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A Comparative Taxonomy of Medieval and Modern Approaches to Liar Sentences

2008; Taylor & Francis; Volume: 29; Issue: 3 Linguagem: Inglês

10.1080/01445340701614464

1464-5149

Catarina Dutilh Novaes,

Historical Philosophy and Science

Abstract Two periods in the history of logic and philosophy are characterized notably by vivid interest in self-referential paradoxical sentences in general, and Liar sentences in particular: the later medieval period (roughly from the 12th to the 15th century) and the last 100 years. In this paper, I undertake a comparative taxonomy of these two traditions. I outline and discuss eight main approaches to Liar sentences in the medieval tradition, and compare them to the most influential modern approaches to such sentences. I also emphasize the aspects of each tradition that find no counterpart in the other one. It is expected that such a comparison may point in new directions for future research on the paradoxes; indeed, the present analysis allows me to draw a few conclusions about the general nature of Liar sentences, and to identify aspects that would require further investigation. Acknowledgements This research was funded by a grant from the Niels Stensen Foundation (The Netherlands). Notes 1 For reasons of space, I shall focus on recent literature on the Liar paradox. However, it must be noted that there is a rich older tradition on this topic. A. Rüstow's Der Lügner. Theorie, Geschichte und Auflösung (1910), for example, is a classic. 2 For the present purposes, I focus on insoluble sentences of the Liar family, that is, sentences that assert the falsity of a sentence (itself or other), leading to paradox. The typical case is of course 'This sentence is false', but I also consider quantification over sentences such as 'Every sentence is false' or even 'Some sentence is false' under the assumption that there are no other sentences, as well as reciprocal Liar sentences: Socrates says 'What Plato is saying is false', while Plato says 'What Socrates is saying is true'. But to be sure, there is a considerable range of insoluble sentences, including epistemic and practical paradoxes, which shall not be directly dealt with here. As for the terminology, I will use the terms 'Liar sentence' and 'insoluble' interchangeably, usually (but not necessarily) the medieval term for medieval contexts and the modern term for modern contexts. When this does not occur, it should not be interpreted as having any kind of deeper meaning. 3 But notice that, while I may occasionally disagree with Spade's interpretation of the material, anyone working on medieval insolubles (in fact, on later medieval logic and semantics in general) is profoundly indebted to his work. Not only did he catalogue manuscripts and edit/translate many of the insolubilia treatises which are now available in modern editions, but his analyses of the material are also exceptionally profound and challenging. In fact, it is hard to imagine what working in this field would even be like if it were not for Spade's impressive contribution to it. 4 Edited in Roure Citation1970, but currently in the course of receiving a critical edition with translation by S. Read, to appear with Dallas Medieval Texts and Translations. 5 For the present analysis, I rely on Read's forthcoming edition of Bradwardine's text and on Spade's description of Paul's text in his Spade Citation1975. Simmons Citation1993, ch. 5 also relies significantly on Paul's text; he was able to use an unpublished edition/translation of the text (by M. McCord Adams). The text is otherwise unfortunately of difficult access. 6 A text representing the position that Bradwardine describes, that is, the distinguentes position but not associated with a fallacy, is Insolubilia Monacensia—see Nuchelmans Citation1988. 7 In his Quaestiones Elenchorum, Question LIII (John Duns Scotus 1891). 8 I change the order of presentation with respect to Paul's list because his thirteenth position is chronologically the last one to have appeared. He presents Swyneshed's position as the last one in his list because it is the one he favours himself, but chronologically Swyneshed's position should even come before Heytesbury (indeed, Spade believes Swyneshed's treatise to have been written before Heytesbury's—cf. Spade Citation2005). 9 Other influential texts tend to list roughly the same positions. Ralph Strode, for example, analyses the positions of Bradwardine, Heytesbury, Swyneshed, and Robert Fland, basically accepting Fland's position (a variation of Bradwardine's and Heytesbury's solutions). 10 The fact that this term, imported from the obligational framework, is used to name the earliest medieval position with respect to Liar sentences is one of the strongest pieces of evidence presented by Martin (Citation2001) to corroborate his (very convincing) thesis that the origin of the medieval insolubilia literature is to be found in connection with obligations. 11 Simmons (Citation1987, Citation1993, 5.2.2) has made a similar point with respect to Pseudo-Sherwood's and Ockham's respective solutions, both overtly cast in terms of this fallacy. 12 Some of the other authors who explicitly use this fallacy in their solutions to the Liar are Simon of Faversham, Lambert of Auxerre/Lagny, and Duns-Scotus, as already mentioned. For the first two, see the relevant texts in Pozzi Citation1987. 13 One should not exclude the possibility of it saying yet other things. 14 The relevant texts can all be found in Pozzi Citation1987 (in Latin and Italian). 15 In the 14th century, it was very common to spell out the truth-conditions of sentences in terms of the supposition of its terms. In the first chapters of the second part of his Summa Logicae (Citation1974), Ockham presents the following truth-conditions: in the case of affirmative sentences, they are true if the predicate supposits (at least) for the same thing(s) as the subject; in the case of negative sentences, they are true if this does not occur—which can happen in two ways, namely if the subject does not actually supposit for anything (in the case of empty names) or if it does but there is no intersection between its supposita and the supposita of the predicate. 16 What comes closest to such a realization is the position known as transcasus (essentially in the same spirit as the restrictionist approach) (see Spade Citation2005, 2.2; Bradwardine comments on it in 5.4 of his treatise), which consisted in maintaining that, when uttering (L) 'This sentence is false', what is referred to is a sentence previously uttered by the speaker, and not the very sentence (L). 17 Notice that, in the quotations of translations of medieval texts, the term 'proposition' appears in its Latin acceptation of propositio, i.e. equivalent to what I refer to elsewhere in this text as declarative sentences. 18 On the use of the notion of what a sentence signifies to define truth, in particular in connection with insolubles, see Read Citation2002. 19 Given the overview purpose of this piece, this remains a project for future research. 20 In §§28–30, Swyneshed also examines and rebuts a possible objection to this truth-value gap approach based on an authoritative text by Aristotle; he presents an alternative reading for the Aristotelian text according to which the necessity of bivalence does not necessarily follow from it. 21 Unfortunately, the word for 'falsity' in Greek is significantly less nice than the word for 'truth'. 22 Here I refer to the non-classical position with respect to truth-values, namely the rejection of bivalence and of non-contradiction. Earlier in this paper, I have referred to the non-classical approach which consists of reformulating or rejecting classical rules of inference; obviously, these two non-classical approaches are related, since a rejection of the traditional view on truth-values often (but not necessarily) leads to a revision of the classical rules of inference; but these are nevertheless essentially different non-classical approaches. 23 As is well known, the concept of truth-value gaps is only one of the elements of Kripke's mathematically sophisticated theory of truth. The other key concept in the construction is that of a fixed-point, but this latter concept has no particular relevance for the comparison being presented here. 24 It has been argued that the problem that gap-solutions have with the strengthened Liar in fact arises just as much with the weaker versions of the paradox (Rieger Citation2001). The real problem for any version of the paradox would be indeed that of revenge. 25 By contrast, I am not aware of any medieval discussion on what differentiates sentences lacking a truth-value from the 'well-behaved' ones. 26 Buridan's notion of virtual implication is related to his staunch commitment to actually formed tokens: 'The point of the qualification "virtual" in this phrase seems to be that for this implication to hold the consequent need not actually be formed, although, of course, the antecedent has to exist (for otherwise it could not be said to be true)' (Klima Citation2004, p. 105). 27 Buridan had argued in a similar fashion for this position in his Quaestiones Elenchorum, p. 92. 28 But notice that, as noted by Pironet (forthcoming), the range of application of Heytesbury's rules for solving insolubilia was not reduced to performing in an obligational disputation. The notion of casus, for example, while arguably originally stemming from the obligational framework, was widely used for sophismata as well (cf. Pironet forthcoming, p. 245). But Pironet (forthcoming, p. 243) confirms that Heytesbury was indeed the first in the 14th century to analyse insolubilia explicitly against the obligational background. Pironet (forthcoming) is an in-depth analysis of the background of Heytesbury's treatise as well as of its influence in later authors; it also offers several examples of specific insoluble sentences and the casus required for each of them to be a real insoluble. 29 In his quietism with respect to Liar paradoxes, Heytesbury goes as far as acknowledging that even the solution he favours (the fourth solution examined by him) is not entirely satisfactory (§9, §43), and that insolubles are indeed insoluble (§64). Among modern authors, such quietism is rarely found, precisely because we now think that there is an awful lot at stake with such paradoxes (the very possibility of a theory of truth and semantics). But perhaps the revision theory of truth (proposed independently by Herzberger and Gupta—see Kremer Citation2006) is what comes closest to Heytesbury's attitude towards insolubles; rather than solving these paradoxes, the revision theory of truth simply seeks to model the reasoning leading to paradox. 30 An example from Ockham: Socrates begins to talk and says 'Socrates says a falsehood', and then says nothing else (in Pozzi Citation1987, p. 138). Without the specification that he says nothing else, the sentence may not be paradoxical after all. 31 This is a terminology borrowed from Gaifman's (Citation1992, Citation2000) pointer semantics, which is particularly suitable to help us see what is going on in this example. 32 Notice that, in this particular aspect, Glanzberg's approach has a remarkable similarity with Bradwardine's analysis. The difference is that, for Bradwardine, a Liar sentence typically expresses not one but several contents. 33 Gauker (Citation2006) offers a sustained critique of the contextualist approach to paradox. According to him, the essence of such (modern) solutions is what he calls 'stepping back': 'Every one of these authors, as far as I am aware, has supposed that what we have to explain, above all, is how it is that once we judge that a liar sentence such as "This sentence is not true" is not true, we can in some sense 'step back' and judge that the liar sentence is after all true. This is supposed to be possible because the context in which we judge that the liar sentence is true is not the context in which the liar sentence says of itself that it is not true' (Gauker Citation2006, p. 393). Gauker claims that this description fits in particular the proposals by Simmons, Glanzberg, and Barwise and Etchemendy. To me, though, it seems that, while it is indeed a powerful critique of the 'stepping back' approach, the different contextualist approaches, even these three mentioned by him, go well beyond this basic principle, and thus his critique is not as far-reaching as it claims to be. 34 See Spade Citation2005, note 26. 36 Latin text in Spade Citation1979, §§14–15; (translation from Spade Citation1983, p. 105). For an explanation of the notions involved in this definition which are not essential for the present discussion, such as 'to signify principally', 'to signify naturally' etc., see Spade Citation1983. 35 Thus, with this definition of truth, just as much as with Bradwardine's, the standard correspondentist semantic ascent from 'p' to 'p is true' is not valid, as noticed by Spade (Citation1983, p. 105). 37 Latin text in Spade Citation1979, §5; my translation. 38 One of Swyneshed's conclusions in his treatise that was viewed with much skepticism in his time was precisely the claim that, in some valid inferences, falsehood follows from truth, in the case of inferences which are valid according to the modal criterion (things cannot be as the antecedent signifies them to be without being as the consequent signifies them to be), but which have a false consequent that does signify things as they are and is thus false on account of being self-falsifying (Spade Citation1979, §26). In sum, the very notion of 'follows from' must be reconsidered in view of his redefinition of the notion of truth so as to include non-self-falsification, but the very notion of self-falsification presupposes some previous notion of 'follows from' to be defined. There is thus an imminent threat of circularity in these definitions. 39 I owe the reference to Van Benthem Citation2004 to Sara Uckelman. 40 Notice that this psychological 'flip-flop' is also found in the revision theory of truth (cf. Kremer Citation2006), where a Liar sentence is taken successively to be false, true, false, true, etc. 41 This is obviously not true of dialethic theorists; however, the reason why they reject the idea that a theory explodes if a contradiction is derived from it is related to their views on the very notion of contradiction, and not to a non-foundational view of logic and semantics as that held by the medieval authors.