the sophism as an insoluble or not. But whichever we do, B is valid, since the contradictory of its conclusion is incompatible with its premise, even though its premise may be true and conclusion false.

Conclusion
Paul of Venice, writing at the end of the fourteenth century, presented two different solutions to the insolubles in his Logica Parva and Logica Magna. Between them, they serve to illustrate the major lines of approach to insolubles in the preceding century. Those two lines of approach divide between those, following Bradwardine, who aim to solve the insolubles by postulating a hidden, additional signification in insolubles; and those who attempt to solve them while simply taking them at face value. The former include Heytesbury, Albert of Saxony, John Buridan, Gregory of Rimini, Pierre d'Ailly, John of Holland, John Hunter and many others. At Oxford, Heytesbury simplified Bradwardine's approach by merely postulating an additional signification, but denying that we need to speculate what it is. This led to the 'modified Heytesbury solution ', combining Heytesbury's incorporation of the theory of obligations with Bradwardine's claim that the hidden signification is of the insoluble's own truth. Among those rejecting Bradwardine's approach was Roger Swyneshed, who claimed that insolubles falsify themselves and so are false. Swyneshed's solution has three dramatic and themselves paradoxical consequences, including the claim that pairs of contradictories can both be false. 49 Paul presented his students with an intriguing sophism in his Quadratura, threatening contradiction in classing a certain inference, B, as both valid and invalid. Assuming some proposition A will signify only that everything true will be false, the conclusion of B infers that A will be false. Like the sequence of statements in Yablo's paradox, B looks ahead to a moment when A will be false, yet that moment may never come. In the Quadratura, Paul solved it in line with the modified Heytesbury solution, claiming that at the moment when A becomes false its signification will change and so the premise of the inference will be false too. The sophism can also be solved by his preferred approach in the Logica Magna, following Swyneshed's lead, retaining the univocality of A but on pain of accepting that a valid inference can have true premises and false conclusion.