The Philosophical Quarterly Vol. No. ISSN –
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The Scots Philosophical Association and the University of St Andrews
LESSONS ON TRUTH FROM MEDIAEVAL SOLUTIONS TO THE LIAR PARADOX B C D N Some fourteenth-century treatises on paradoxes of the liar family offer a promising starting-point for the formulation of full-fledged theories of truth with systematic relevance in their own right. In particular, Bradwardine’s thesis that sentences typically say more than one thing gives rise to a quantificational approach to truth, and Buridan’s theory of truth based on the notion of suppositio allows for remarkable metaphysical parsimony. Bradwardine’s and Buridan’s theories both have theoretical advantages, but fail to provide a satisfactory account of truth because both are committed to the thesis, fatal for both, that every sentence signifies/implies its own truth. I close with remarks on Greg Restall’s recent model-theoretic formalization of Bradwardine’s theory of truth.
I. INTRODUCTION One of the traits common to later mediaeval logic and to contemporary investigations in (the philosophy of ) logic is a vivid interest in paradoxes that are now known as paradoxes of the liar family (referred to as ‘insolubilia’ by mediaeval authors). However, while these paradoxes have been among the main spurs of the development of (formal) theories of truth in recent times, it is often said that ‘the mediaevals did not draw great theoretical lessons from the insolubles’.1 It is true that the general mediaeval attitude towards the paradoxes was significantly more relaxed than the feeling of imminent threat that most modern philosophers and logicians seem to experience;2 nevertheless, to say that the mediaevals did not draw any theoretical lessons from insolubles seems quite off the mark. In fact, in many mediaeval insolubilia treatises we find analyses of the notions of truth and sentential meaning which offer a promising starting-point for the formulation of full-fledged theories of truth and meaning with systematic relevance in their own right. Several authors 1 P.V. Spade, ‘Insolubilia’, in N. Kretzmann et al. (eds), The Cambridge History of Later Medieval Philosophy (Cambridge UP, ), pp. –, at p. . 2 One recent exception is J. Beall, Spandrels of Truth (Oxford UP, ), which proposes an attitude towards paradoxes similar in spirit to the mediaeval ‘zen’ mindset.
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LESSONS ON TRUTH FROM MEDIAEVAL SOLUTIONS TO THE LIAR
did realize that in order to treat such paradoxes, a deep analysis of these notions was required. Bradwardine’s treatise3 on insolubilia is particularly significant, but in several other treatises, such as Buridan’s4 and Albert of Saxony’s,5 one encounters equally interesting material. In a previous article in this journal, I discussed some of the implications of these analyses for a general account of sentential meaning.6 In what follows, I undertake a similar endeavour, but this time concerning truth. In particular, the sentential pluralism developed in Bradwardine’s treatise and discussed in my article LSM, i.e., the thesis that sentences may and in fact typically do signify several different things at once, requires a series of modifications to the notion of truth, as I shall show: one must now quantify over the things that a sentence says when defining truth and falsity. But the quantificational–correspondentist theory of truth which emerges from Bradwardine’s treatise did not go uncontested. As discussed in my LSM, Buridan rejects the notion of ‘what a sentence signifies’ (the cornerstone of Bradwardine’s account), apparently on ontological grounds. Accordingly, he formulates his notion of truth within an entirely different framework, supposition theory, which requires far less ontological commitment and is thus more suitable for a nominalist like Buridan. However, while in my previous paper Buridan came out as something of a winner, in so far as his theory of sentential meaning appeared as ontologically better supported and semantically more detailed than Bradwardine’s, here I shall show that Buridan’s theory of truth based on supposition runs into a series of difficulties. Surprisingly, though, Bradwardine’s theory of truth (as it stands) also allows for the derivation of undesirable implications similar to those afflicting Buridan’s theory. The difference is that while in Buridan’s account 3 M.L. Roure, ‘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 age, (), pp. –, is an unreliable edition, since it is based on only two of the available manuscripts. In contrast, S. Read’s edition, Thomas Bradwardine, Insolubilia (Dallas Medieval Texts and Translations; Leuven: Peeters, ), is based on all thirteen available manuscripts and includes a translation and an introduction. Here I refer to Read’s edition and quote his translations. 4 Buridan’s most extensive treatment of insolubilia is to be found in his treatise Sophismata, of which many modern editions and translations exist. The most recent, and the ones I shall use, are (translation) John Buridan, Summulae de Dialectica, tr. G. Klima (Yale UP, ), and (Latin text) John Buridan, Summulae de Practica Sophismatum [Sophismata], ed. F. Pironet (Turnhout: Brepols, ). 5 There is an English translation of the chapter on insolubilia from Albert’s Sophismata (also to be found in his Perutilis Logica) in Albert of Saxony, ‘Insolubles’, in N. Kretzmann and E. Stump (eds), The Cambridge Translations of Medieval Philosophical Texts, Vol. (Cambridge UP, ), pp. –. 6 C. Dutilh Novaes, ‘Lessons on Sentential Meaning from Mediaeval Solutions to the Liar Paradox’, henceforward LSM, The Philosophical Quarterly, (), pp. –.
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the undesirable implications follow directly from the postulate that every sentence (virtually) implies its own truth, Bradwardine makes no such explicit claim. But the combination of a notion of sentential signification closed under implication with a quantificational definition of truth (the cornerstones of Bradwardine’s theory) inevitably leads to the proof that every sentence signifies its own truth, which in turn leads to an ineffective definition of truth, as I shall argue. So Buridan’s revision of a Bradwardinian theory of truth (which he himself adhered to early in his career), on ontological grounds, offers only a partial diagnosis of the latter’s shortcomings, leaving its logical deficiencies untouched. In what follows, I first analyse how Bradwardine’s pluralism with respect to sentential meaning necessitates a quantificational account of truth, if it is indeed to be a correspondentist account of truth. I then contrast the two most influential accounts of truth in the later mediaeval period, the correspondentist account and the account based on the notion of supposition. Next, I discuss the matter of whether all sentences signify or (virtually) imply their own truth, and show that both Buridan’s and Bradwardine’s theories of truth (as they stand) are ineffective in virtue of the requirement that signification must be closed under implication, combined with a quantificational notion of truth. I close with some considerations on the formalization of Bradwardine’s theory of truth recently proposed by Greg Restall, arguing that it fails to capture some of the essential aspects of Bradwardine’s analysis, but allowing that it does make the accurate prediction that within Bradwardine’s theory, just as within Buridan’s, every sentence implies/ signifies its own truth.
II. BRADWARDINE’S QUANTIFICATIONAL NOTION OF TRUTH In my LSM I argued in favour of a pluralist account of the meaning of sentences, an account which considers the possibility that sentences may (and in fact typically do) say several things at once, as contrasted with unitary views according to which sentences say at most one thing. One of the upshots of sentential pluralism is that the common Tarskian approach to truth, the gist of which can be traced back to Aristotle’s Metaphysics and De Interpretatione, must be revised.7 This is why: the (slightly reformulated) Tarskian adequacy condition ‘p is true iff P’, where p is the name of a sentence and P, under the correspondentist notion of truth examined here, 7 In his ‘The Semantic Conception of Truth and the Foundations of Semantics’, Philosophy and Phenomenological Research, (), pp. –, Tarski explicitly says that his theory of truth is to do justice to the classical Aristotelian conception of truth.
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is what sentence p says/means/asserts, can be glossed as ‘a sentence is true iff things are as it signifies them to be’. If P is all that p says, then everything seems fine. But the problem is that according to the pluralist view under consideration here, even if p says P (that is, p : P),8 it may very well be that p says something else, say Q (that is, p : Q ), and in addition it may be that P is indeed the case but Q is not. Is p then true or not true? It says two things, namely, one which obtains and another which does not obtain; but according to the standard Tarskian definition of truth, the fact that P obtains is sufficient for the truth of p. However, p also says something that does not obtain, namely Q; it seems unwarranted to conclude without further reflection that p is true even though it says at least one thing that does not obtain. Therefore if one seeks to avoid unreflective endorsement of the substantive assumption of sentential semantic unitarianism, while leaving open the possibility of sentential semantic pluralism, then it is clear that a different account of the notion of truth is required. Given that what distinguishes sentential pluralism is the idea that a sentence typically says several things, an approach that suggests itself naturally is to quantify over these things, thus yielding what one could call quantificational notions of truth and falsity. In another article9 I have discussed in more detail different combinations of quantifiers (universal and existential) for appropriate definitions of truth and falsity, commenting on Andjelkovic´ and Williamson’s analysis of the issue.10 The candidates are ∀T. ∀F. ∃T. ∃F.
∀s[T(s)] ⇔ [∃P(s : P) & ∀P[(s : P) ⇒ P]]11 ∀s[F(s)] ⇔ [∃P(s : P) & ∀P[(s : P) ⇒ ¬P]] ∀s[T(s)] ⇔ ∃P[(s : P) & P] ∀s[F(s)] ⇔ ∃P[(s : P) & ¬P].
Clearly, if the dual truth–falsity is determined by definitions having the same quantifier, then the definitions are jointly too weak or jointly too strong, and bivalence and the principle of non-contradiction are not preserved. That is, if the universal quantifier is used for both definitions, then all the sentences 8 This is the notation introduced by Stephen Read to render the notion of saying, and used in several of his articles on Bradwardine’s solution to the liar: see S. Read, ‘The Liar Paradox from John Buridan back to Thomas Bradwardine’, hereafter LPB, Vivarium, (), pp. –; ‘Symmetry and Paradox’, History and Philosophy of Logic, (), pp. –; ‘The Truth-Schema and the Liar’, and ‘Further Thoughts on Tarski’s T-Scheme and the Liar’, both in S. Rahman et al. (eds), Unity, Truth and the Liar (Berlin: Springer, ), pp. –, –; ‘Plural Signification and the Liar Paradox’, Philosophical Studies, (), pp. –. 9 C. Dutilh Novaes, ‘Tarski’s Hidden Theory of Meaning’, in Rahman et al. (eds), Unity, Truth and the Liar, pp. –. 10 M. Andjelkovic´ and T. Williamson, ‘Truth, Falsity and Borderline Cases’, Philosophical Topics, (), pp. –. 11 The clause to the effect that p does say at least one thing avoids the consequence that the definition is vacuously verified for things that do not say anything.
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that say several things, some of which obtain while others do not, are neither true nor false. Similarly, if the existential quantifier is used for both definitions, then those very sentences that say things some of which obtain and some of which do not obtain are both true and false. So (contrary to Andjelkovic´ and Williamson’s intuitions) it seems that we need mixed pairs of quantifiers in the definitions, one with a universal quantifier and the other with an existential quantifier – that is, either (∀T)/(∃F) or (∃T)/(∀F) – if we are to preserve bivalence and the principle of non-contradiction. Which pair should it be? From Bradwardine’s definition of the notion of truth in his Insolubilia, it is clear that according to him, truth should be formulated in terms of universal quantification, while falsity, being its dual, would correspond to existential quantification: A true proposition is an utterance signifying only as things are. A false proposition is an utterance signifying other than things are (Bradwardine, Insolubilia, .).
Another formulation of the same idea, found in different authors, which makes the quantificational approach even more explicit, is [A] sentence is true when things are however it signifies them to be.... A sentence is false when things are not however it signifies them to be (Albert of Saxony, Insolubilia).12
These are both variations of the traditional (Aristotelian) correspondentist notion of truth (operating with the mediation of a third level, that of the signification of a sentence), often formulated by the mediaeval authors as For the truth of a proposition it is necessary and sufficient that things are as it signifies.13
The main difference from the traditional Aristotelian formulation is clearly the move to the quantificational approach, from ‘as a sentence signifies’ to ‘however a sentence signifies’. It is possible that the reformulation of the traditional Aristotelian definition was prompted precisely by the analysis of liar-like insolubilia; but there is no reason why this definition should be used only in insolubilia contexts, and in some later treatises on the truth and falsity of sentences, such as Paul of Venice’s, variations of the quantificational approach to truth are discussed outside the context of insolubilia.14 In fact, 12 Translated from the Latin text in L. Pozzi, Il Mentitore e il Medioevo (Parma: Zara, ), p. . 13 Paul of Venice mentions this thesis, but in the form of its denial. See Paul of Venice, Logica Magna, part II, fascicule : Tractatus de veritate et falsitate propositionis, ed. F. del Punta, tr. M. McCord Adams (The British Academy/Oxford UP, ), p. . 14 Other terms used to express the idea of quantification in Paul’s treatise are these: a sentence signifies precisely (praecise), or wholly (totaliter), how things are.
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as I shall argue shortly, it is a more fundamental notion of truth than the Tarskian unquantified notion. As illustrated by Paul of Venice’s treatise and as also noted by Gyula Klima,15 variations of the quantificational formulation were current in the fourteenth century. They can be found in many authors of the Buridanian circle, including Albert of Saxony and Buridan himself: indeed, it is the definition of truth which the latter endorses in his earlier Questiones Elencorum.16 Bradwardine is the earliest source I am aware of where one finds traces of quantification in a definition of truth; this does not necessarily mean that it was his invention, but Ralph Strode, writing in the second half of the fourteenth century, explicitly credits Bradwardine with the introduction of a genuinely novel solution to insolubilia (see Read, LPB), and this recast definition of truth is one of its cornerstones. However, a thorough history of such formulations of the notion of truth in the later Latin Middle Ages is still to be written. But it is not (only) for the sake of historical faithfulness towards Bradwardine that we should prefer the pair (∀T)/(∃F) rather than the pair (∃T)/(∀F) to characterize truth and falsity. There are compelling arguments (some of which I have discussed in ‘Tarski’s Hidden Theory’) substantiating the view that truth corresponds to universal quantification over the things that a sentence says, while falsity corresponds to existential quantification. Indeed, in several of his papers on Bradwardine’s solution to the liar, Read has stressed the idea that truth is a limit or peak of success; any falling off from it is failure. A sentence is true if things are only as it says they are; if things are not so even in a minimal respect, it is false. An analogy may be helpful here: for a picture of any kind to be a faithful depiction of what it depicts, all of its features must correspond with features of what it depicts. Within the unitarian paradigm, sentences are thought to be rather selective representations of states of affairs, representing only a very small portion of reality; but if one considers more complex forms of representation, it becomes apparent that the intuitive notion of ‘faithful representation’ is usually associated with complete correspondence. For example, a portrait of a man that represents accurately only some of his features (if it represents the correct eye colour, but not the correct hair colour) cannot be said to be a true portrait of the man, even though some features are correctly represented. In this vein, once we accept the pluralist tenet that sentences typically say several (different) things, it becomes clear that sentences too are complex forms of 15 G. Klima, ‘Logic Without Truth: Buridan on the Liar’, in Rahman et al. (eds), Truth, Unity and the Liar, pp. –. 16 Buridan, Questiones Elencorum, ed. H. Braakhuis and R. van der Lecq (Nijmegen: Ingenium, ).
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representation, and thus that for a sentence to be true, complete correspondence is required. Hence it seems manifest that one should settle the competition between the two pairs (∀T)/(∃F) and (∃T)/(∀F) by associating truth with universal quantification, and accordingly, falsity with existential quantification. Moreover, the quantificational notion of truth proposed here is also more fundamental than the usual Tarskian one, since it does not take sides on the issue of sentential semantic pluralism. If it turns out that sentences indeed say only one thing, as the unitarian holds, then the quantificational definitions work just as well, but in that case the domain of quantification would be restricted to the one single thing that a sentence says. As noted by Andjelkovic´ and Williamson, and as discussed at length in my ‘Tarski’s Hidden Theory’, the usual Tarskian definition behaves more like a theory than like a definition, since it presupposes that sentences say only one thing. The quantificational definition of truth, in contrast, is neutral in this respect and can accommodate both the unitarian and the pluralist positions: if it turns out that sentences say exactly one thing, then (∀T) and (∃F) simply collapse into the usual unquantified definitions of truth and falsity.
III. THE NOTION OF TRUTH AS CO-SUPPOSITION But the Aristotelian correspondentist definition of truth (and its quantificational variations) was not unanimously adopted by fourteenth-century logicians: an alternative conception, based on the notion of supposition, more specifically, on the notion of co-supposition of subject and predicate (in the case of affirmative sentences), was possibly even more influential at the time.17 Definitions of truth in terms of co-supposition were pervasive in the fourteenth century, and can be found, for example, in Ockham, Buridan, Albert of Saxony, among many others. The following passage from Ockham is particularly clear concerning the semantic character of this notion of truth, as opposed to the ‘metaphysical’ correspondentist definition: On this point it should be noted that for the truth of such a singular proposition ... it is not required that the subject and the predicate be really identical, or that the predicate be in reality in the subject or that it really inhere in the subject, or that the predicate be united to the subject itself outside the mind. Thus, for the truth of ‘This is an angel’, it is not required that the general term ‘angel’ be really identical with what is posited as the subject, or that it be really in the subject, or anything of this 17 That these two notions co-existed in the fourteenth century can be inferred from Paul of Venice’s examination of nine different views on truth in his Tractatus de veritate et falsitate propositionis, among which there are variations of each of them.
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sort. Rather, it is sufficient and necessary that the subject and predicate supposit for the same thing. And, therefore, if in ‘This is an angel’ the subject and predicate supposit for the same thing, the proposition will be true. Thus, it is not asserted [denotatur] that this thing has angelhood or that angelhood is in it – or anything of this sort. Rather, it is asserted that this thing is truly an angel – not, indeed, that it is the predicate, but that it is that for which the predicate supposits.18
Ockham emphasizes that the truth of a sentence does not depend primarily on the nature and properties of things in reality, but simply on the semantic properties of its terms, their supposition; this is why this conception of truth can be said to be semantic rather than metaphysical. Naturally, the supposition of a term ultimately does depend on the nature of things, since the term ‘angel’ can be truly predicated only of those individuals having certain characteristics (in short, angels). But the crucial move here is to ascribe truth to the semantic level of supposition, while the metaphysical connection between terms and things is taken care of by the notion of signification, which is precisely somewhere in between semantics and metaphysics/epistemology. Moreover, another important difference with respect to the correspondentist conception of truth is the switch to the term/ individual perspective. Within the correspondentist conception the basic units are sentences and their truthmakers (although the correspondentist notion of truth in Bradwardine’s treatise is rather vague concerning what must obtain in reality for a sentence to be true); but within the cosuppositional notion of truth the basic units are terms and the individuals which are their supposita. Buridan is also explicit in his rejection of the notion of truth based on what a sentence ‘signifies’, in favour of the co-suppositional definition. As discussed in my LSM, he accepts the use of the locution ‘however a sentence signifies things to be’, but only in so far as it is an abbreviation referring to the real causes of truth or falsity of sentences, namely, the several suppositional clauses defining the truth or falsity of a sentence as a function of its syntax (whether it is affirmative or negative and whether it is universal or particular), as spelt out in ch. , conclusion of his Sophismata (Buridan, Summulae de Dialectica, p. ). As I argued in LSM, Buridan’s notion of the ‘signification’ of sentences is made so meagre (for ontological reasons) that it can no longer be used to handle truth. The co-suppositional notion of truth is remarkable in its simplicity and ontological economy; in fact, it seems to me that of all the conceptions of truth proposed in the history of philosophy, it is the one that comes closest to Tarski’s ideal of a ‘purely semantic conception of truth’. A semantic 18 William of Ockham, Summa Logicae, part II, tr. A. Freddoso (South Bend: St Augustine’s Press, ), p. .
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conception of truth, according to Tarski, is one which dispenses with intermediary entities and operates directly from linguistic entities to things, without the mediation of the meaning or signification of terms or sentences.19 This is exactly what the co-supposition conception of truth has to offer, a non-mediated passage from words to things, thus yielding remarkable ontological parsimony. The obvious question to be asked now is how these two different notions of truth, the correspondentist and the co-suppositional, relate to each other. Are they extensionally equivalent and only conceptually different, in that their respective truth predicates have the same extension? (Of course, Buridan has a peculiar understanding of the signification of a sentence, but his account of the truth of a sentence does not depend on it.) Or are they also extensionally different? In fact, it seems that in ‘well behaved’ cases, a Bradwardinian notion of sentential signification, yielding a correspondentist definition of truth, and a Buridanian/Ockhamist notion of truth, based on co-supposition, produce truth predicates with the same extension. However, liar sentences disrupt the serenity of this picture. While in normal cases the co-suppositional criterion is a sufficient criterion for truth, it is put under pressure when faced with the usual liar paradoxes. This is why. Suppose ‘What I am saying now is false’ is true. Then according to the co-suppositional criterion, since this is an affirmative sentence, subject and predicate supposit for the same, that is, ‘what I am saying now’ and ‘false’ co-supposit, and thus what I am saying now is false; hence the whole sentence (the suppositum of the subject) is false. But if the sentence is false, then its terms do not co-supposit (given that it is affirmative); ‘what I am saying now’ does not supposit for the same thing as ‘false’ does, and thus the sentence is true after all. In other words, the co-suppositional definition of truth takes us straight into paradox; therefore it cannot be a sufficient condition for truth, as Buridan himself recognizes (see Read, LPB, p. ). Clearly, like any theory of truth, mediaeval as well as modern, the cosuppositional conception of truth requires amendments in order to avoid the threat of liar-like paradoxes, and different strategies were proposed by mediaeval authors to deal with it.20 One of them was the ban on self-reference in 19 Tarski, ‘The Semantic Conception of Truth and the Foundations of Semantics’. Interestingly, Tarski explicitly refers to the mediaeval concept of supposition when explaining his criterion for the material adequacy of a definition of truth in § of the paper. The reference, however, does not go very far in the comparison of mediaeval theories of supposition with Tarski’s semantic conception of truth. See C. Panaccio, ‘Tarski et la suppositio materialis’, Philosophiques, (), pp. –, for a discussion of Tarski’s ‘misuse’ of the notion of material supposition. 20 See C. Dutilh Novaes, ‘A Comparative Taxonomy of Medieval and Modern Approaches to Liar Sentences’, History and Philosophy of Logic, (), pp. –.
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such cases, proposed by Ockham and other restringentes.21 Buridan criticizes this solution,22 and proposes instead that the co-suppositional criterion must be satisfied not only for the sentence in question, but also for all the sentences that are (virtually) implied by it, for the original sentence to be true.23 This is what I shall call the strengthened co-suppositional condition (SCSC). While rejecting the view that every sentence signifies its own truth, Buridan argues that every sentence (virtually) implies its own truth, and this is the core of his solution to the liar paradox. 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’ is affirmed of the subject that supposits for [the original proposition].24
With respect to liar sentences, Buridan accepts as valid, for example, the implication from ‘What I am saying now is false’ to ‘What I have just said is true’. In more formal terms, we have L. <(L)> is false25 T. <(L)> is true and the claim that (L) implies (T). What happens now is that (L) cannot satisfy (SCSC): co-supposition cannot hold of it as well as of all the sentences it implies (in the case of affirmative sentences), because <(L)> cannot supposit for both ‘false’ (in (L)) and ‘true’ (in (T)). Hence (L) is false, since it does not meet the necessary and sufficient conditions for truth in terms of (SCSC). Buridan’s reformulated notion of truth in terms of co-supposition is apparently able to withstand the threat of paradox by means of the requirement that co-supposition must hold of the putatively true sentence as well as of its implications, while maintaining the basic idea of truth as cosupposition. The question is now whether (SCSC) should apply to all sentences, or if it applies only to insoluble sentences. From Buridan’s text, one is left with the impression that he still views the simple co-suppositional 21 P.V. Spade, ‘Ockham on Self-Reference’, Notre Dame Journal of Formal Logic, (), pp. –. 22 Buridan, Summulae de Dialectica, p. (Sophismata ch. , th sophism). 23 The force of ‘virtually’ in ‘virtually implies’ is that the second sentence would be implied by the first only if the first is actually formulated. Buridan endorses a token-based semantics: see G. Klima, ‘Consequences of a Closed, Token-Based Semantics: the Case of John Buridan’, History and Philosophy of Logic, (), pp. –; C. Dutilh Novaes, ‘Buridan’s consequentia: Consequence and Inference within a Token-Based Semantics’, History and Philosophy of Logic, (), pp. –. 24 Buridan, Summulae de Dialectica, p. (Sophismata ch. , th sophism). 25 ‘<...>’ is a nominalization device, similar to the ‘dictum’ construction used by Buridan and other mediaeval authors (subject and predicate in the accusative, verb in the infinitive); these are the constructions which typically supposit for sentences within sentences.
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condition as a sufficient condition for truth in normal cases, and that (SCSC) is required as a sufficient condition for truth (only) in self-referential cases. And this is why it is said that when a proposition has or can have reference to itself, it does not suffice for the truth of an affirmative [proposition] that its terms supposit for the same [thing or things], as I have said elsewhere, but it is also required that even in this consequent the terms supposit for the same [thing or things].26
If, as argued by Read in LPB, Buridan indeed restricts (SCSC) to insolubilia, then the move is ad hoc: he would block the inference that a liar sentence is true from the premise that it is false, but he would do so by undermining the uniformity of his account of truth. But if Buridan held (SCSC) to apply to all sentences, it is his whole theory of truth that is jeopardized, as I shall show in the next section.
IV. WHICH SENTENCES SIGNIFY/IMPLY THEIR OWN TRUTH? While there are obvious similarities between Bradwardine’s, Buridan’s and Albert of Saxony’s respective solutions to the liar paradox, two points of dissimilarity stand out – Bradwardine’s and Albert’s acceptance but Buridan’s rejection of the notion of what a sentence signifies in their definitions of truth, and the question of which sentences signify/imply their own truth. Buridan and Albert defend the view that all sentences signify/imply their own truth, while Bradwardine restricts this claim (at least explicitly) to insolubilia. As I have remarked, he proves that sentences saying of themselves that they are false also say of themselves that they are true, on the basis of his fundamental principles. The trouble is that the same fundamental principles also allow for the proof that all sentences say of themselves that they are true, as I shall now show. The problem is now that if (SCSC) is to be applied to all sentences as a definition of truth, some very counter-intuitive results follow (see also Read, LPB). In short, Buridan’s notion of truth either winds up being ad hoc if (SCSC) applies only to insolubilia, or it seems to turn every sentence into a semi-truthteller, as I shall argue. First, a few general points. We know that Bradwardine makes no general claims about sentences signifying their own truth, but rather proves that a very specific group of sentences, namely, those that say of themselves that they are false, also say of themselves that they are true (but he does not prove, or say, anything explicit about other sentences not signifying their own truth). The backbone of his solution to liar-like paradoxes is precisely 26
Buridan, Summulae de Dialectica, p. (Sophismata ch. , th sophism); my italics.
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the claim that such sentences say contradictory things, and thus cannot be true (and hence are false). Buridan, in contrast, rejects the idea (which he had held earlier) that (all) sentences signify their own truth, seemingly on ontological grounds, but (as quoted above) he makes an equally general claim: every sentence virtually implies its own truth. Therefore while he seems to have strong ontological objections to the notion of the signification of a sentence (which is admittedly rather frail in the context of Bradwardine’s treatment of insolubilia), Buridan seems less careful when it comes to the notion of truth. Regarding his own earlier theory presented in his Questiones Elencorum, he rejects the notion of sentences signifying their own truth, but not the large scope given to this thesis, i.e., as applying to all sentences; it is now the property of virtually implying their own truth that applies to all sentences, and this is essentially presented as a postulate, with no particular argumentation given. Albert of Saxony (who does not reject the notion of the signification of a sentence, as Buridan does in the Sophismata) does offer an argument (though not a very strong one) to support the claim that every sentence signifies its own truth, based on the notions of supposition and signification. He says that for a sentence to be true amounts to the co-supposition of its subject and predicate, and that every affirmative sentence signifies that its subject and predicate co-supposit; hence every affirmative sentence signifies itself to be true (the same reasoning is used to prove that every negative sentence signifies itself to be true, and thus that every sentence signifies itself to be true).27 However, the general claim that every sentence signifies/virtually implies its own truth has problematic implications. In the case of Buridan, we have the requirement that for the truth of a sentence, the co-suppositional criterion must hold of all the consequences of a sentence for it to be true. So, with ‘CS(s)’ standing for the co-suppositional criterion holding of sentence s, (SCSC) can be formulated as ∀s[T(s) ⇔ (CS(s) & ∀p((s → p) ⇒ CS(p)))].
But in particular, ∀s[s → T(s)]
and thus, instantiating the universal quantifier ranging over p with T(s), we have ∀s[T(s) ⇒ (CS(s) & CS(T(s)))]. 27
See Latin text in Pozzi, Il Mentitore e il Medioevo, p. .
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For the co-suppositional criterion to be verified of T(s) amounts to its subject term, which is a term suppositing for sentence s, and its predicate term, the predicate ‘true’, both to supposit for the same thing(s) – that is, the cosuppositional criterion is satisfied iff s is true: CS(T(s)) ⇔ T(s). Thus, by substitution, ∀s[T(s) ⇒ (CS(s) & T(s))].
But of course the necessary conditions for s to be true are precisely what the right-hand side of the definition was supposed to specify; yet T(s) is one of its clauses. However, this definition is not entirely uninformative, since it does provide a procedure for identifying sentences that are not true: if one of the co-suppositional clauses is violated, either for s itself or for one of its implications, this is sufficient for the non-truth (or falsity, under bivalence) of the sentence. It may also indicate potentially strong candidates for truth, namely, those sentences whose co-suppositional clauses can all be verified except one (i.e., s itself and all its implications except T(s)). But it can never show in a definitive way that a sentence is true, since one of the necessary conditions for the truth of a sentence is precisely that it is true: the definition corresponds to a looping, non-terminating procedure. It would be nonterminating anyway if for a given sentence s there are infinitely many things that s signifies/virtually implies. But if (for every s) the truth of s is one of these things, then the procedure is looping (and thus non-terminating) for all sentences s. It is indeed a quasi-circular definition, and it turns every sentence into a semi-truthteller. Albert of Saxony’s definition of truth fares no better than Buridan’s. He requires that for a sentence to be true, all that the sentence signifies must obtain; but if every sentence signifies that it is true, then for every sentence to be true, one of the things that must obtain is that it is true. Hence quasicircularity ensues again. As for Bradwardine, quite surprisingly, the general principles laid down in his Insolubles actually allow for the derivation of the unrestricted claim that every sentence signifies its own truth, not only sentences that say of themselves that they are false.28 The gist of the proof is as follows: a sentence s signifies (among other things) the conjunction of all things P that it signifies, and the conjunction of all P that s signifies in turn implies the truth of s (by the definition of truth: a sentence is true iff all the things it signifies are the 28 The proof is in Read, ‘The Validity Paradox’, in M. Pelis (ed.), The Logica Yearbook, (London: College Publications, ), and ‘Field’s Paradox and its Medieval Solution’, forthcoming in History and Philosophy of Logic.
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case). But then, by transitivity, sentence s signifies its own truth. This is quite a blow for the project of developing a full-fledged theory of truth on the basis of Bradwardine’s analysis of insolubilia; it means that the core of the objections raised against Buridan and Albert of Saxony here and elsewhere (in particular in Read’s LPB) also applies to Bradwardine. In other words, within Bradwardine’s framework it would be just as impossible to determine that a sentence is true (although it would be possible to determine that it is false). Given that every sentence signifies itself to be true, one of the things which according to the quantificational principle must obtain for it to be true is that it is true. More formally, the definition of truth is ∀s[T(s) ⇔ ∃P(s : P) & ∀P[(s : P) ⇒ P]].
But, as proved by Read, ∀s(s : T(s)).
So, just as with Buridan, instantiating P with T(s), we obtain ∀s[T(s) ⇒ T(s)].
The lesson to be learnt from these observations seems to be that if closure under implication is present in an account of truth (immediately, as in Buridan’s analysis, or indirectly, as in Bradwardine’s and Albert’s analyses via the notion of signification), then the thesis that every sentence implies its own truth undermines the effectiveness of the definition of truth. Buridan and Albert of Saxony manœuvre themselves into this situation by postulating that every sentence signifies/virtually implies its own truth; Bradwardine does not have such a postulate, but is dragged into a similar predicament by his own fundamental principles. Thus in order to formulate a theory of truth inspired by Bradwardine’s treatment of insolubilia, one would have to revise his fundamental principles so as to block the derivation of the unrestricted claim. The quantificational definition of truth combined with a notion of signification closed under implication inevitably leads to the conclusion that every sentence signifies its own truth, which in turn turns every sentence into a semi-truthteller. Hence the two principles simply cannot be combined if one is to formulate a complete, effective theory of truth. It is also clear why Bradwardine’s general solution to paradoxical sentences has so far proved to be so effective in the treatment of a wide range of paradoxes (as shown in several of Read’s articles): all these paradoxical sentences come out as false, according to the Bradwardinian analysis, never as true. As noted, Bradwardine’s theory, like Buridan’s, is perfectly able to show that a sentence is false; what it cannot do is show that a sentence is true. © The Author
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This does not mean, however, that Bradwardine’s account of truth is entirely doomed to failure; in fact, the insights related to semantic pluralism and to his quantificational approach to truth seem quite valuable. But something has to go, and I believe that it is the notion of sentential signification closed under implication which needs to be revised. Alternatively, one may prefer to maintain both principles, bite the bullet, and allow that it is indeed not possible to formulate an effective non-circular definition of truth. (As they say, one person’s modus ponens is another person’s modus tollens.) Klima (in ‘Logic without Truth’) still attempts to rescue Buridan by claiming that while it may be true that in the end he is left with no definition of truth at all, this is not a problem for him, because his logic does not require a notion of truth. Indeed, his notion of validity is defined not in terms of truth but in terms of the ‘however a sentence signifies things to be’ clause (plus the familiar modal notion of impossibility), in both his Treatise on Consequences and his Sophismata: ... a proposition is antecedent to another which is related to it in such a way that it is impossible for things to be however the first signifies them to be without their being however the other signifies them to be, when these propositions are formed at the same time.29
But actually the ‘however a sentence signifies things to be’ clause is for Buridan an abbreviation for the satisfaction of the (appropriate) co-suppositional clauses, so the basic definition of a valid consequence can be glossed as ‘it is impossible for the appropriate co-suppositional clauses for the truth of the antecedent to be satisfied while the appropriate co-suppositional clauses for the truth of the consequent are not satisfied’ (the qualification of ‘appropriate’ clauses is necessary because sentences have different co-suppositional clauses depending on whether they are affirmative or negative, universal or particular, etc.). Therefore similar problems for Buridan’s account seem to re-emerge; if (SCSC) is to be applied to all sentences and not only to liar sentences, then the question of their truth resurfaces, because ‘antecedent is true’ and ‘consequent is true’ are two of the sentences of which it must be determined whether the co-suppositional condition holds (since they are implied by antecedent and consequent, respectively) in order to assess the relative modal value of consequent with respect to antecedent. This may not be a problem if (SCSC) is to be applied only to liar-like sentences, given that a liar sentence is in practice an impossible sentence (not only is it not true, it cannot be true; Buridan accepts the ‘ex impossibili’ rule), so its relative as well 29 Latin text in Buridan, Treatise on Consequences: H. Hubien, ‘Iohannis Buridani tractatus de consequentiis’, Series Philosophes Médiévaux, Vol. (Université de Louvain, ), p. .
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as absolute modal value is already well established. But if (SCSC) is to be applied to all sentences, then we are again in a situation of underdeterminacy and circularity, given that one needs to know to start with whether B follows from A (i.e., one needs to know whether it is possible that ‘A is true’ is the case while ‘B is true’ is not) in order to apply (SCSC) (relative to each other), and thus to determine whether B follows from A. Be that as it may, it is clear that Buridan’s attempts to avoid ontological commitment to the ‘signification’ of sentences (as discussed in my LSM) still do not make his account immune to the logical difficulties into which any account of truth combining quantification and signification defined in terms of closure under implication seems to run. If it was indeed Bradwardine’s account and his ontological commitments that Buridan was targeting, then he seems to have missed the logical weakness of Bradwardine’s theory. This also means that it is not Buridan’s nominalistic ontology that makes his definition of truth ineffective, but rather (as with Bradwardine) the combination of closure under implication with a quantificational approach to truth. V. RESTALL ON BRADWARDINE In some recent papers, Greg Restall has offered intriguing model-theoretic formalizations of the theory of truth that emerges from Bradwardine’s treatment of insolubilia.30 From the point of view of a historian of logic, it is of course very gratifying that Bradwardine’s writings are seen as containing sufficient insight, beyond mere historical interest, so as to justify systematic interest in them. Moreover, Restall’s work has certainly elucidated some aspects of Bradwardine’s theory which are more appropriately studied by means of formal tools rather than by conceptual analysis alone. Nevertheless, I believe that some crucial aspects of Bradwardine’s theory are misrepresented in Restall’s formalization. A formalization is of course always a simplification, but a simplification which promises to offer further insight precisely because it separates what is really crucial from what is secondary, allowing for a more uniform analysis. In any formalization, some elements of the target phenomenon are represented by certain features of the model – what Shapiro called the representers31 – while other features of the model are artefacts (again in 30 G. Restall, ‘Models for the Liar in Bradwardine’s Theory of Truth’, in Rahman et al. (eds), Unity, Truth and the Liar, pp. –, and ‘Modal Models for Bradwardine’s Theory of Truth’, Review of Symbolic Logic, (), pp. –. 31 S. Shapiro, ‘Logical Consequence: Models and Modality’, in M. Schirn (ed.), Philosophy of Mathematics Today (Oxford UP, ), pp. –.
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Shapiro’s terminology), introduced for convenience of study, not corresponding to anything really present in the target phenomenon; similarly, some aspects of the latter are left out completely. So a good formalization is not one where every aspect of the target phenomenon is represented, but rather one where the trade-off between simplification and accuracy of representation is favourable. (So my criticism here should not be understood as directed towards the very enterprise of formalizing theories whose original presentation is somewhat more informal.) But what are the criteria for what is to count as a good formalization? This is a very delicate issue, but one which does not seem to receive the attention it deserves.32 In particular, if a formalization makes a prediction which is not made by the informal theory being formalized (or vice versa), i.e., if there is a mismatch between formalization and what is formalized, then this may mean one of two things: either the formalization is not sufficiently faithful to the informal theory, in which case it is a ‘bad’ formalization, or the formalization in fact ‘sees’ something the original theory had failed to see, in which case it is a ‘good’ formalization in that it makes a non-trivial prediction. If, however, the predictions of the informal theory and of the formalized version match completely, then on the one hand one may say that the formalization is entirely accurate, but on the other hand one may also say that it is uninformative, in that it has brought no further insights other than those already made explicit by the informal theory itself. So there is a sense in which the cases of mismatch are precisely the interesting ones; when mismatch occurs, further analysis is required in order to establish whether this is indeed a novel result revealed by the formalization or rather a sign that there is something wrong. In Restall’s formalization of Bradwardine’s theory of truth we have exactly such a situation of mismatch. Bradwardine’s theory predicts (in fact, proves) that every sentence which says of itself that it is false also says of itself that it is true; but Restall claims to provide a counter-example to this claim, i.e., the construction of a simple model where (according to his framework) there is a declarative object which says of itself that it is not true (a liar sentence, therefore), yet does not say of itself that it is true. Restall seems to imply that the model which he sees as a counter-example (‘Modal Models’, p. ) suggests that there is a problem with Bradwardine’s proof: ‘At the very least, we can use them [Bradwardine models] as a tool for finding counter-examples to certain claims, such as the claim that λ : Tλ follows 32 For some of the (surprisingly few) discussions of this issue, see C. Dutilh Novaes, Formalizing Medieval Logical Theories (Berlin: Springer, ), ch. ; G. Brun, Die richtige Formel (Frankfurt: Ontos, ); M. Baumgartner and T. Lampert, ‘Adequate Formalization’, Synthese, (), pp. –.
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from λ : ¬Tλ under the Bradwardine axiom’. It was Bradwardine himself who had claimed (proved) that from the fact that a sentence says of itself that it is not true it follows that it says of itself that it is true, and more importantly, this proof is the backbone of his solution to the liar paradox: a liar sentence is a sentence that says two things which cannot obtain simultaneously (namely, that it is true and that it is not true), and therefore it is an impossible (and a fortiori false) sentence. If Restall is right in claiming that not every liar sentence says of itself that it is true (besides saying of itself that it is false), then it seems that Bradwardine’s whole solution to liar sentences falls apart. There seems to be a lot at stake here, so we must try to figure out whether there is a problem with Restall’s formalization or whether it is Bradwardine’s approach to the liar that is ultimately flawed. As a matter of fact, closer inspection reveals that the supposed countermodel is a counter-model mainly in virtue of a particular characteristic of Restall’s model-theoretic construction, namely, that the worlds are construed as (maximally) consistent. The notion of saying is construed as a modal notion, i.e., in terms of accessibility relations to other worlds, in the following way: a declarative object t in a world w says everything that holds in all the worlds v to which w is related by means of the accessibility relation defined by the particular declarative object t (so every declarative object defines its own accessibility relation). But if worlds are construed as consistent, then this simply means that no declarative object will ever (be allowed to) say contradictory things – unless of course the accessibility relation is empty, in which case it says everything, vacuously. But the liar does not seem to say everything. Moreover, this would imply that sentences that say contradictory things all say the same thing, namely everything, which also seems wrong. What we need is a notion of saying which is sufficiently fine-grained to make a distinction between different sentences saying contradictory things, yet not all saying the same. For a declarative object in w to say something contradictory, say A and ¬A, A and ¬A would have to hold in all the worlds accessible from w by means of the appropriate accessibility relation; but there simply are no worlds where A and ¬A both hold. How can we ever have A and ¬A in all worlds accessible from w by means of the appropriate accessibility relation (except vacuously) if there are no worlds at all where A and ¬A hold? Hence the fact that in Restall’s formalization possible worlds are construed as (maximally) consistent immediately implies that no declarative object can ever say contradictory things (except vacuously). Is this a fair requirement for the notion of saying? It is clear that it automatically blocks Bradwardine’s solution to the liar, but not because it shows it to be internally inconsistent; rather, it seems that the notion of saying underlying © The Author
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Bradwardine’s solution is much too fine-grained to be handled by models containing only consistent worlds. In personal communication, Restall has remarked that it is not so much of a problem if his models overgenerate with respect to Bradwardine’s theory, i.e., if they yield results that Bradwardine might not want to endorse, because the aim is to show that certain things do not follow from basic principles. ‘If a model (a strange one, which overgenerates) satisfies all of the axioms Read finds in Bradwardine, but which does not validate something else, then this shows that this something else doesn’t follow from merely those axioms’ (Restall, personal communication). Here, however, we seem to have a case of undergeneration; Bradwardine’s theory predicts that sentences may say contradictory things, but Restall’s framework blocks this possibility from the start. So while overgeneration may not be a problem for his purposes, the same does not seem to hold of undergeneration. In this sense, it appears that the requirement of consistency for worlds is not a representer in Shapiro’s terms; it is an artefact, and more specifically an artefact that entails undue oversimplification. Models with worlds that are (maximally) consistent are convenient to work with, but they seriously misrepresent the notion of saying, and this is essentially why Restall’s liar ‘fails’ to say of itself that it is true, and so not necessarily in virtue of any deeper shortcoming in Bradwardine’s own argumentation. So in Restall’s model the liar cannot say of itself that it is true, simply because the possibility of its saying contradictory things is blocked from the outset. But even if we consider the notion of saying in general, i.e., apart from its role in Bradwardine’s solution to the liar, again the requirement of consistency appears to be much too strong: there is no reason why there should be a ban on saying contradictory things (in fact they seem to be said quite often). If the notion of saying is to be construed as modal, i.e., in terms of accessibility relations between worlds, then clearly these worlds should not be defined as consistent. What might be required is a notion of ‘impossible worlds’ (i.e., worlds where contradictory things hold – for example, that a given liar sentence is and is not true) which is sufficiently fine-grained to differentiate impossible worlds from one another. If every impossible world is identical with every other impossible world, then all the sentences which say contradictory things mean exactly the same thing; for example, a sentence which says of itself that it is true and that it is not true, and another sentence which says that grass is green and that grass is not green, have exactly the same meaning, which again would be an undesirable result. (It is, by the way, also what happens in Restall’s framework, but for different reasons: for a sentence to say contradictory things, the relevant accessibility relation must be empty, in which case it says everything.) The upshot is © The Author
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again that the notion of saying appears to be extremely fine-grained, and a formalization of it must take this element into account. Indeed, the idea of providing counter-models to certain key claims is prima facie a good one; but what would be required in this particular case is a semantic framework, satisfying the basic ‘Bradwardine axioms’, in which it is possible for sentences to say contradictory things, yet a liar sentence which says of itself that it is false but does not say of itself that it is true can be construed. This would be a true counter-example to Bradwardine’s claim, precisely because the possibility of a sentence saying contradictory things exists and yet is not realized with respect to at least one liar sentence. But that is not what is happening in Restall’s construction. (E. Zardini suggested in personal communication that one possibility is to adopt a neighbourhood rather than a relational semantics to model the notion of saying, so that (s : P) and (x : ¬P) could occur and be accounted for in a non-trivial way, i.e., not by an empty accessibility relation.) Interestingly, though, Restall also claims that what he calls ‘Buridan’s axiom’ (that every sentence says of itself that it is true) holds of all declarative objects in his ‘Bradwardine models’ (p. ).33 Here the mismatch between informal theory and formalization actually reveals something about the informal theory which is not immediately apparent, namely, that even within Bradwardine’s theory it can be proved that every sentence signifies itself to be true (not only liar sentences). Indeed, Read’s recent proof to this effect sheds new light on the ‘dispute’ between Bradwardine and Buridan. Read and I have both claimed that Bradwardine’s theory of truth was vastly superior to Buridan’s precisely because it avoided the pitfalls related to all sentences saying of themselves that they are true. It is now clear that things are not so straightforward: Bradwardine’s theory is not immune to these pitfalls after all. Restall’s model-theoretic formalization has already suggested this much, and it was right.
VI. CONCLUSIONS • Bradwardine’s quantificational approach to truth has the advantage of not making substantial assumptions concerning the meaning of sentences: it works just as well whether sentences say at most one thing or say several things. Given this, it is to be preferred over Tarskian approaches, which 33 As I have just described, this is not the case of the liar sentence in his alleged counterexample (it does not say of itself that it is true), so the claim that Buridan’s axiom holds in all Bradwardine models is somewhat strange. There seems to be an internal inconsistency between these two claims.
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presuppose that sentences say exactly one thing. Of course, in Tarski’s original formulation, this assumption makes perfect sense, as his theory of truth is originally designed for formalized (i.e., ‘sanitized’) languages. But when it comes to formulating a general theory of truth for all languages on the basis of the T-schema (as a number of philosophers have done after Tarski), the assumption of uniqueness of meaning is much too contentious, and therefore the quantificational approach is to be preferred. • Mediaeval theories of truth based on the notion of supposition are ontologically parsimonious, which is generally speaking a good trait in a theory. But these theories seem not to be able to treat the phenomenon of liar sentences adequately. At any rate, the attempts which have been made to solve the liar paradox within the framework of supposition theory all seem to have failed: e.g., Ockham’s restriction of self-reference is ad hoc, and Buridan’s strengthened co-supposition criterion runs into the difficulties described above. Surprisingly, though, a significant portion of the objections raised against Buridan’s theory of truth also apply to Bradwardine’s theory in so far as it also predicts that all sentences signify their own truth. So a fully satisfying quantificational approach to truth still needs to be developed, and the ontological disagreements between Buridan and Bradwardine seem not to be at the heart of the matter after all. • A recent objection raised by Greg Restall against Bradwardine’s treatment of liar sentences, that at least some of these sentences do not in fact seem to say of themselves that they are true, is dissolved by the observation that it is rather the manner in which he construes the notion of what a sentence says that is not an accurate rendering of this notion. Nevertheless, Restall’s claim to the effect that what he calls ‘Buridan’s axiom’ also holds in Bradwardine’s framework has subsequently been confirmed, which suggests that Bradwardine’s fundamental principles would have to be revised if they are to constitute the backbone of a full-fledged theory of truth.34 University of Amsterdam
34 This research was partially funded by the Niels Stensen Foundation (the Netherlands). Thanks to Stephen Read, Gyula Klima, Greg Restall and Elia Zardini for comments and helpful discussions.
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