The Rule of Contradictory Pairs, Insolubles and Validity

The Oxford Calculator Roger Swyneshed put forward three provocative claims in his treatise on insolubles, written in the early 1330s, of which the second states that there is a formally valid inference with true premises and false conclusion. His example deployed the Liar paradox as the conclusion of the inference: ‘The conclusion of this inference is false, so this conclusion is false’. His account of insolubles supported his claim that the conclusion is false, and so the premise, referring to the conclusion, would seem to be true. But what is his account of validity that can allow true premises to lead to a false conclusion? This article considers Roger’s own account, as well as that of Paul of Venice, writing some sixty years later, whose account of the truth and falsehood of insolubles followed Roger’s closely. Paul endorsed Roger’s three claims. But their accounts of validity were different. The question is whether these accounts are coherent and support Paul’s claim in his Logica Magna that he endorsed all the normal


The Rule of Contradictory Pairs
In his treatise on insolubles, that is, the logical paradoxes, written in Oxford in the early 1330s, Roger Swyneshed, one of the Oxford Calculators, put forward three surprising, even paradoxical, claims:1 1.
There is a false proposition which principally signifies as things are 2. There is a formally valid inference with true premises and false conclusion 3. There is a pair of contradictory propositions both of which are false.2 The third claim, (3), if correct, is a counter-instance to the Rule of Contradictory Pairs (RCP), that of any pair of mutually contradictory propositions, one is true and the other false.
I have discussed Swyneshed's rejection of the RCP elsewhere, and whether it is entailed by Aristotle's account of contradictories in his De interpretatione.3 RCP follows immediately from Boethius' definition of 'contradictory' in his Second Commentary on Aristotle's De interpretatione: "Contradiction is then the opposition of an affirmation and a denial in which neither can both be false nor both true, but one is always true the other false."4 But that is not how Aristotle defined contradictories. He did so syntactically, contrasting affirmations with denials: Since it is possible to state both of what is that it is not, of what is not that it is, of what is that it is and of what is not that it is not and also in the times which are outside the present, it must similarly be possible to deny whatever anyone has affirmed and to affirm whatever anyone has denied. And so it is clear that for every affirmation there is an opposite denial and for every denial there is an opposite affirmation. Let us call this a contradiction when an affirmation and denial are opposed.5 1 For biographical information on Swyneshed, see J. Weisheipl, "Roger Swyneshed OSB, Logician, Natural Philosopher, and Theologian," in Studies Presented to Daniel Callus (Oxford, 1964), 231-252. 2 Note that throughout this article, I will use 'proposition' to refer to what the medievals referred to as 'propositiones' , that is, concrete token sentences, whether spoken, written or mental. 3 See S. Read Whitaker, in the study where he coined the label Rule of Contradictory Pairs,6 argues that not only is Aristotle's syntactic definition not equivalent to the semantic one, but that Aristotle realised this and presented counterinstances to RCP in chapters 7-9 of De interpretatione, in chapter 7 cases where a pair of contradictories are both true, in chapter 8 cases where they are both false, and in chapter 9 cases where they are neither -or at least, not definitely either. Aristotle writes at the end of chapter 9: "Clearly, then, it is not necessary that of every affirmation and opposite denial one should be true and the other false. For what holds for things that are does not hold for things that are not but may possibly be or not be; with these it is as we have said."7 Quite what Aristotle's claim about future contingents was has puzzled philosophers for millennia. His claims in chapter 7-8 that contradictories can both be true or both be false, however, have for the most part been overlooked, or reinterpreted as remarks about contraries or subcontraries, not contradictories, as the text has it. For example, Thomas Aquinas notes that Aristotle wrote, when summarising chapter 7: We have explained, then, that a single affirmation has a single denial as its contradictory opposite, and which these are; that contrary statements are different, and which these are; and that not all contradictory pairs are true or false, why this is, and when they are true or false.8 Aquinas's comment is: He also says here that it has been shown that not every contradiction is true or false, 'contradiction' being taken here broadly for any kind of opposition of affirmation and negation; for in enunciations that are truly contradictory one is always true and the other false. In other words, he corrects what Aristotle says to ensure that the Rule of Contradictory Pairs is maintained.

Swyneshed on Insolubles
It is not surprising, therefore, that Roger Swyneshed ran into fierce opposition when he claimed that some pairs of contradictories are both false. Ralph Strode protested that Roger's solution to the Liar paradox turns on two notions. First, a proposition falsifies itself if it entails its own falsehood.12 This is clearly true of 'This is false' , he says, for from 'This is false' , 'This is false' immediately follows, provided 'this' refers to the same proposition, in this case, to the premise.13 Accordingly, Roger tightens up the definition of truth to require of true propositions that not only should they signify as things are, but in addition, they should not falsify themselves. Then a false proposition is one that either does not signify as things are or, even if it does signify as things are, falsifies itself. All self-falsifying propositions, like the self-referential 'This is false' , are false, though not all selfverifying propositions, that is, those which entail their own truth, are truethink of 'Every proposition is true' , for example, which entails its own truth, but is clearly falsified by every false proposition.
'This is false' also serves to establish Roger's first iconoclastic claim, that some false proposition signifies as things are. For 'This is false' signifies that it is false, and it is, he says, false (since it falsifies itself), so it is a false proposition that signifies as things are. It also provides a neat example to establish his second claim, that there is a formally valid argument with true premises and false conclusion: "This inference is formally valid (bona et formalis): 'The conclusion of this inference is false, so the conclusion is false' ."14 Call it ( †). The conclusion is an insoluble, falsifying itself, and the premise truly and correctly says that it is false. So the premise is true and the conclusion false. Nonetheless, ( †) appears to be a transparent case of validity, in that the conclusion merely repeats 12 See Rogerus Swyneshed, Insolubilia, §17 (ed. Spade, 186): "Next there follow eight basic principles (suppositiones), of which the first is this: every proposition relevant to inferring itself to be false is one falsifying itself. This is clear from the intention of the Philosopher in the fourth book of the Metaphysics, where he claims that some utterances destroy themselves. And it is plain there that the only utterances that he takes to destroy themselves are utterances relevant to inferring themselves to be false, which is signified by this phrase 'falsifying itself' . And it sounds better in the Latin language to say that such a proposition falsifies itself than destroys itself in that no utterance is destructive of itself nor of anything other than itself. But it sounds better in the Greek language to say that such a proposition destroys itself than falsifies itself. And so the translator uses the verb 'destroy itself' where we use 'falsify itself' in that it applies more truly to such a proposition. Roger's text reveals little about how Roger understands 'signify' , though he usually accompanies it by the adverb 'principally' . As Spade observes,16 nothing seems to be lost by ignoring the qualification, but it does serve to connect Roger's terminology with comments by his contemporaries and critics, such as William Heytesbury. As we will see in §5, Roger's solution to the insolubles was taken up by Paul of Venice (though with some changes). As Paul makes clear, the motivation for Roger's solution is to avoid the postulation, by Thomas Bradwardine and Heytesbury, among others, of an additional signification beyond what Heytesbury characterises as "what the words usually mean" or "as the words commonly suggest" (sicut verba communiter pretendunt).17 Roger's proposal is to take the terms in insolubles at face value and amend the account of truth and falsehood, rather than propose hidden significations in a seemingly ad hoc manner, as Heytesbury does.
Indeed, there is a common approach in Bradwardine's, Heytesbury's, and Roger's solutions to the insolubles. The contradiction which appears to ensue from admitting self-reference is internalised in such a way as to render the insoluble false and so disclaim anything that might follow. This is literally so in Bradwardine's and Heytesbury's case: they both impute a contradictory signification to the insolubles that renders them false.18 But a similar phenomenon lies at the heart of Roger's account: the ensuing contradiction shows that the insoluble cannot be true, so it falsifies itself, and accordingly is false. 15 For example, G. Klima, "Consequences of a Closed, Token- One might pause, however, when Roger says that ( †) is formally valid. What is the account of validity which he appeals to here? In this case, the conclusion says the same thing about the same thing as the premise does. But not all valid inference steps are instances of identity. What is the general account which allows a valid inference to fail to preserve truth? It cannot be truthpreservation, often taken as a criterion of validity in both medieval and modern times,19 since Roger claims that ( †) fails to preserve truth. Does Roger have an account of validity which explains how such inferences can be valid but fail to preserve truth?

Validity
Among medievals, Roger was not alone in this rejection of the truthpreservation criterion. We find a similar rejection in, e.g., John Buridan, Albert of Saxony and Pseudo-Scotus -though the much-repeated counterexample seems to have originated with Buridan in his early Questions on the Prior Analytics.20 Consider the inference (*) Every proposition is affirmative, so no proposition is negative.
This seems to be valid, given the meanings of 'affirmative' and 'negative' , as is the contrapositive inference Some proposition is negative, so not every proposition is affirmative.
But whereas this second inference preserves truth, the first does not, Buridan observes. For the premise can be true (if, for example, God has annihilated all negative propositions), but the conclusion cannot, for it falsifies itself in the non-Swyneshedian but natural sense that it is a counter-instance to itself.

Read
Vivarium 58 (2020) 275-304 itself a negative proposition, but it must exist in order to be true or false, yet if it does exist, it is false. Nonetheless, Buridan says, the rule of Contraposition "is common to every valid consequence."21 So validity of inference cannot be a matter of preserving truth.22 Example (*) is not, for Buridan, formally valid, which he defines as validity in all (non-logical, that is, categorematic) terms. Nor are any of the above inferences syllogistic inferences, for a syllogism requires (at least) two premises. However, consider the following syllogism in the second figure: Every true proposition is a premise Not every conclusion is a premise So not every conclusion is true and suppose that God has annihilated all propositions other than these three. Then the premises are true (the first, since the premises are the only true propositions) and the conclusion is false because it falsifies itself by Roger's criterion -it entails its own falsehood, since it is the only conclusion. So the premises of this syllogism can be true and the conclusion false, yet it is a formally valid syllogism in Baroco.
However, as Klima notes,23 the Parisian account of validity of consequence represented by Buridan and others was radically different from the English tradition. Moreover, in both cases, formal consequence was a special case of validity in general.24 So the Parisian definition of formal consequence as truthpreservation under substitution of all non-logical terms was a definition of 'formality' , not of validity. Where truth was preserved but not formally, the inference was said to be materially valid.25 Validity itself was truth-preservation. So too in the English tradition: in Ockham, for example, we find many divisions of consequence, but what is common to them is that in a valid consequence 21 Iohannes Buridan, Summulae de dialectica, "Sophismata" c. It must be known that for the correctness of a conditional no more is required than that if the antecedent is true, the consequent is true … Some conditionals hold as-of-now and others simply … and there is consequence as-of-now when the premise cannot be true as-of-now without the conclusion.27 Burley proceeds to distinguish what he called natural from accidental consequences, in the way later used to distinguish formal from material consequence:28 Simple consequences are divided like this: some are natural and others accidental. Natural consequence is when the conclusion is understood in the premises (de intellectu antecedentis), nor can the premises be true unless the conclusion is true, e.g., 'If it is a man it is an animal' . Accidental consequences are of two sorts: some hold on account of the terms or on account of the matter, e.g., 'It is true that God exists, so it is necessary that God exists' , which holds on account of the terms or the matter, for in God truth and necessity are the same. Other accidental consequences are 'From the impossible anything follows' and 'The necessary follows from anything' … Accidental consequence is when the conclusion is not understood in the premises.
Weber shows how the phrases intelligitur in antecedente and de ( formali) intellectu antecedentis pervade English treatments of consequences in the The contrast between the Parisian and English notions of formal consequence is not as extreme as it may seem. Indeed, to contrast formal validity as "truth-preservation under all substitutions of non-logical terms" with "a containment principle … requiring that the understanding of the antecedent should contain the understanding of the consequent,"31 as Klima and others do, is to some extent misleading. The containment principle explains why truth is preserved (when it is), whereas (despite the arguments given by Tarski32) truthpreservation under substitution does not.33 Rather, in both cases, validity is ensured by the connection of meaning between premises and conclusion, in the Parisian case the meanings of the logical, or syncategorematic, terms alone, in the English case, all the constituent terms. Both accounts give a reason why the consequence is valid, though the English account of formal consequence includes more consequences, with ad impossibile sequitur quodlibet and necessarium sequitur ad quodlibet being for the most part the only inferences classed as materially valid in the English tradition, where the understanding of the conclusion is notoriously not contained in the understanding of the premises.34

Firmness
But what of Roger? Weisheipl records a reference in a fourteenth-century commentary on William Sutton's treatise on consequence to a treatise on consequences by Roger,35 but this treatise seems never to have been identified, although it may still exist.  34 We will return to the matter of formality in §6. 35 Weisheipl, "Roger Swyneshed OSB," 245.
to truth-preservation as the account of validity is preservation of "signifying as is the case" or signifying as things are (significans sicut est) -Spade invents the neologism 'firm' for the property of signifying as things are.36 He infers this37 from Roger's response to the objection that, since Roger's opinion "claims that some false proposition principally signifies as things are, then for the same reason it should claim that some true proposition signifies other than things are."38 Roger countered: The inference is not valid. But the premise [that every proposition falsifying itself is false] is true. And even though, if from some propositions each of which signifies principally as things are, some proposition follows, it signifies as things are, however, if from some propositions one of which signifies other than things are and all others as things are, some proposition follows, it does not follow that it signifies as things are.39 Here Roger certainly seems to equate validity with preservation of signifying as things are (significans sicut est) -firmness-preservation, we might say -though it does literally only give firmness-preservation as a necessary condition of validity. Note that ( †) preserves firmness, even though it does not preserve truth: the premise and the conclusion both signify that the conclusion is false, which it is. A similar move was made by John Buridan. Buridan was responding to the sophism (*) mentioned above: 'Every proposition is affirmative, so no proposition is negative' . In the fifth conclusion in chapter 8 of his Sophismata (repeating similar remarks in his Treatise on Consequences), Buridan wrote: The fifth conclusion is that for the validity of a consequence it does not suffice for it to be impossible for the antecedent to be true without the consequent, if they are formed together … Therefore, something more is required, namely, that things cannot be as the [ Preservation of firmness as the generic criterion for validity is also found in some authors in the English tradition, e.g., in Strode: A consequence is said to be valid (bona) when things cannot be as is exactly signified by the premises unless they are as is exactly signified by the conclusion … But a consequence is said to be valid in two ways: for some consequences are said to be valid in form (bona de forma) and some valid only in matter (bona de materia). A consequence is said to be valid in form when, if the way things are exactly signified by the premises is understood, the way things are exactly signified by the conclusion is also understood … and so it is said that in such consequences the conclusion is formally understood in the premises.41 Miroslav Hanke rejects the attribution of preservation of firmness as a criterion of validity to Roger.42 For consider ( † †) The conclusion of this inference is not firm, so the conclusion is not firm.
This inference is not considered by Roger, though he does consider its conclusion later in his treatise and realises it is paradoxical:43 for if it were firm to ground the notion of causes of truth in the basic subject-predicate (a.k.a. "categorical") propositions using Ockham's nominalist formula concerning the various supposita of the subject and the predicate. See his Sophismata in Iohannes Buridan, Summulae de dialectica, c. 2, conc. 14 (trans. Klima, 858-859 (that is, if things were as it signifies), it would not be firm, so by reductio ad absurdum, it is not firm. However, since the conclusion signifies that it is not firm, it does signify as things are and so is firm. Contradiction. Roger does not class the paradox, strictly speaking, as an insoluble, since it does not falsify itself.44 In fact, it undermines itself more radically, for Roger believes it neither signifies as things are nor other than they are.45 So it is not firm.
But there is an equivocation here over the phrase 'not firm' , between 'signifying other than things are' and 'not signifying as things are' . When we infer that the conclusion is not firm, we are concluding that it does not signify as things are. But although Roger concedes that if it signifies other than things are then it does not signify as things are, he rejects the converse implication: [The proposition 'This proposition signifies other than things are'] should be denied. And this inference should be denied: "it does not signify other than things are, and it signifies complexly, therefore, it signifies as things are." But it is necessary to add that it is not relevant to inferring itself not to signify as things are. And if that is added, it should be denied. For it follows directly from 'it signifies other than things are' that it does not signify as things are.46 On the other hand, 'This proposition does not signify as things are' should be granted, he says,47 since for the same reason, it does not signify as things are. Even so, both 'This proposition signifies other than things are' and 'This proposition does not signify as things are' do signify in some way. He claims that the conclusion of ( † †) neither signifies as things are nor other than things are, and so, he infers, it is neither true nor false.
Hanke finds the inference ( † †) itself in the work of John Mair, written some two centuries after Roger.48 Mair considered two arguments against defining valid consequence as preservation of things being as is signified (that is, firmness): But that means that ( † †) is problematic for Roger, for the premise is firm (and indeed true, since it does not undermine itself). Thus the inference has a firm premise and infirm conclusion. If Roger really took preservation of firmness to be necessary for validity, he would have to say the inference was invalid.
Hanke50 backs up this judgment by considering Roger's reaction to an objection he considers,51 namely, that his second claim is incompatible with Aristotle's rules in the first book of the Prior Analytics that if the conclusion is false, so is the premise, and if the premise is true, so is the conclusion.52 Roger replies that Aristotle's rules must be understood to concern firmness, not truth: Here it must be said that what Aristotle means by the first rule is this: if the conclusion does not signify as things are and neither the premise nor the conclusion is relevant to inferring itself not to signify as things are, then the premise does not signify as things are. the premise nor the conclusion is relevant to inferring itself not to signify as things are, then the conclusion signifies as things are.53 Hanke's point is that if Roger really believed that firmness-preservation sufficed for validity, then the qualification in the repeated clause 'and neither the premise nor the conclusion is relevant to inferring itself not to signify as things are' would be unnecessary. Its presence implies that an inference preserving firmness which did not satisfy the condition, such as inference ( † †) above, might not be valid. So is ( † †) valid or not, and what was (or should be) Roger's view? Is his account of validity given in §35, where he writes, "if from some propositions each of which signifies principally as it is, some proposition follows, it signifies as it is" (that is, validity preserves firmness), as Spade inferred,54 or in §37, where he adds the qualification?55 One problem with Hanke's proposal that we should follow what is given in §37 is that it seems to leave it open whether inferences like (*) are valid or not -Roger's theory will be incomplete. Indeed, reading §37 as saying that validity is preservation of firmness provided neither premise nor conclusion undermines itself suggests that firmness-preservation may not be necessary or sufficient when they do, and so Roger might reject ( † †) as invalid. Mair states several times that ( † †) is valid because it proceeds from one synonym to another (nempe illa consequentia est valida, quia a synonimo ad synonimum proceditur) even though it does not preserve firmness.56 However, we noted at the end of §2 that while proceeding from one synonym to another, that is, propositions with the same signification, might plausibly be sufficient for validity, it can hardly be necessary.57 But Strode and Eland, in arguing passionately against Roger's position, claim that it has nine impossible consequences, the fifth of which is that "Some inference is formally valid whose premise is true and conclusion neither true nor false" (Strode) and "Some inference is formally valid where the premise is true58 and the conclusion neither signifies as things are nor other than they are" (Eland).59 They do not say what this inference was. They may, of course, be speaking in propria persona when they say the inference is formally valid (bona et formalis), and not attributing this view to Roger, but when they cited Roger's second claim (they dubbed it the third: "In a formally valid inference the false follows from the true") they were voicing Roger's own view. So it is reasonable to assume the same for the fifth impossibility they attribute to him. In that case, Roger cannot be wedded to the unqualified account in §35, whereby ( † †) would be invalid. Strode and Eland's fifth impossibility also suggests that inference ( † †), or something like it, may have already been known in the fourteenth century, if not when Roger wrote his treatise, at least soon afterwards. There is good reason, then, to conclude that Roger's view of ( † †), which he does not himself consider explicitly, is that it is valid, and presumably formally so. This means that he will be committed to the claim that some formally valid argument has a firm premise and infirm conclusion, as Strode and Eland aver. However, it does leave us wondering what Roger's generic account of validity was, given that it is neither truth-preservation nor firmness-preservation. The example is at least consistent with the idea that synonymy between premise and conclusion does suffice for validity. We noted that we had such identity of what premise and conclusion signified in inference ( †).
But quite how the English inclusion criterion ("if the way things are exactly signified by the premises is understood, the way things are exactly signified by the conclusion is also understood," as articulated by Strode above) should be applied in the case of insolubles is still unclear. Recall the distinction between formally and materially valid consequences: they are different ways of being valid, but there is nonetheless a common feature to their validity, be it preservation of truth or of firmness or whatever. The danger with Roger's or Mair's account is that validity for insolubles can come apart from that for non-insolubles, so that, e.g., firmness-preservation is necessary and sufficient for validity for non-insolubles while synonymy is sufficient, but presumably not necessary, for insolubles. Then what does validity for insolubles have to do with validity for non-insolubles? Why call it 'validity' at all? What we need is a uniform account of validity, which is then satisfied differently by formal consequences and material consequences, and so too for non-insolubles 58 Spade follows the (single) manuscript in reading falsum here. But verum not only matches Strode's text, but makes better sense. 59 Spade, "Robert Fland's Insolubilia": Strode §1 (p. 76) ; Eland, there referred to as 'Fland' , §35, (p. 68). Cf. Read, "Paradoxes of Signification," 345-357. and insolubles. It turns out that such an account is found in Paul of Venice's Logica Magna.

Paul of Venice on Insolubles and Consequence
As mentioned above, Paul of Venice follows Roger in claiming that insolubles falsify themselves, though he amends the surrounding theory and updates it, especially to make it accord with his own theory of signification. Unlike Roger, Paul believes that the total significate of a proposition includes far more than what the terms commonly mean (sicut termini communiter pretendunt),60 but Paul restricts the criterion for truth to the proposition's exact significate (significatum adequatum). Paul's exact significate matches Roger's signifying principally: The first assumption is this: A true proposition is one whose exact significate is true and for which it is not incompatible that the proposition is true. This is clear from what has been said in the treatise 'On the truth and falsity of propositions' .61 The second assumption: A false proposition is one which either falsifies itself or whose falsity does not arise from its terms, but from its false exact significate.62 Thus the truth or falsehood of propositions is dependent on the truth or falsehood of their exact significates.63 In including that extra clause, 'and it is not incompatible that the proposition, signifying exactly in that way, is true' in his definition of truth in the treatise 'On the truth and falsity of propositions' , Paul had anticipated the problems thrown up by the insolubles. In the treatise on insolubles, Paul proceeds to draw several iconoclastic Conclusions from his account of truth and falsehood, including the three famous claims of Roger's that we noted earlier. In particular, he states in his fifth Conclusion that "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."64 Again, we may wonder what account of validity Paul endorses which permits such a situation, and how it is still possible to reason and argue coherently. His account of validity is, in fact, different from Roger's. Paul writes in the chapter in the Logica Magna on consequence and inference (De Rationali): A valid inference which signifies in accordance with the composition of its parts may be defined as one where the conclusion's contradictory would be incompatible with its premise, given that these signify as they do; and by 'as they do' I refer to what they customarily signify (ipsorum significata consueta).65 He reiterates this in the treatise on insolubles: Briefly, I assume all the rules of formal inferences in which it is stated that an argument from this to that is formally valid. This assumption is clear since, otherwise, it would be possible both that some inference is not falsitas non consurgit ex terminis sed ex adequato significato falso." Text and translation from Paul's treatise on 'Insolubles' are from an edition currently in preparation by Barbara Bartocci Oxford, 1990, 80). The significance of the qualification "which signifies in accordance with the composition of its parts" is that Paul believes that, e.g., 'conversely' in 'A man is an ass so an ass is a man and conversely' is an inference (consequentia) but does not signify by the composition of its parts, and has neither premises nor conclusion.
valid and that the conclusion's opposite is formally incompatible with the premise, the opposite of which, however, I assume strongly. For, in agreement with everybody, I mean that if there is an inference that signifies by the composition of its parts and the conclusion's opposite is formally incompatible with the premise, that inference is formally valid.66 Such an account of validity may look implausible for one who endorses Roger's third claim (that pairs of contradictories can both be false), as Paul does in his second Conclusion. Indeed, the fifth Conclusion appears flatly to contradict what he had written in the fourth section of the chapter on consequence earlier in the Logica Magna. He wrote there: The Third Rule is this:67 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. The Rule is proved as follows: If the premise of a valid inference is true and its conclusion is false, and if the conclusion's contradictory exists, then it is true; it is therefore consistent with the premise, since every independent truth is compatible with any other independent truth; but if that is so, the inference is not valid.68 The proof of this third rule appeals overtly to the Rule of Contradictory Pairs: "if the conclusion is false … then its contradictory is true." But his second Conclusion in the treatise on insolubles denies this, repeating Roger's third claim: "Some contradictories, that is, which contradict one another, are simultaneously false."69 We noted above that when formulating his account of truth in the treatise 'On Truth and Falsehood' , Paul anticipated the issue with insolubles, but at this (**) This is unknown to you, therefore this is unknown to you,70 where each occurrence of 'this' refers to the conclusion. In the treatise on insolubles, Paul cites this inference in the proof of his fourth Conclusion: "There are some inferences that are formally valid, and known by you to be so, which signify by the composition of their parts, where the premise is known by you, yet the conclusion is not known by you."71 His proof runs: 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. And indeed the conclusion is not known by you. Proof: because if it is known by you, then it is true and it signifies exactly that it is unknown to you, therefore it is true that it is unknown to you, therefore you do not know the conclusion. And thus whatever one says, it follows that the conclusion is not known by you.72 The conclusion ('This is unknown to you') is an example of the Knower Paradox.73 If you know it, then it must be true, and so you do not know it -and so, by reductio ad absurdum, you do not know it. But that is what it says, and you have just proved it, so you do know it. Paradox. Paul is rather conflicted about this sophism. In chapter 2 of the 'Insolubles' , just quoted, he calls it an insoluble. But he includes it in chapter 8 of the same treatise, which is headed 'On merely apparent insolubles' .74 Moreover, whereas in chapter 2 he says that the sophismatic inference is known by you to be valid, that its premise is known by you, and that the conclusion is not known by you (and he agrees in chapter 8 that indeed the conclusion cannot be known by you), in the earlier chapter on 'Consequences' he infers from the fact that the conclusion is not known by you that the premise is not known by you either. For his Ninth Rule states: 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.75 Paul retains the validity of the Ninth Rule in the face of the inferential Knower Paradox by denying the premise: In the given posited case, [I] grant that all the clauses in the [Ninth] Rule are satisfied, but [I] deny that the premise is known, the reason being that just as it is incompatible for the conclusion to be known, so it is for the premise. But more about this when we come to deal with the insolubles.76 In the Insolubles, it transpires that the final qualification in the Ninth Rule ("you know that what is false does not follow from anything that is true") fails, as noted above in his fifth Conclusion.
It remains to be seen, however, whether anything hangs on the inconsistency between the chapter on consequence and the treatise on insolubles. For the Third Rule cited above only follows from his account of validity given the RCP: so once that is rejected in the second Conclusion of chapter 2 of the Insolubles, Read Vivarium 58 (2020) 275-304 perhaps the way is clear for the fifth Conclusion to accord with the earlier account of validity.
Paul considers three objections to the Third Rule of consequences, but none raises the issue of insolubles. Hughes notes that Paul is following Strode's Consequences here and responding to some of Strode's problems with the Rule,77 but Strode was not a follower of Roger on insolubles -indeed, he was a major critic, as we saw in §2. As we noted in §4, Strode defined validity in terms of preservation of signifying as things are, so Strode's proof of the rule (his first, Paul's third) is different from Paul's, and in fact does not invoke the RCP.78 Let us, then, return to the earlier example from Roger discussed in §2: ( †) The conclusion of this inference is false, so this conclusion is false.
The premise is (for Paul, as for Roger) true and the conclusion false, but Paul (like Roger) endorses the validity of the inference. For the conclusion's contradictory (viz 'This conclusion is not false') is also false, and clearly incompatible with the premise, which truly and correctly says that the conclusion is false. But care is needed here in interpreting Paul's formula 'contradictorium consequentis repugnat antecedenti ' , which I have translated as 'the conclusion's contradictory is incompatible with the premise(s)' . Paul explained 'repugnat' ('is incompatible') in the chapter on conditionals: "Two propositions are mutually incompatible when, signifying as they do, they cannot be nor can have been nor could be true together, or at least their significates cannot be nor can have been nor could be true together."79 And he emphasizes in the chapter on inference that "two things [are] incompatible with each other when the conjunction formed from them is per se impossible."80 How does this relate to the treatment of validity as firmness-preservation? As we noted in §3, propositions were taken by the medievals to be concrete tokens, and that they subscribe to this conception is clear from one of the objections that is standardly made to the definition of validity as the impossibility of the premises being true and the conclusion false, by stipulating a case where the conclusion does not exist. Paul considers such an objection to his Third Rule in the chapter on Given this conception, each proposition has many contradictories. For most thinkers, this is not a problem, for they will all be equivalent and have the same truth-value. But for Paul, this will not be true. As Paul points out in the seventh Conclusion in chapter 2 of his 'Insolubles' , there are pairs of convertible propositions (which for him means those with the same exact signification, predicating the same thing of the same subject) one of which can be true and the other false: "Seventh Conclusion: some pairs of propositions are convertible, but one is true or possible and the other is false or impossible. It is clear as regards these propositions: 'This is false' and 'This is false' , both referring to the second."82 Nonetheless, though some mutually convertible propositions will be true and some false, they will all be contradictories of another proposition, in predicating the same thing of the same thing, and so affirming of it what the other denies or vice versa.
This Aristotelian, syntactic conception of contradictory opposition, denying of the same what the other affirms and vice versa, serves for Paul to define validity, despite contradictory pairs sometimes having the same truth-value, or pairs of convertible propositions sometimes having different truth-values. What are incompatible are their exact significates, where the exact significates are what are customarily signified by the premise and conclusion. So the requirement for validity is that the contradictory significate of the conclusion and the significate of the premise cannot both be true -where being true or false for significates is their obtaining or not.83 Recall that, for Paul as well as Roger, contradictories cannot both be true, though they can both be false. Hence any inferences, such as ( †) or ( † †), where the exact significates of premise and conclusion are the same, must be valid, for the contradictory significate of the conclusion will also be a contradictory significate of the premise. This account of validity cannot guarantee either truth-preservation or firmness-preservation for valid inferences, for the contradictory significate of the conclusion of ( †), viz that the conclusion is not false, is incompatible with the significate of the premise, viz that the conclusion is false, so ( †) is aside), it is not how Paul defines it. His definition, which applies to noninsolubles and insolubles alike, turns on the incompatibility of the premise with the contradictory of the conclusion. The premise of ( † †) and the contradictory of its conclusion are indeed incompatible, since their exact significates cannot both be true. So ( † †) is valid for Paul. But it does not follow that if the exact significate of the premise is true then so is that of the conclusion. For as we have just seen Paul observe, the premise's exact significate is true -the conclusion neither signifies as things are nor other than they are -so the conclusion is neither true nor false. Here Paul offers an alternative response to the paradox generated by the conclusion of ( † †), closer to Roger's, denying both that its significate is true and that its significate is false. So the inference is valid, the premise is firm, and the conclusion is infirm.

Validity in Form
Recall from §3 that Roger's and Paul's account of insolubles can be adapted to yield syllogistic counterexamples to truth-preservation as necessary for validity. How then do they avoid inconsistency? Note that the standard medieval view is that the syllogistic is an example of formal validity, that is (for Paul), where the contradictory of the conclusion is formally incompatible with the premises.87 Paul states this explicitly: "The regular syllogism has three figures containing moods such that an argument in any of these moods is formally valid."88 He notes that both Baroco and Bocardo are reduced to the first figure per impossibile, as Aristotle proposed in De interpretatione, chapters 5-6: The fourth mood of [the second] figure, namely Baroco, is composed implicitly or explicitly from a universal affirmative and a particular, indefinite, or singular negative constructed normally, concluding directly or indirectly in a particular, indefinite, or singular negative either implicitly or explicitly. For example, from these premises: 'Every man is an animal' and 'Some stone is not an animal' , it follows that some stone is not a man, and conversely indirectly. where N is 'true proposition' , M is 'premise' and X is 'conclusion' . We take the major premise ('Every N is M') with the opposite of the conclusion ('Every X is N'), as Paul said in the quotation above from his treatise on 'Syllogisms' ,102 and infer the opposite of the minor premise ('Every X is M') by Barbara. But that contradicts the minor premise ('Not every X is M'), so we can infer the opposite of the hypothesis ('Not every X is N'), the conclusion of Baroco. Of course, in this strange case, both the final conclusion, 'Not every conclusion is true' , and its contradictory opposite, the hypothesis, 'Every conclusion is true' , are false, the hypothesis because things are not as it signifies (the only conclusion is false), and the conclusion because it falsifies itself.103 Paul's statement, cited above, that he "assume[s] all the rules of formal inference in which it is stated that an argument from this to that is formally valid" Premise Hypothesis Negation-Introduction (1) figure 1 Aristotle's reduction of the paradoxical example of Baroco turns out to be rather rash. For it commits him to asserting falsehoods, though only the falsehoods manifested in insolubles. We might grant that there is a certain honesty in bravely asserting the Moorean paradox, 'What I am saying is false. What I have just said was false' , asserting a falsehood followed immediately by a truth, namely, that the falsehood was false.104 This should perhaps be no surprise, given Roger's second claim. For the second claim could be (correctly) qualified as saying that there is a valid inference whose premise is true and whose conclusion is an insoluble implying its own falsehood, and so false. This is no more than an immediate consequence of strengthening the criterion for truth so as to deem all insolubles simply false.

Conclusion
Roger Swyneshed rejects the definition of validity of an inference as requiring the preservation of truth. This is because, following his diagnosis of the paradoxes thrown up by the insolubles, a valid inference can issue in a false conclusion from true premises simply because the conclusion falsifies itself, despite things being as it signifies. The classic example is the Liar paradox, 'This is false' , which, in saying of itself that it is false, falsifies itself, so that it is false even though things are as it signifies. His three iconoclastic claims come together: the first recognises that the Liar is false despite things being as it signifies; the second that the Liar, though false, follows validly from the true claim that it is false; and the third that its contradictory, 'This is not false' , saying of 'This is false' that it is not false, is also false, so that a pair of contradictories can both be false. This last is contrary to received wisdom endorsing the Rule of Contradictory Pairs, that in any such pair, one is true, the other false, but is arguably in accordance with Aristotle's remarks in De interpretatione, although their counterexamples to the rule differ. What account of validity Roger adopted in place of truth-preservation is a matter of speculation. At one point he suggests preservation of firmness is necessary, only to qualify it a few sentences later. Why he does so is unclear, but he may be prescient in recognising that not only does inference ( †) show that truth-preservation is not necessary, but inference ( † †) shows that preservation of firmness is not necessary either. ( † †) is not mentioned by Roger or Strode or Eland, but one of the latter two's objections to Roger's theory is that it accepts that an inference can be valid but fail to preserve firmness, suggesting Read Vivarium 58 (2020) 275-304 that they had an example in mind. Indeed, Hanke credits to Paul, or perhaps to his reading of Paul, the realisation that whatever value one proposes should be preserved to ensure validity, there will be a paradox like ( †) and ( † †) which will show that its preservation cannot be necessary, assuming that identity (or synonymy) is sufficient.105 Paul of Venice follows Roger in endorsing all three claims, although Paul's theory of signification and account of validity are different. But he shares with Roger the aim of solving the paradoxes without postulating hidden meanings in them, as their contemporaries Bradwardine and Heytesbury had done. Paul agrees that things are as the Liar signifies -as he puts it, its exact significate is true. But that is not enough to make it true, since it falsifies itself. So it is false, while at the same time following validly from the true claim that it is false. For the premise asserting that it is false does not falsify itself, while its exact significate, the same as the exact significate of the conclusion, is true. So the premise is true. Why, then, is the inference valid? For Paul, the reason is that the premise is incompatible with the contradictory of the conclusion, namely, 'This is not false' . On either definition, Swyneshed's inference concluding in the Liar turns out to be valid. So too do all instances of (by definition, valid) second-and third-figure moods of Aristotle's assertoric syllogism, even those whose validation proceeds, for Aristotle, by reduction per impossibile to the first figure, and even though their premises may be true and conclusion false -and not only those which are self-falsifying. For Aristotle, this situation cannot arise, since syllogistic propositions obey the RCP. The reason Aristotle's method of reduction continues to work for Roger and Paul is that for them contradictory pairs cannot both be true, though they can both be false, contrary to the RCP.