Ernesto Perini-Santos - John Buridan’s Theory of Truth and the Paradox of the Liar

© Koninklijke Brill NV, Leiden, 2011 DOI: 10.1163/156853411X590499

Vivarium 49 (2011) 184-213 brill.nl/viv

vivarium

John Buridan’s Theory of Truth and the Paradox of the Liar*

Ernesto Perini-SantosUFMG, Brazil

AbstractThe solution John Buridan offers for the Paradox of the Liar has not been correctly placed within the framework of his philosophy of language. More precisely, there are two important points of the Buridanian philosophy of language that are crucial to the correct understanding of his solution to the Liar paradox that are either misrepre-sented or ignored in some important accounts of his theory. The first point is that the Aristotelian formula, ‘propositio est vera quia qualitercumque significat in rebus significa-tis ita est’, once amended, is a correct way to talk about the truth of a sentence. The second one is that he has a double indexing theory of truth: a sentence is true in a time about a time, and such times should be distinguished in the account of the truth-conditions of sentences. These two claims are connected in an important way: the Aristotelian formula indicates the time about which a sentence is true. Some interpret-ers of the Buridanian solution to the paradox, following the lead of Herzberger, have missed these points and have been led to postulate truth-values gaps, or surrogates of truth-value gaps, when there is nothing of this sort in his theory. I argue against this tradition of interpretation of Buridan and propose an interpretation of his solution to the Liar.

KeywordsParadox of the Liar, John Buridan, medieval theories of truth, medieval semantics, medieval pragmatics

* Earlier versions of this paper were read at conferences in Bologna and Rio de Janeiro. I thank the audiences on both occasions for helpful comments. I am also grateful for the support of the Conselho Nacional de Desenvolvimento Científico e Tecnológico.

E. Perini-Santos / Vivarium 49 (2011) 184-213 185

1. Introduction

The solution John Buridan offers for the Paradox of the Liar is certainly among the more oft-discussed medieval approaches to the subject.1 In spite of this, I think that it has not been entirely understood, nor has it been correctly placed within the framework of his philosophy of language. More precisely, there are two important (and related) points of the Buridanian philosophy of language that are crucial to the correct understanding of his solution to the Liar paradox that are either misrepresented or ignored in some important accounts of his theory. The first point concerns the interpretation of the formula ‘propositio est vera quia qualitercumque significat in rebus significatis ita est’—hereafter ‘the Aristotelian formula’. The second point concerns the distinction between the time about which ( pro quo) a sentence is true and the time in which (in quo) it is true, a distinction that is at the heart of the seventh chapter of the Soph-isms, and which is taken up at the beginning of the eighth chapter, in which he discusses the Liar. I propose two main theses that are, to my mind, essential to the correct interpretation of his solution to the Liar paradox:

(i) The Aristotelian formula, once amended, is a correct way to talk about the truth of a sentence.

(ii) A sentence is true in a time about a time, and such times should be distin-guished in the account of the truth-conditions of sentences.

These two claims are connected in an important way: the Aristotelian formula indicates the time about which a sentence is true. I defend these claims in sec-tions 2 and 3 of this paper. Following this, I examine some important inter-pretations of the Buridanian solution to the Liar. In the light of the results of the previous sections, we will see that they are wanting. Finally, I present my own interpretation to the Buridanian solution to the paradox of the Liar.

The Buridanian theory of truth turns out to have a pragmatic aspect. In very general terms, pragmatics deals with linguistic phenomena that can be explained only by considering the context of production of linguistic items. Such phenomena can be divided in two very broad kinds: the different types of “speech acts and speech products,” on the one hand, and the way the content

1) “The task Buridan sets for himself in the eighth chapter is that of dispelling the air of paradox surrounding certain self-referential propositions. His analyses of these sophisms pushed termin-ist logic to new heights, and remain impressive quite apart from their historical context, as it evidenced by the large secondary literature on the topic.” (Jack Zupko, John Buridan (Notre Dame, 2003), 131).

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of a given utterance depends on the context of its production, on the other.2 The first domain of pragmatics deals with the different kinds of acts performed by an utterance, their effects, conditions of satisfaction etc. I will have nothing to say about it in this paper. On the other hand, Buridan’s theory of truth recognizes the need to identify the context of utterance of a sentence and to distinguish it from its circumstances of evaluation, to give its adequate truth-conditions. And this point, suggested by (ii), is crucial to his solution to the Paradox of the Liar. As I will try to show, even if there are proximities between Buridan and some contemporary theories that make room for contextual effects in the determination of the content of an assertion, there are also some important differences. As a result, we will find an interesting and original pic-ture of a pragmatic theory of truth.

2. The amended Aristotelian formula is a correct way to talk about the truth of a sentence

In his Sophisms, Buridan states explicitly that the Aristotelian formula does not offer a correct definition of truth:

The second conclusion is that it is not required for the truth a spoken sentence that howso-ever it signifies [things to be] outside [the mind] so should things signified be outside [the mind]. Or let us put this conclusion in following form: some sentence is true, yet it is not the case that howsoever it signifies [things to be] outside, so are the things that are signified outside.3

There are three arguments supporting this claim. The first and more impor-tant argument4 is based on the fact that sentences that are not affirmative, in

2) See Robert Stalnaker, ‘Pragmatics’, in Context and Content (Oxford, 1999), 34.3) John Buridan, Summulae de Dialectica, transl. G. Klima (New Haven-London, 2001), 850 (hereafter, ‘transl. Klima’). “Secunda conclusio est quod non requiritur ad veritatem propositio-nis vocalis quod qualitercumque significant ad extra, ita sint quantum ad res significatas ad extra. Vel ponatur conclusio sub hac forma: aliqua propositio est vera, tamen non est qualitercumque significant ad extra, ita est in rebus quae significantur ad extra.” (Summulae de Practica Sophisma-tum, ed. F. Pironet (Turnhout, 2009) II, 2 conclusio, 38, l. 7-10—hereafter, ‘Soph.,’). In the quotations of Klima’s translation, I replace ‘proposition’ by ‘sentence’ as a translation of proposi-tio, so as to have a uniform vocabulary throughout the text.4) It is the only argument present in the following texts dealing with the formula: the commen-taries to the Metaphysics, Lectura Erfordiensis in I-V Metaphysica, ed. L.M. de Rijk (Turnhout, 2008), qu. 23A, § 624, p. 145, l. 10-19, the Abbrevitationes Caminenses, ed. L.M. de Rijk (Turn-hout, 2008), § 92, p. 197, l. 29-33 (hereafter ‘Lect. Erf.’ and ‘Abbrev. Cam.’) and the Quaestiones in Aristotelis Metaphysicam (Paris, 1518) [reprint, Frankfurt am Main, 1964], VI, qu. 9 (hereaf-


the present tense and de inesse may be true without being in things signified as they signify them to be. Let us consider the following sentence:

(1) Antichrist will walk

(1) signifies the Antichrist and his future walk, but such things are nothing. According to the Aristotelian formula, a sentence is true if, and only if, how-soever it signifies things outside to be, so they are. In this case, the things signi-fied do not exist, so cannot be in some way or another. The tense of the last verb of the formula does not represent the fact that a future tense sentence is true because things will be in a certain way. The same argument applies to past tense sentences, to de possibili sentences and, in a modified way, to negative sentences. None of such sentences can be true because things are the way they signify things to be

This line of thought has two important features. On the one hand, affirma-tive de inesse present tense sentences remain untouched by the argument: nothing in it prevents the application of the Aristotelian formula to such sen-tences. On the other hand, a rather straightforward amendment of the for-mula seems to make it fit other kinds of sentences. Both consequences appear in more than one text. The early Erfurt commentary on the Metaphysics states the conclusion as follows:

And I state briefly [the conclusion] that we should speak differently about present tense sentences and about past tense and future tense sentences, and [we should speak] differently about affirmative sentences and about negative sentences. I state therefore that an affirma-tive atomic present tense [sentence] is true because howsoever it signifies [things] to be, so are the thing or the things signified. But a future tense [sentence] is true because howsoever it signifies [things] coming to be, so [they] will be, and a past tense [sentence] is true because howsoever it signifies [things] having been, so [they] have been.5

ter, ‘Quaest. Met.’)—, the Tractatus de consequentiis, ed. H. Hubien (Louvain-Paris, 1976) I, 1, 14-27, p. 17 (hereafter, ‘Tractatus’) and the Quaestiones in Analytica Posteriora, ed. H. Hubien (unpublished) II, qu. 10a (herafter, ‘Quaest. Analyt. Post.’)—in this latter text, there are four arguments against the Aristotelian formula, two versions of the present argument, and two argu-ments based on the insolubilia. 5) “Et dico breviter quod aliter oportet loqui de propositionibus de presenti et aliter de istis de preterito et de futuro, et aliter de istis affirmativis et de negativis. Dico ergo quod affirmativa kathegorica de presenti ex eo est vera quia qualitercumque significat esse, ita est in re significata vel in rebus significatis. Sed illa de futuro ex eo est vera quia qualitercumque significat fore, ita erit, et ista de preterito vera est quia qualitercumque significat fuisse, ita fuit.” (Lect. Erf., qu. 23A, 625, p. 145, l. 22-28. See also de Abbrev. Cam., § 92, p. 197, l. 33-42).


This is also the result of the Quaestiones In Analytica Posteriora,6 of the Tracta-tus de Consequentiis,7 and it is taken up more extensively in the Quaestiones in Aristotelis Metaphysicam: the Aristotelian formula is true for affirmative pres-ent tense de inesse sentences,8 but false if taken universally,9 yet it can be amended so as to apply to all kinds of sentences.10 Let us call this result ‘the amended Aristotelian formula’.

The amended formulation captures the fact that, in the account of the truth of a sentence that is not de inesse, affirmative and in the present tense, we should refer to items that do not (or may not) exist at the time of the utter-ance, and therefore cannot (or may not) be in some way or another in the time of the utterance. A correction in the tense of the verb in the right-hand side of formula is enough to make it fit all kinds of sentences. The amended formula-tion is accepted in many places, and the original presentation is correct, as far as affirmative present tense de inesse sentences are concerned.

The second argument against the formula is the refusal of complexe significa-bilia, that is, of a specific correlate of sentences, over and above what is signi-fied by its parts. I will not examine it here, for it concerns an interpretation of the formula, or rather a way it should not be interpreted.11 It does not lead to an amendment of the formula, nor does it say that there is no correct interpre-tation of it. Its main result is to block the idea that a theory of truth could be couched in the vocabulary of signification—that we could explain what it is for a sentence to be true, identifying a proper correlate of the whole sentence. This is also the central result of the next argument.

6) Quaest. Analyt. Post., II, qu. 10a. 7) Tractatus, I, I, 14-53, p. 17-18. 8) “. . . omnis affirmativa vera que est de presenti et de inesse ex eo est vera: quia qualitercumque significat esse ita est in re significata vel in rebus significatis . . .” (Quaest. Met., VI, qu. 8, XXXVIIIvb). 9) “Dico quod illud commune dictum si dicatur universaliter est falsum: sed illud commune dictum est verum quantum ad veritates affirmativarum de presenti et de inesse: de aliis autem oportet alio modo loqui: et quia secundum diversitates propositionum essent diversi modi dicendi ut patuit: ideo consuevimus loco illorum diversorum modorum uti primo modo qui debet convenire illis affirmativis de inesse et de presenti: et est tunc impropria locutio: sicut sepe accipimus unam propositionem loco alterius.” (Quaest. Met., VI, qu. 8, XXXIXra).10) “. . . propositio de preterito vera ex eo est vera: quia qualitercumque significat fuisse ita fuit et propositio de futuro: quia qualitercumque significat fore ita erit et propositio de possibili: quia qualitercumque significat posse esse fuisse vel fore: ita potest esse fuisse vel fore.” (In Met. Arist. Quest. Met., VI, qu. 8, XXXVIIIvb).11) Soph., II. See Alain De Libera, ‘Aliquid, aliqua, aliqualiter. Signifiable complexement et théo-rie des tropes au XIVème siècle’, in Chemins de la Pensée Médiévale, eds. P. Bakker, E. Faye and Ch. Grellard (Turnhout, 2002), 27-45.


The eighth conclusion of the second chapter of the Sophisms states that the formula is not adequate for affirmative present tense de inesse sentences:

. . . every true affirmative sentence about actuality [de inesse] and about the present [de praesenti] is not true on the ground that whatever and howsoever it signifies as being, so it is, for the sentence ‘A man is an animal’ is true, by which, however, that Aristotle exists is signified [Aristotelem esse significatur, i.e., Aristotle is signified to be], whereas it is not so where the thing [signified] is concerned.12

This follows from the precedent conclusions:

a. a (categorematic) term signifies that what it signifies exists, since the signi-fication of a term extends indifferently to all things it signifies—e.g. ‘homo’ signifies that men exist;13

b. a sentence signifies whatever its components signify—let us call it ‘the inheritance principle’14—, therefore, the sentence ‘homo est asinus’ signifies that men exist;15

c. since signification extends indifferently to the past, to the future, and to the possible—‘signification’ is an ampliative term—, a term also signifies that what it signifies has existed, will exist and may exist;16

d. the extension of signification covers past, present, future and possible individuals;17

e. (i) a true affirmative de inesse present sentence signifies individual things that do not exist when it is true, therefore it is not on things as it signifies; (ii) whatever and howsoever (quidquid et qualitercumque) is signified by ‘homo est homo’ and ‘asinus est asinus’ is signified by ‘homo est asinus’—and, of course, the former are true, and the latter, false. [from (b), (c) and (d)]18

12) Transl. Klima, 854. “. . . omnis propositio affirmativa de inesse et de praesenti vera non ex eo est vera quia quidquid vel qualitercumque esse significatur per eam ita est, quia haec propositio ‘homo est animal’ est vera, per quam tamen Aristotelem esse significatur, et non est ita in re.” (Soph., II, 4 conclusio, 41, l. 26-42, l. 2).13) Soph., II, quarta conclusio, 40.14) Soph., I, ad tertium soph., 30, l.11-22.15) Soph., II, quinta conclusio, 40-41. This conclusion follows from (a) and the inheritance principle.16) Soph., II, sexta conclusio, 41.17) Soph., II, septima conclusio, 41.18) Soph., II, octava conclusio, 41-42.


A true sentence signifies many things that are not, in the world, as it signifies them to be, for it signifies past, future and possible things. We are led back to the unwelcome consequences of the first argument, and we cannot help our-selves with the amended formulation, since the reasoning depends on sub-sentential features of sentences, i.e., (a), the inheritance principle and (c), that are not touched by the amendment. Moreover, a true sentence and a false sentence may signify the same things.

Let us consider the sentence:

(2) A person is reading a paper on John Buridan.

(2) is true, now, of you. But (2) also signifies

(3) Aristotle has existed,

for a part (2), namely, ‘person’, signifies (3). Of course, things are not as (3) signifies them to be, for Aristotle does not exist, so cannot be in some way or another. Therefore, it is not on things howsoever (2) signifies them to be.

This argument is not aimed exactly at the Aristotelian formula, but at the Aristotelian formula plus ‘quidquid ’. Does it make any difference? The argu-ment leading from (2) to (3) depends crucially on the inheritance principle: a sentence signifies what its terms signify; (2) does not signify (3) directly, but through what is signified by part of it. My suggestion is that this extension in the account of the truth-conditions of a sentence is authorized by the inclu-sion of ‘quidquid ’ in the formula: only if we allow a sentence to signify things to be in a way signified by part of it will we go from (2) to (3).

It may seem a bit far-fetched. Isn’t (3) signified by (2)? For Buridan, the answer is, of course, ‘yes’. What is at stake, however, is not whether (2) signi-fies (3) or not, but whether this is relevant to the truth of (2). We should notice first that this is the only place in the Sophisms in which ‘quidquid ’ is added to the Aristotelian formula—crucially, it is not present in the eighth chapter, where the formula is used approvingly. It seems reasonable to assume that the addition of ‘quidquid ’ is connected with this particular argument. Moreover, it is not the case that a term can supposit for whatever it signifies19—and since truth will be explained in terms of supposition, we

19) Soph., I, Undecima conclusio, 27, l. 23-25. Notice that what is at stake is what is signified ad extra, and not the fact that a written or spoken term signifies a mental term.


cannot expect everything signified by a term to be taken up in a formula that aims to indicate the truth of a sentence of which it is part.

Finally, in the fourth and the fifth sophisms of the eighth chapter, Buridan argues that a sentence does not make an assertion (apparently) made by part of it.20 Consider the following sequences of words:

(4) I say that a man is an ass

(5) <. . .> a man is an ass.

If (5) says something false, and is part of (4), does (4) say something false? It doesn’t, for a part of a sentence does not say anything truth-evaluable— inasmuch as we consider (5) to be part of (4), it cannot be evaluated as true or false. This holds also, of course, for isolated words, that cannot be evaluated as true or false. It seems reasonable to consider that a truth-evaluable item signi-fied by a non truth-evaluable part of a sentence is not part of the truth- conditions of this sentence. Even if we accept that an isolated word signifies something truth-evaluable, what is so signified is not something that a sentence in which such a word figures states to be true or false. If it is legitimate to say that the assertion made by a sentence is the way it signifies things to be (and what follows from the way it signifies things to be), and not the things it signifies (nor what follows from what it signifies), then (3) is no more an assertion made by (2) than (5) is assertion made by (4), although (3) is signified by (2).21

What is the point of the argument? I think it restates a claim already estab-lished, i.e., that signification does not offer the proper vocabulary in which a truth theory could be developed. This is, indeed, what the eighth conclusion says explicitly:

And thus, it seems to me that in assigning the causes of truth or falsity of sentences it is not sufficient to deal with significations, but we have also to take into account the suppositions concerned.22

20) Soph., VIII, soph., 4-5; see especially soph., 4, 147, l. 14-15. See also Summulae de Proposi-tionibus, ed. R. van der Lecq, 1.7.1. (Turnhout, 2005), 70-72.21) The analogy, of course, is less than perfect, since (5) is not signified by (4). But the point is not what is signified by a sentence—(3) is undoubtedly signified by (2)—, but what is relevant to its truth.22) Transl. Klima, 854. “Ut sic, ut videtur mihi, in assignando causas veritatum vel falsitatum propositionum, non sufficit ire ad significationes, sed etiam oportet ire ad suppositiones.” (Soph., II, octava conclusio, 42, l. 6-8).


This should come as no surprise: once the signification of terms is determined, by whatever theory of signification, supposition theory will take up the job of providing truth-conditions for sentences. Accordingly, the last five conclu-sions of this chapter give the rules for the truth of sentences, the so-called modes of personal supposition.

The overall result, therefore, is that there is no reason not to accept the amended Aristotelian formula, but it is not a theory of truth. More precisely, a compo-sitional theory of truth cannot be stated in the vocabulary of signification.23 The fact that the theory of truth cannot be couched in the vocabulary of sig-nification doesn’t mean that there is an opposition between truth as it is explained by the supposition theory and as it is explained by the Aristotelian formula—the latter does not offer any criteria to which other criteria could be opposed. Therefore, it is not correct to say that the Aristotelian formula “departs from”24 the criteria of truth as established by supposition theory. The arguments concerning the Aristotelian formula are presented in the second chapter of the Sophisms before any presentation of the supposition theory, therefore before any suppositional criterion to which the formula could be opposed, contrary to what Stephen Read says.25 Indeed, the fact that the

23) Following Stalnaker’s distinction, we can say that supposition theory is a “descriptive seman-tic theory”, while signification is a “foundational semantic theory”. A descriptive semantic theory “assigns semantic values to the expressions of the language, and explains how the semantic values of the complex expressions are a function of the semantic values of their parts,” while a founda-tional semantic theory is concerned with questions about what gives “expressions their semantic value, or more generally, about what makes it the case that the language spoken by a particular individual or community has a particular descriptive semantics.”, Robert Stalnaker, ‘Reference and Necessity’, in A Companion to the Philosophy of Language, eds. Bob Hale and Crispin Wright (Oxford, 1997), 535.24) Stephen Read, ‘The Liar Paradox from John Buridan back to Thomas Bradwardine’, Vivar-ium, 40 (2002), 195.25) “But what is the relation between these two criteria of truth, first, that “things are however it signifies”, secondly, that, for example, for affirmatives, subject and predicate supposit for the same? Do the two criteria always give the answer? Is it possible to prove that they do? Buridan tackles the issue directly in ch. 2 of the his Sophismata. His conclusions that the criterion in terms of signification is inadequate, and does indeed depart form the second.” (Read, ‘The Liar Paradox’, 195). The compatibility between supposition theory and the Aristotelian formula is very clear in the question 10a of the Quaestiones in Posteriorium Analyticorum, the conclusions of which present the truth-conditions of sentences in terms of supposition (conclusions 1 to 4), and then uses the Aristotelian formula (conclusions 5-9). “Quinta conclusio est quod ad veritatem propositionis de inesse et de praesenti requiritur et sufficit quod qualitercumque ipsa significat esse ita est in re significata vel in rebus significatis. Et hoc ultimum est notum ex vi nominis. Si


amended formulation is accepted in many places makes it hard to believe that there is an opposition between the two (supposed) sets of criteria.

But if the Aristotelian formula does not offer any criteria, and if it is rejected before any criterion to which it could be opposed is presented, how can we argue against it? The unwelcome consequences of this, such as the fact that (1) is true about future objects that are not presently in some way or another (leading to the amended formulation), or that (2) is true without things being as it is signified by (3) (that can be avoided with the exclusion of ‘quidquid ’), are compared to our previous, pre-theoretical, understanding of the objects about which such sentences are true. Supposition theory itself is not intended to explain truth to someone unable to grasp the truth-conditions of a sen-tence. On the contrary, its results are thought to be close to what we can grasp before any theoretical explanation. That is what Buridan says after stating, for the first time in the Sophisms, supposition criteria for the truth of affirmative categoric sentences:

And perhaps this is not a conclusion but rather an indemonstrable principle, or, if it is a conclusion, it comes close to an indemonstrable principle.26

Such “indemonstrable principles” were available in the unfolding of the argu-ments concerning the Aristotelian formula.

3. The distinction between the time about which a sentence is true and the time in which it is true and self-referential paradoxes

The compositional theory of truth can only be developed in the vocabulary of the theory of supposition. Buridan says that the suppositional account of truth conditions fails exactly in the case of insolubilia.27 One might think, therefore, that once we abandon the idea that truth can be explained in terms of signifi-cation, and see that only supposition theory can state the truth-conditions of a sentence (in the text of the Sophisms, by the end of the second chapter),

enim dico quod homo est albus, significo idem esse hominem et album, et ita est si propositio sit vera. Et ex hoc infertur <quod> si non sit res significata, vel etiam res pro qua terminus supponit, propositio talis non est vera, sed falsa.” (Quaest. Analyt. Post., qu. 10).26) Transl. Klima, 855. “Et forte quod haec non est conclusio, sed est principium indemonstra-bile, vel, si est conclusio, ipsa est propinqua principio indemonstrabili.” (Soph., II, 10 conclusio, 42, l. 22-24).27) See, e.g., Soph., II, tredecima conclusio, 45, l. 10-12; Quaest. Met., VII, XXXVIIIva.


everything can be dealt with within supposition theory, until we reach the insolubilia.

This description is, of course, not untrue. It may be, however, a bit mislead-ing. It is not exact that, given a sentential token, its truth-conditions can be determined using only the vocabulary of supposition theory. For any senten-tial token, we have to determine the time about which it is intended to be true, and the time of evaluation of a sentence is not determined by supposi-tional rules plus the time of its utterance alone—or at least not for present tense sentences. The time of utterance is the time in which the sentence is true; we need to fix independently the time about which it is true. The truth- conditions are stated using the vocabulary of the supposition theory, but the work of supposition theory cannot get off the ground without fixing the time of evaluation.28

Once the time of evaluation is fixed, suppositional rules fail exactly in the case of insolubilia. However, supposition rules cannot succeed nor fail without the time of evaluation being fixed. This remark is of the utmost importance for the understanding of self-referential paradoxes, not only because it is crucial in the Buridanian account of truth in general, but also because the distinction between the time pro quo a sentence is true and the time in quo it is true should be brought into the foreground, when we consider self-referential paradoxes involving more than one sentence, and also utterances of sequences of words that may be a benign assertion or an instance of the paradox of the Liar.

Let us consider the sentence:

(6) (2) is true.

For (6) to be true, we need to evaluate at a time in which (2) exists. Since there is no instant of time in which the whole sentence (2) exists, how can we deter-mine the interval of time about which (6) should be evaluated, i.e., in which the whole sentence (2) is said to exist? We can use as present “as much as we want,” and in uttering (6) we use as present an interval of time covering the existence of the whole sentence (2)—an interval of time that doesn’t include

28) See Ernesto Perini-Santos, ‘John Buridan on the Bearer of Logical Relations’, Logica Univer-salis 2 (2008), 69-79 and Id., “When the inference ‘p is true, therefore p’ fails: John Buridan on the evaluation of propositions”, in Medieval Supposition Theory Revisited, ed. E. Bos (Leiden, forthcoming).


the time of utterance of (6).29 The time of evaluation of (6) is not determined automatically from the time of its utterance.

One might think that this follows from the fact that the utterer of (6) should wait until (2) is complete, in order to declare it true or false, and there-fore not only (6) will never exist exactly in the time in which (2) exists, but also the time of evaluation will have a certain extension.30 This is in itself an interesting result, for a present tense sentence, namely, (6), is true about a time that doesn’t include the time of its own utterance. This conclusion does not follow, however, from the fact that (6) talks about another sentence—it may be the case of (2) itself. Let us suppose that you are the only person reading a paper on Buridan at a given moment, so (2) is true exactly about you. You decide to utter (2), and at the very moment you utter it, you take your eyes off the text. Were (2) to be evaluated at the exact moment in which you utter it, (2) would be false. (Likewise, you can imagine a situation in which you utter ‘I am reading a paper on John Buridan’ while talking on the phone, answering the question ‘what are you doing now?’) But, of course, (2) need not be evalu-ated exactly at the time in which it exists—you may want it to be true about the interval of time immediately preceding the moment of its utterance, or about intervals of time immediately preceding and immediately following its utterance. Again, the time of evaluation of a present tense sentence is not determined automatically from the time of its utterance—it depends on the intention of the utterer, and it may not include the time of its utterance.

For any sentence φ, we should distinguish the situation sn in which it is uttered, therefore in which it can be true or false, and the situation sm about which it is intended to be true, or at which it is evaluated (considering times as situations).31

φ sn

,sm

There are many relations between sn and sm, more or less constrained by the tense of the verb. If φ is a present tense sentence, the time at which it should be evaluated is not automatically determined from the time of utterance: sm may be identical to sn, may contain it as a proper part, or may not contain it as a proper part. The latter is the case of (6), and also of (2) in the story proposed above: as you utter (2) in s1, you intend it to be true about a situation

29) Soph., VII, soph., 1, 129, l. 1-2.30) Soph., VII, soph., 2, 129-130.31) I will consider times as situations to extend more naturally this account to de possibili sentences.


s 2 that does not contain s1 as a proper part—in the exact time of your utterance of (2), you are not reading a paper on Buridan, but that does not prevent (2) from being true. Since (2) can be true only in a situation in which it exists, (2) is true in s1, but it is true about s 2.

Let us suppose that

. . . during the first hour of this day there is no true sentence, but that every sentence is false, and that after the end of this hour Socrates says that every sentence is false, and that he refers not to the time at which he speaks but to the time of that first hour, then in this way his sentence would be true, for an induction over the sentences of that first hour would be exhaustive.32

In this case, Socrates utters the following sentence:

(7) Every sentence is false.

(7) is uttered in a situation s1 and is intended to be true about a situation s 2:

(7) Every sentence is false s1,s

2

In the case above, Socrates intends (7) to be true about a situation that doesn’t include the moment of utterance of (7), so this sentence does not say, of itself, that it is false. There is nothing self-referential about it, nor is it paradoxical. Of course, this is not the hard case. The difficulty would arise only if (7) were intended to be true about a situation that included (or were identical with) the time of utterance of (7) itself, that is, if s 2 ⊇ s1. Since the relation between s1 and s 2 can vary, they have to be different parameters in the account of truth of sentences, as it is shown by (7).33

32) Transl. Klima 965-966. “. . . per totam primam horam hujus diei nulla sit propositio vera sed omnis falsa, et post finem huius horae Socrates dicat quod omnis propositio est falsa, et quod loquatur non pro tempore in quo loquitur, sed pro tempore illius primae horae. Sic propositio sua esset vera: quia inductio esset sufficiens quae fieret in propositionibus illius primae horae.”, Soph., VIII, soph., 7, 152, l. 19-23.33) “Sed haec solutio, quamvis sit vera in dicto casu, non aufert difficultatem sophismatis secun-dum alium casum, scilicet quod ipse loquatur pro illomet tempore in quo ipse loquitur.” (Soph., VIII, 7, 152, 24-26). The paradox could arise in the situation described above—for instance, if someone in s 2 were to utter:

(8) (7) is true.

But, then again, both for (7) and for (8) we need to fix the time of evaluation, a parameter that is not given by the time of their utterance.


Is any utterance of (7) paradoxical? We cannot answer this question without fixing the time of evaluation that is not given by the mere tokening of the words—i.e., cannot be calculated automatically from the time of its utterance plus the form of the words. That is the reason why, without the distinction between the time pro quo a sentence is true and the time in quo it is true, we do not know whether supposition rules fail or not for (7). We can see why it is misleading to only focus on supposition rules, to the exclusion of the dis-tinction between the time pro quo a sentence is true and the time in quo it is true, for only when we help ourselves with this distinction can we decide whether or not supposition rules fail for sentences like (7).

We need a double indexing of truth, because a sentence can be true about a situation different from the situation in which it is true, and we should avoid the assimilation of the two roles. There are different developments of the idea that truth theory requires double indexing. David Kaplan is one of most influ-ent proponents of this idea: we have to distinguish the context that fixes the proposition expressed from the circumstance of evaluation of the proposition expressed.34 In a sense, the parallel is quite straightforward: a sentence is true about a situation (the time pro quo a sentence is true / the circumstance of evaluation of a proposition) that may not coincide with the situation in which we can fix what it says (the time in quo a sentence is true / the context of a sentence; notice that the parameters of a Kaplanian context are taken from the concrete situation in quo the sentence is (said to be) true).

Nevertheless, we should be careful here. To begin with, we have to distin-guish an utterance, a concrete event, from a sentence-in-a context.35 However, the main problem does not lie in the difference between the parameterized, abstract notion of a context and the concrete event of an utterance. According to Buridan’s theory, truth-bearers are sentences, not propositions; in his frame-work, there is no available truth-bearer that could do without a determined time in quo it is said to have a truth-value. In contrast, the content we reach in a Kaplanian framework is an abstract entity to which no contextual parameter is attached. Moreover, it is not clear (at least to me) that Buridan had any interest in the set of all the utterances of a sentence-type, so as to identify what would be the logical consequences (if any) of all of its instances, nor in the relation between different utterances of a given sentence-type and the different situations about which each utterance could be true.

34) David Kaplan, ‘Demonstratives’, in Themes from Kaplan, eds. J. Almog, J. Perry and H. Wet-tstein (Oxford, 1989), 509-510.35) Kaplan, ‘Demonstratives’, 522.


More generally, one may be skeptical as to the representation of Buridan’s double indexing truth-theory as a version of two-dimensional semantics. The central idea of two-dimensional semantics is the need for two world parame-ters, one to determine the proposition expressed and the other to evaluate it, and it is especially important to the understanding of different cases in which the two worlds may come apart, crucially, in modal contexts.36 It may well be the case that Buridan’s treatment of modal phenomena is represented in the two-dimensional framework, but it is not what is at stake in the distinction between the time in quo a sentence is true and the time pro quo it is true, that motivates the double indexing of truth in the Buridanian framework.37

The eighth chapter extends to consequences of the semantic result of the seventh chapter. The central point is that the antecedent and the consequent should be evaluated at—be true about—the same situation, when they may fail to exist in, and therefore be true in, the same situation. Consider the sentences:

(9) Every sentence is affirmative

and

(10) No sentence is negative.

36) See Robert Stalnaker, ‘Assertion’, in Context and Content (Oxford, 1999), 81. See also David Chalmers: “The core idea of two-dimensional semantics is that there are two different ways in which the extension of an expression depends on possible states of the world. First, the actual extension of an expression depends on the character of the actual world in which an expression is uttered. Second, the counterfactual extension of an expression depends on the character of the counterfactual world in which the expression is evaluated. Corresponding to these two sorts of dependence, expressions correspondingly have two sorts of intensions, associating possible states of the world with extensions in different ways. On the two-dimensional framework, these two intensions can be seen as capturing two dimensions of meaning. These two intensions corre-spond to two different ways of thinking of possibilities. In the first case, one thinks of a possibil-ity a representing a way the world might turn out to be <. . . > In the second case, on acknowledges that the actual world is fixed, and thinks of a possibility as a way the world might have been but is not <. . .>” (David Chalmers, ‘Foundations of Two-Dimensional Semantics’, in Two- Dimensional Semantics, eds. M. Garcia-Carpintero and J. Maciá (Oxford, 2006), 59.37) It seems to me therefore that the parallel between Buridan’s theory and two-dimensional semantics proposed by Catarina Dutilh-Novaes is not very deep; Catarina Dutilh-Novaes, ‘Buri-dan’s Consequentia: Consequence and Inference within a Token-Based Semantics’, History and Philosophy of Logic 26 (2005), 287.


I utter (9) in a situation s1, and I intend it to be true about a situation s 2 that includes s1 as a proper part. (10) is true about every situation about which (9) is true, so the consequence is valid. As I utter (10), however, I create a new situation s3 about which neither (9) nor (10) are true:

For let us posit that the following sentence is true: ‘Every sentence is affirmative’; then it can entail a false one, namely, ‘No sentence is negative’. But when this is concluded, the former is no longer true, but false.38

(9) and (10) should be evaluated at—should be true about—the same situa-tion. Evaluated at s 2, both are true, and evaluated at s3, both are false. It remains the case that (10) is true about every situation about which (9) is true, so the consequence is valid. Nothing forces the situation at which both sentences are evaluated to include a situation in which any of those sentences do exist. We are past that point since the seventh chapter. We can try a first definition of validity as follows:

[validity] An inference from p1, . . . pn to c is valid iff for every situation s about which p1, . . . pn are true, c is true about s.39

The explanation of this fact lies entirely in the distinction between the situa-tion in which a sentence is true and the situation about which it is true, for the sentences need not be true in the same situation, while it must be the case that (10) is true about every situation about which (9) is true. In other words, the sophism is solved by the semantic result of the seventh chapter. However, Buridan doesn’t use this distinction but rather the Aristotelian formula. A consequence is valid if, and only if,

. . . things cannot be as the antecedent signifies without being as the consequent signifies.40

38) Transl. Klima, 955. “Ponamus enim quod haec propositio sit vera ‘omnis propositio est affirma-tiva’, tunc ex ea potest sequi falsum, scilicet ‘nulla propositio est negativa’. Sed quando concluditur illa, tunc prima non amplius est vera, sed falsa.” (Soph., VIII, soph., 1, 142, l. 23-143, l. 3). 39) Contrast with Kaplan: “From the perspective of [the Logic of Demonstratives], validity is truth in every possible context. For traditional logic, validity is truth in every possible circum-stance.” (Kaplan, ‘Demonstratives’, 549). Buridan is clearly on the side of the “traditional logic”: a consequence is valid iff its conclusion is true at every circumstance at which its premises are true. Nothing in Buridan’s framework seems to correspond to validity as truth in every possible context, as in the Kaplanian Logic of Demonstratives.40) Transl. Klima, 955. “. . . non possit esse qualiter antecedens significat quin etiam ita sit quali-ter consequens significat.” (Soph., VIII, soph., 1, 143, l. 11-12).


The first, obvious, thing to notice is that the Aristotelian formula is not ban-ished from the Sophisms. As I have tried to show in the preceding section, nothing in his arguments against the Aristotelian formula bars its use for affir-mative present tense de inesse sentences, nor, in general, the use of the amended formula for all sentences (or at least to all sentences dealt with in the texts about the Aristotelian formula).

More importantly, the Aristotelian formula plays here a specific role. If we think that the sophism ‘Every proposition is affirmative, therefore no proposi-tion is negative’ can be solved with the distinction between the time pro quo a sentence is true and the time in quo it is true, as it seems to be the case, we can give a rather direct interpretation of the formula: the Aristotelian formula designates the situation about which a sentence is true, that is, the time pro quo it is true.41

However, the definition proposed above is not correct. Truth-bearers are, for Buridan, concrete contingent entities, and not atemporal abstract entities, such as contemporary propositions—Buridan has a “token-based” semantics.42 The antecedent may be true without the truth of the consequent, if the latter doesn’t exist.43 We should therefore add the condition that the antecedent and the consequent exist:

[validity] An inference from p1, . . . pn, to c is valid iff, if p1, . . . pn, and c exist in some situation sn, then for every situation sm

about which p1, . . . pn are true, c is true about sm.

The requirement of a “token-based” semantics is fulfilled in the modified defi-nition by the double indexing of the notion of validity, which is an extension of the double-indexing theory of truth.

41) It seems to me misleading to say that the Aristotelian formula is a “place-holder for the condi-tions of the truth of the several types of propositions” (transl. Klima, 858), if it is meant by this that the Aristotelian formula is a place-holder for truth-conditions of sentences as described by supposition theory. The determination of the time pro quo a sentence is true is not guided by rules, suppositional or others, and it should be determined before the statement of truth- conditions in terms of supposition theory (see Perini-Santos, ‘John Buridan on the Bearer of Logical Relations’, and Id., ‘When the inference “p is true, therefore p” fails’).42) See Gyula Klima, John Buridan (Oxford, 2009), 210-233; and Dutilh-Novaes, ‘Buridan’s Consequentia’. 43) Soph., VIII, soph., 1, 141, 28-142, 2.


4. Some misleading interpretations of the Buridanian solution to the Liar

Some important interpretations of the Buridanian solution to the Liar have failed to see the role of the distinction between the time in quo and the time pro quo a sentence is true, and therefore have been led to suggest that Buridan has made room for truth-value gaps in his logic (if not abandoned the very notion of truth), while what is at stake is a double indexing of truth. The first interpreter to have suggested this line of thought is, to my knowledge, Hans Herzberger (a logician, not a medievalist!):

In discussing his Insolubilia, Jean Buridan remarked that “those propositions are indeed in conflict with regard to the case being as they signify.” Put more positively, the case may be as a sentence signifies, and yet the sentence not be true. I read this to mean that correspon-dence conditions can be satisfied independently of pressupositions. Indeed this seems to be the case. The Liar sentence is not true, but that’s exactly what it says, thereby engendering that all too familiar semantic collision between a component of veracity and a component of nonveracity in the same sentence. <. . .>

Let us say that a sentence is secure in case “things are as the sentence signifies,” and con-trasecure otherwise. If furthermore the requisite pressupositions are satisfied, the sentence will be true in the former case and false in the latter.44

Herzberger identifies “correspondence conditions” that may be satisfied with-out the sentence being true, when some unidentified pressuposition is not fulfilled. The expression ‘correspondence conditions’ designates the situation about which a sentence is true—that is what is aimed at by the text quoted by Herzberger.45 If the correspondence conditions are fulfilled, the sentence is secure. If, on top of that, the requisite pressupositions are satisfied, a secure sentence is true.

This description of the Buridanian philosophy of language is untrue. There are no truth-value gaps in the Buridanian framework,46 nor does the distinc-tion between the time in quo and the time pro quo a sentence is true establish, in any sense, a presuppositional theory of truth. Nowhere in his Sophisms does Buridan say, or even suggest, that a sentence may lack truth-value. In the texts in the seventh and eighth chapters in which he explains the distinction, he

44) Hans G. Herzberger, ‘Truth and Modality in Semantically Closed Languages’, in The Paradox of the Liar, ed. R. Martin (New Haven, 1970), 31.45) Herzberger quotes from Scott’s translation a line from Soph., VIII, soph., 1, 143, l. 20-23.46) See Sirridge’s remark against Herzberger: Mary Sirridge, “Buridan: ‘Every Proposition is False’ is False”, Notre Dame Journal of Formal Logic XIX (1978), 404.


does not say that one parameter may be defined for a linguistic item, and the other missing, so that the item would be undefined as to its truth-value.

Indeed, if anything is conceivable in this framework, it is an existing sen-tence, or rather an existing sequence of words that could be used as a sentence, without any intention of a speaker fixing the time of its evaluation. In this case, such a sequence of words would lack a truth-value—although it is not sure that it would count as a sentence, for the reasons pointed out in the sec-tion 2 above. Be that as it may, this is, of course, the opposite of the situation imagined by Herzberger, when we consider the words he quotes from the Buridanian text: the situation described by a sentence holds (and therefore the situation about which it is true is fixed, otherwise there would be no condi-tions to be fulfilled), without the time in which this very sentence is true being determined.

That is the way Paul-Vincent Spade reads Herzberger:

. . . the world may be as a certain sentence-token would signify it to be, even though that sentence-token might not in fact exist. Thus, if Socrates is running, then the world is as would be signified by a sentence-token of the type ‘Socrates is running’, even if that sen-tence-token did not in fact exist. We shall say that the sentencen-token is “secure” in this case. If in addition the sentence-token happens to exist, then it is true. A sentence-token is true iff it is secure and exists. The term ‘secure’ is taken from [Herzberger (1970)]. Buridan’s revised notion of consequence, then, amounts to saying that the valid consequences are just the transformations that always preserve security.47

It may seem that, in the Buridanian framework, it is presupposed that truth-bearers do exist, and that this presupposition follows from the fact that truth-bearers are contingent entities. A presuppositional failure would be a situation in which the situation about which a sentence p is true holds, but there is no situation in which p is true. But that does not make sense: the existence of a sentence-token is not a presuppositional condition. We can only ask for the presuppositional conditions of existing sentences. We can ask, for instance, what are the presuppositions (if any) of ‘the present king of France is bald’, ‘John stopped smoking’ or ‘John regrets not having introduced presuppositional con-ditions in his theory of truth’. But such sentences, of course, do exist. We cannot ask about the presuppositions of the sentence in the box below, nor whether it is “secure” or “contrasecure” (of course, if a sentence-token must exist to be true, it must also exist to be “secure” or “contrasecure”):

47) Paul Vincent Spade, ‘John Buridan on the Liar: A Study and Reconstruction’, Notre Dame Journal of Formal Logic XIX (1978), 582.


We can notice that ‘Socrates is running’ does exist in Spade’s text, so we can-not add the presupposition that it exists! Again, it is an entirely different mat-ter, whether or not it exists in the situation about which it is said to be true—this is not a presupposition, but a double indexing to every truth- evaluable item in the Buridanian framework.

Spade uses Herzeberger’s suggestion to say that what is preserved in a con-sequence is security. Since there is no such notion in the Buridanian frame-work—neither Herzberger nor Spade have been able to identify plausible presuppositional conditions that should be added to a “secure” sentence to make it true—, this is not the correct explanation of the Buridanian definition of validity. Once more, what seems to be missing is the distinction between the time in which a sentence is true and the time about which it is true.

In Read’s paper, we have also seen the idea that “correspondence conditions” may be fulfilled without the sentence being true. In this regard, Read can be seen as part of this tradition. A very similar idea has been taken up more recently by Gyula Klima. Using the conclusions of the sophism ‘Every sen-tence is affirmative, therefore no sentence is negative’, Klima says that:

<. . .> Buridan’s logic as such has simply no use for a theory of truth. What it really needs is just the set of “correspondence-conditions” briefly indicated by the Aristotelian formula.48

His conclusion follows from the fact that the conditions established by a sen-tence may be fulfilled without the existence of the sentence, therefore, without its truth.49 For this reason, Buridan, according to Klima, has a “logic without truth”.50 Again, the only point Buridan wants to establish is that a sentence

48) Klima, John Buridan, 223.49) “What Buridan gains, therefore, by returning to the (reinterpreted) Aristotelian formula is a way of expressing the satisfaction of the correspondence conditions of a proposition in a given situation independently from its truth, indeed, independently from its existence in that situa-tion. This is most obvious in Buridan’s discussion leading to his final definition of logical valid-ity.” (Klima, John Buridan, 223).50) “But then, understanding the issue of validity in this way, as definable without any reference to the truth-values of the antecedent and consequent which they can only have in those situa-


may be true about a situation in which it does not exist. Klima’s wording is compatible with this distinction: the correspondence conditions of a sentence p may be fulfilled by a situation s without p existing “in that situation”. But from this remark it does not follow that p does not exist in any situation.

Klima’s proposal is another version of Herzberger’s interpretation, and it fails for the same reason: we cannot determine fulfillment conditions for unex-isting sentences, although we can say that, for a given sentence, it may fail to exist in a given situation about which it is true. This distinction is dealt with by Buridan’s double indexing theory of truth, a theory that is independently motivated and that does not lead to any revision of central theses in logic—such as truth-value gaps, validity without truth, or “secure” and “contrasecure” sentences. We can notice, for instance, that in the Tractatus de consequentiis, Buridan accommodates the need for the existence conditions of truth-bearers with the definition of validity in terms for truth (using both the vocabulary of supposition theory and the Aristotelian formula).51

There is no “logic without truth” in Buridan’s philosophy, any more than truth-value gaps or secure and contrasecure sentences. The Aristotelian for-mula is not a way to designate “correspondence conditions” without truth, and there is no argument against its amended version nor against its use for present tense de inesse sentences.52 Everything supposedly explained by a pre-suppositional framework is accounted for, in his theory, by the distinction between the time in quo a sentence is true and the time pro quo it is true.

This conclusion may seem to overlook the fact that Buridan says explicitly that it may be the case that things are the way a sentence signifies them to be with-out the sentence being true.53 In the wording of Prior, a sentence may be pos-sible without being “possibly-true”:

tions in which they exist, Buridan has a logic without truth, a logical theory that works for determining the validity of inferences, and yet one that can do so without checking the truth-values of propositions in any situation. Thus, Buridan’s logic does not have and does not need a definition of truth.” (Klima, John Buridan, 225).51) Tractatus, I, chapters 2 and 3.52) Notice that Arthur Prior, ‘Some Problems of Self-reference in John Buridan’, in Papers in Logic and Ethics, eds. P. Geach and A. Kenny (Amherst, 1976), 130-146; Sirridge, “Buridan: ‘Every Proposition is False’ is False”; and Read, ‘The Paradox of the Liar’, do not think that the use of the Aristotelian formula implies the abandon of the notion of truth; they fail however to see the connection between this formula and the semantic result of the chapter 7 (which has been masterly exposed by Prior, in ‘The Possibly-True and the Possible’, Mind 78 (1969)!). 53) Soph., VIII, 2 soph., p. 142, l. 8-9.


A sentence on a sheet may be said to be possibly-true on that sheet if and only if there is some sheet (that one or another) on which it is true, and possibly-false on that sheet if and only if there is some sheet (that one or another) on which it is false.54

A possibly-true sentence should be contrasted with a possible one:

the important point to notice is that for a sentence S on a sheet X to be “possible” in virtue of what is on Y, the sentence S does not itself have to be on Y.55

In terms of Prior, a sentence can be possible, without being possibly-true, if it exists on a sheet, but is true in virtue of what exists on some other sheet.56 Of course, every true sentence does exist on some sheet or other, although it may be true of a sheet on which it doesn’t exist (contrast this point with Spade and Klima). This vocabulary can be extended to the definition of validity:

The most satisfactory definition of validity (on a sheet in a set of sheets) is to say that a sentence on a sheet may be validly inferred from other sentences on this sheet if and only if there is no sheet (in the set) of which all the premiss-sentences are true but of which the conclusion sentence is false.57

This definition accounts for the validity of the inference from (9) to (10). Notice that validity is defined for sentences on a sheet—and therefore there is a sheet on which all the sentences that constitute the consequence do exist. The double indexing theory of truth is, I think, is another way of stating Prior’s interpreta-tion of Buridan, based on the two first sophisms of the eighth chapter.

Let us take stock:

a. Buridan does not abandon the Aristotelian formula as a way to talk about truth, although a compositional theory of truth can only be couched in the vocabulary of supposition theory;

b. The distinction between the situation in quo a sentence is true and the situ-ation pro quo it is true leads to a double indexing theory of truth, and this

54) Prior, ‘The Possibly-True and the Possible’, 485.55) Prior, ‘The Possibly-True and the Possible’, 486.56) See also the definition of ‘necessary’ and ‘necessarily-true’: “We may say similarly that a sen-tence is necessary on the sheet on which it occurs if it is true of every sheet, and that it is necessarily-true on any sheet on which it occurs if it is true on every sheet on which it occurs.” (Prior, ‘The Possibly-True and the Possible’, 487). 57) Prior, ‘The Possibly-True and the Possible’, 489.


result should be extended to other areas of his semantics, such as the defini-tion of validity;

c. The Aristotelian formula is a way to designate the situation pro quo a sen-tence is true;

d. There are no truth-value gaps, nor anything of the sort, in Buridan’s phi-losophy of language.

With these results in mind, we should reconsider Buridan’s solution to the paradox of the Liar.

5. A brief statement of Buridan’s solution to the paradox of the Liar

The results established so far should be taken into account in any explanation of Buridan’s solution to the paradox of the Liar. The main problem of the interpretations in the Herzberger tradition is precisely missing the points (a)-(d) above. I will present briefly my interpretation of Buridan’s solution to the paradox of the Liar. I should say right away, however, that I do not think that my interpretation of Buridan leads to an acceptable solution to the para-dox. And I hasten to add that I do not really care, at least if caring about that means presenting Buridan’s theory as a viable position on this issue today. The fact that my proposal has unwelcome consequences, of which Buridan does not seem to be aware, is uncomfortable and maybe a sign of misinterpretation of an extremely sharp philosopher, but not in itself an argument against the interpretation proposed.58

Buridan states his solution to the paradox as follows:

Therefore, every sentence asserting itself to be false, either directly or implicitly, is false, for although things are as it signifies, insofar as it signifies itself to be false, nevertheless, things are not as it signifies insofar as it signifies itself to be true. Therefore, it is false and not true, since for its truth it is required not only that things be as it signifies but also that they be in whatever way it signifies [them to be].59

58) My proposal is, to a certain extent, in accordance with Prior, ‘Some Problems of Self-refer-ence in John Buridan’; Fabienne Pironnet, ‘John Buridan on the Liar Paradox: Study of an Opinion and Chronology of the Texts’, in Argumentationstheorie, ed. K. Jacobi (Leiden, 1993), 293-300, and Read, ‘The Paradox of the Liar’, inasmuch as the falsity of the Liar sentence is due to the falsity of what it virtually implies (contrast with Spade and Klima).59) Transl. Klima, 967-968. “Ideo omnis propositio asserens se esse falsam, sive directe sive con-secutive, est falsa, quia licet qualiter significet esse ita sit quantum ad hoc quod significat se esse


We can argue for this solution as follows:

i. (i) is false [ex hypothesi]ii. ‘(i) is false’ exists [ex hypothesi]iii. (p) (p ∧ p exists) ⇒ p is true60

iv. (p) (q) (p ⇒ q) ⇒ (p:q) 61

v. ((i) ∧ (ii)): (i) is true [(iii), (iv)]vi. (p) (q) ((p:q) ∧ ¬ q) ⇒ ¬ p62

vii. ¬ (i) is true [≡ (i)]viii. ¬ ((i) ∧ (ii)) [(v), (vi), (vii)]ix. ¬ (i) ∨ ¬ (ii) [(viii), De Morgan]x. ¬ (i) [(ii) (ix), Disjunctive syllogism]

If we assume that (i) is false, we have a direct equivalence with the denial of ‘p is true’, and the reasoning goes as above. I take it that the conclusion of the argument proposed above is what is aimed at by Buridan: whether (i) is true or false, ‘(i) is true’ is false, and therefore (i) is false. The crucial point is that if (i) is false, so things are the way it signifies them to be, and it is true,63 but we cannot argue back from the falsity of ‘(i) is true’ to its truth—we can say that it is stably false. If ‘(i) is true’ is false,

. . . things are not as the implied consequent signifies, even if this locution is understood in the required sense.64

falsam, tamen non qualiter significat esse ita est quantum ad hoc quod significat se esse veram. Ideo est falsa et non vera, quia ad veritatem requiritur non solum quod qualiter significat ita sit, sed quod qualitercumque significat ita sit.” (Soph., VIII, soph., 7, 154, l.25-155, l. 3. See also soph., 8, 158, l. 17-23; soph., 9, 162, l. 7-18; soph., 11, 165, l. 16-19).60) “. . . ad omnem copulativam ex aliqua propositione et alia dicente quod ipsa est, sequatur quod illa est vera.” (Soph., VIII, 2 soph., p. 144-145).61) “Non enim sufficit ad hoc quod propositio sit vera quod sit ita quod sit ita sicut significant secundum formalem significationem, immo etiam oportet quod sit ita sicut significaret consequens quod in ea virtualiter implicatur. . . ” (Soph., VIII, 7 soph., p. 156). ‘p:q’ means ‘p says that q’. I follow Read’s notation in Stephen Read, ‘Further Thoughts on Tarski’s T-Scheme and the Liar’, in Unity, Truth and the Liar, eds. S. Rahman, T. Tulenheimo and E. Genot (Berlin, 2008), 206. 62) “. . . ad veritatem requiritur non solum quod qualiter significat ita sit, sed quod qualiter-cumque significat ita sit.” (Soph., VIII, soph., 7, 155, l. 1-3). 63) “Item, si sit falsa, tunc qualitercumque significat ita est, quia non significat nisi quod omnis propositio est falsa; et ita est; ergo ipsa est vera.” (Soph., VIII, soph., 7, p. 152, l. 8-9).64) Transl. Klima, 970. “. . . non est qualiter per illud consequens implicatum significatur, etiam si huius modi locutio fiat secundum debitum sensum.” (Soph., VIII, soph., 7, p. 156, l. 17-18; see also soph., 9, 163, 10-22; soph., 11, 165, l. 20-27).


The sophism is false because what (or at least one of the things) it virtually implies is stably false, and its falsity implies the falsity of the sophism itself.

I think that this is the solution that we can read in Buridan.65 However, I do not think that it is a good solution. It probably won’t resist a more sophisti-cated version of the paradox,66 and it is not clear how we could block the argument from starting anew, once we reach (x), so that we could prove the sophism true again. Moreover, this solution has some other problems of its own. However, to see this point, we have to consider this interpretation in the general framework I have proposed in the first section of the paper.

The section 4 of my paper has been devoted to showing that some previous interpretations of Buridan’s solution are wrong, because they have missed the importance of the distinction between the time in quo a sentence is true and the time pro quo it is true. I have not mentioned the distinction in my own interpretation. Why was it important?

The first important thing is to establish that the Aristotelian formula indi-cates the truth of a sentence—it is present in the texts corresponding to the premises (iv) and (vi) above.67 Moreover, it is also important to establish the correct reading to the second premise—it doesn’t concern the existence of (i) simpliciter, but the existence of (i) in the situation in which (ii) is evaluated—and since all the premises should be evaluated at the same situation, it estab-lishes the existence of (i) in the situation at which the inference is evaluated. This point deserves a closer look.

65) Buridan proposes another solution in the Quaestiones on the Metaphysics, according to which the sophism is false because there is a contradiction between what the sentence formally signifies (that it is itself false) and what it virtually implies (that it is itself true). See, e.g.: “. . . ad omnem propositionem asserentem se esse falsam sequitur ista propositio quod ipsa est falsam; et, per consequens quod ipsa est non est vera. Et sic statim concluditur quod ad omnem propositionem asserentem se esse falsa sequuntur propositiones contradictorie, scilicet, quod ipsa est vera et quod ipsa non est vera.” (Quaest. Met., VI, qu. XI, fol. XLIra). The common feature of both solu-tions is that Liar sentences are false in virtue of what they virtually imply.66) See, for instance, Prior’s suggestion: “what is formally signified by this sentence is not the case”, Prior, ‘Some Problems of Self-reference’, 140.67) We can notice that, in some important texts about these issues, Buridan goes from the Aris-totelian formula, or something close to it, to the supposition theory without any suggestion that he is changing the subject; see Soph., VIII, soph., 7, p. 155, l. 29-156, l. 9; see also soph., 11, 165, l. 2-7.


Let us consider the following the following inference. From

(11) Socrates walks

And

(12) (11) exists

it follows:

(13) (11) is true.

If we apply the definition of validity, we have:

[validity’] The inference from (11) and (12) to (13) is valid iff, if (11), (12) and (13) exist in some situation sn, then for every situation sm

about which (11) and (12) are true, (13) is true about sm.

We might think that a more cautious formulation would have different situa-tions in which each premise and the conclusion would exist—the present for-mulation seems to make (12) useless, since it guarantees the existence of (11) and (13) in the same situation sn. Part of the response is that we need a situa-tion in which every sentence of the inference exists, so that it can be valid or invalid (in Prior’s terms, the inference is defined for sentences “on a sheet”)—just as a sentence has to exist to be true or false. To this, we should add that the conclusion should be evaluated at the same situation at which the premises are evaluated, whether or not this situation coincides with a situation in which they exist. Without the premise (12), if (11) were true about a situation in which it doesn’t exist, we would have the truth of the premise and the falsity of the conclusion. This is blocked by (12), that can only be true about a situ-ation in which (11) exists. Were (12) to be evaluated at a situation in which (11) doesn’t exist, it would be false, and therefore the falsity of (13) wouldn’t make the inference invalid.

This is a consequence of the extension of the semantic result of the seventh chapter to consequences at the opening of the eighth chapter. We can present the truth-conditions of (11)-(13) as follows:


(11) Socrates walks s1,s

2

(12) (11) exists s3,s

1

(13) (11) is true s4,s

1

Inasmuch as we take (11)-(13) separately, all we need is that (12) and (13) be evaluated at the situation in which (11) exists, that is, s1. The other parameters vary independently, i.e., (11) can be true about a situation s 2 in which neither (11) nor (12) nor (13) exist, and (12) (and (13)) can be true in a situation s3 (and in s4) about which neither sentence is true. But one of the consequences of the definition of validity above is that all the premises and the conclusion should be evaluated at the same situation (this is the key to the solution to the first sophism of the eighth chapter). Therefore, we have to align the truth-conditions of (11), (12) and (13), so that they are evaluated at the same situation:

(11’) Socrates walks s1,s

x

(12’) (11) exists s3,s

x

(13’) (11) is true s4,s

x

Since all the sentences should exist in the same situation, we should also align the situation in which (11)-(13) exist. Every sentence of a consequence must be true in the same situation about the same situation—in Prior’s terms, a con-sequence is valid on a sheet if there is no sheet about which all the premises are true and the consequence, false.

(11’’) Socrates walks sy,s

x

(12’’) (11) exists sy,s

x

(13’’) (11) is true sy,s

x

This is not enough. Since the situation of evaluation of (12) and (13) should be a situation in which (11) exits, we need x=y:

(11’’’) Socrates walks sx,s

x


(12’’’) (11) exists sx, s

x

(13’’’) (11) is true sx,s

x

The identification of the situation of utterance and the situation of evaluation is not in itself a problem. It is however a bit disappointing. One of the nice features of Buridan’s double indexing semantics is precisely making room for a sentence being true about a situation in which it doesn’t exist, therefore solv-ing quite neatly paradoxes such as ‘every sentence is affirmative, therefore no sentence is negative’. It can also explain quite naturally why not every token-ing of sentences such as ‘every sentence is false’ or ‘I say something false’68 is paradoxical. Now, if we want ‘every sentence is false’ to imply (with the needed extra premise) “ ‘every sentence is false’ is true,” then something must be changed in Buridan’s theory. But maybe this is just as well: such sentences are possible, but not possibly-true.

More important is to notice that, according to Buridan, the theses (iv) and (vi) above will only affect the truth-value of a sentence if the sentence is self-referential:

. . . when a sentence has or can have reference to itself, it does not suffice for the truth of an affirmative [sentence] that its terms supposit for the same [thing or things] <. . .> but it is also required that that even in this consequent <that is, that the sentence itself is true>, the terms supposit for the same [thing or things], and then it is necessary, if this holds, that the sentence be true.69

Why is it so? Here is an explanation, taking (7) as example. Let us follow the steps above, leading to the alignment of the parameters of the sentences in an inference. We start with the sentences taken as separate assertions:

(7) Every sentence is false s1,s

2

(14) (7) exists s3,s

1

68) For the first sentence, see Soph., VIII, 8, 152, 13-23; the same reasoning can be applied to the second sentence.69) Transl. Klima, 969. “. . . ubi propositio propositio habet vel habere potest reflexionem super se, non sufficit ad veritatem affirmativae quod termini pro eodem supponant <. . . .>, sed requiritur quod etiam in tali consequente termini pro eodem supponant, et tunc oportet hoc stante proposi-tionem esse veram.” (Soph., VIII, soph., 7, 156, l. 5-9; soph., 8, 159, l. 10-15; soph., 11, 167, l. 4-7).


(15) (7) is true s4,s

1

As we have seen, (7) can be true about a situation s 2 in which it doesn’t exist, and this is accounted for by the double indexing theory. (14) and (15) are both true about s1. With this distribution of indexes, we cannot infer (15) from (7). We need, first, to align the situations of evaluation (s 2= s1) and of production of the inference itself (s1=s3=s4), and, in this particular case, because the subject of (14) and (15) supposits for (7), we will have the same parameter in both slots of all sentences. That is precisely what happened with the infer-ence (11)—(13) above. But once we do that, (7) can no longer be true, because it will be evaluated at a situation in which it exists. Were it not to exist in the same situation as (14) and (15), the inference could not be done: all the sen-tences need to exist in the situation in which the inference is drawn. On the other hand, (14) and (15) are intended to be true about a situation in which (7) exists, and are therefore false if evaluated at a situation in which this is not the case. (7) is possible, but not possibly true.

This consequence follows from the alignment of the situations of evaluation and of production and from the self-referential character of (7)—that is, from the self-referential character of the words of (7), since it seems difficult to avoid this consequence by intentionally decoupling the situation of evaluation and of utterance (as in the benign use of the sentence ‘every proposition is false’). This is an illustration of the Buridanian claim that the theses (iv) and (vi) will only affect the truth-value of self-referential sentences. In the infer-ence from (11) and (12) to (13) above, the truth-value of (11) is not affected by the alignment of situations. The drawback, however, is that it is not clear how to make this alignment demand compatible with the results of the sev-enth chapter, not only for sentences like (7), but also for (11), that may have tokenings whose evaluation may need to be decoupled from the situation of production.70

6. Conclusion

The Buridanian solution to the paradox of the Liar can only be properly understood if we place it correctly in the framework of his philosophy of lan-guage—as a matter of fact, in its immediate theoretical context, so to speak,

70) See Sophisms, chap. VII, soph., 2-4; see Perini-Santos, ‘John Buridan on the Bearer of Logical Relations’, and Id., ‘When the inference “p is true, therefore p” fails’.


since insolubilia are treated in his Sophisms just after the extension of the results of the seventh chapter to a theory of consequences. In this paper, I have tried to show that Buridan does not abandon the Aristotelian formula, and that it is a way to designate the situation of evaluation of a sentence. This fact is impor-tant to understanding how some previous interpretations of Buridan’s solution to the Liar have been led to postulate truth-values gaps in his theory: they have missed this conceptual point (and what strikes me as a plain textual fact).

For Buridan, a Liar sentence is false because it implies that it is true, with an extra-premise stipulating its existence in the situation at which the infer-ence is evaluated, this conclusion is stably false—that is, it is false and we can-not argue back from its falsity to its truth—and its falsity implies the falsity of the Liar sentence itself. I don’t know whether this solution (or some amended version of it) is a good one or not, or whether it could live up to contemporary standards or not. I think that the answers to both questions are negative, but I have not tried to argue for this position. Anyway, establishing what is Buri-dan’s solution to the paradox and defending his solution as a good one are clearly different tasks, and one cannot shape one’s interpretation so as to make it compatible with what one takes to be a good theory. In this regard, it is surprising to notice that the idea that Buridan has truth-value gaps in his logic, and that this is part of his solution to the Liar can be traced back to a suggestion made by Herzberger, who was not a historian of Philosophy, and who would have certainly been untouched by the suggestion that Buridan is not a predecessor of van Fraassen or of his own solution to the Liar. Not so for historians of Philosophy, who have nonetheless followed Herzberger’s sugges-tion that we can find in Buridan the fulfillment of “correspondence condi-tions” without the truth of the sentence, leading to truth-value gaps, or to conflicting criteria of truth, or even to the abandonment of the very notion of truth, with slender textual basis. It is more important to remark that Buridan’s solution does not seem compatible with some interesting consequences of the distinction between the time in quo a sentence is true and the time pro quo it is true, or at least is less interesting and with a smaller scope than what we might have thought, or less interesting than what I have thought when reading the seventh chapter of his Sophisms. But the effort to interpret a theory is not an effort to make it more interesting.

Ernesto Perini-Santos - John Buridan’s Theory of Truth and the Paradox of the Liar

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