## Abstract

This paper revisits Buridan’s Bridge paradox (*Sophismata*, chapter 8, Sophism 17), itself close kin to the Liar paradox, a version of which also appears in Bradwardine’s *Insolubilia*. Prompted by the occurrence of the paradox in Cervantes’s *Don Quixote*, I discuss and compare four distinct solutions to the problem, namely Bradwardine’s “just false” conception, Buridan’s “contingently true/false” theory, Cervantes’s “both true and false” view, and then the “neither true simpliciter nor false simpliciter” account proposed more recently by Jacquette. All have in common to accept that the Bridge expresses a truth-apt proposition, but only the latter three endorse the transparency of truth. Against some previous commentaries I first show that Buridan’s solution is fully compliant with an account of the paradox within classical logic. I then argue that Cervantes’s insights, as well as Jacquette’s treatment, are both supportive of a dialetheist account, and Jacquette’s in particular of the strict-tolerant account of truth. I defend dialetheist intuitions (whether in LP or ST guise) against two objections: one concerning the future, the other concerning the alleged simplicity of the Bridge compared to the Liar.

### Similar content being viewed by others

## Notes

Mention of the puzzle can be also found in (Prior, 1962). Prior in that paper does not propose an analysis of that specific puzzle, however. Instead he focuses on earlier puzzles in the

*Sophismata*, in particular the Liar. More recent sources citing the paradox include Read (1995) (about Cervantes), and Øhrstrøm and Hasle (2007), p. 36 (about Buridan).See Prior, (1962, p. 283), who writes: “this is a puzzle of some literary interest, since there is a very similar one in

*Don Quixote*”. Prior cites Church (1956, Exercice 15.10) regarding the connection to Cervantes. Church states a version of the puzzle very close to Cervantes’s original, to which he gives credit, but he does not refer to Buridan. As I will argue below, the treatment sketched by Church happens to agree with a central element in the diagnosis proposed by Buridan himself, although Church does not appear to be aware of Buridan’s account. I myself discovered Church’s treatment only after formalizing Buridan’s account. No reference to either Church or Prior appears in the texts by Hughes, Jacquette, or Ulatowski. More generally, the literature on the Bridge paradox is very scattered.Priest (2006, p. 185, fn. 5), in his chapter on norms and the philosophy of law, does not mention the Bridge. However, he briefly mentions the related case of Protagoras and Euathlus (see below, fn. 8) as a case of

*prima facie*dialetheia.I am indebted to an anonymous referee for bringing this fact to my notice. On pg. 458 of his paper, Jacquette reproduces Scott’s translation, which says: “I say that Plato does not speak truly, since Socrates has

*not*fulfilled the condition" (emphasis mine, based on the reviewer’s observation). In the editions of Hughes (1982), Klima (2001), and Pironnet (2004), the problematic negation is missing, viz. Klima (2001): “I say that Plato did not say something true, since Socrates fulfilled the condition”.See Jacquette (1991, p. 457): “Sancho gives an alternative philosophically less satisfactory resolution in this strange but no less edifying History, which it were now tedious to relate”, and his remark “Cervantes is evidently a better novelist than logician.”

See Hughes (1982, p. 161). The Crocodile dilemma: “a crocodile stole a woman’s baby and promised to return it to her if she told him truly whether he would it eat it or not. The woman then replied, “you are going to eat it”." The mother argues that whether what she says is true or false, the Crocodile should keep his word and return the baby; the crocodile argues symmetrically. Protagoras’s paradox: Protagoras sues his student Euathlus, who had promised to pay his fee when winning his first case, but chose never to practice. Protagoras argues that if he loses, Euathlus will have to pay him (because he lost), and if he wins, Euathlus will also have to pay him (to keep his word); Euathlus reasons that if he loses he won’t have to pay (short of winning his first case), and similarly if he wins (because then Protagoras should pay). Hughes does not give the actual sources, but the Crocodile dilemma is attributed to Chrysippus by Lucian of Samosata in his

*Vitarum Auctio*(see Biard, 1993, as well as Wlaker, 1847, part III, commentary to chap. 8, p. 159). The dilemma of Protagoras and Euathlus, also cited in Walker (1847, part III, chap. 8, p. 156), is mentioned in Diogenes Laërtius*Vitae philosophorum*, IX, 56 and in Aulus Gellius,*Noctes atticae*V, 10. The case is also presented by Sextus Empiricus in*Adversus Rhetores*(§96-100), except that Sextus mentions Korax (a Sicilian orator linked to Gorgias) instead of Protagoras, and an unnamed disciple instead of Euathlus. In Sextus’s version, the judges eventually kick Korax and his student out of the court, on the grounds that their arguments have equal strength. Walker in his commentary of Murray’s compendium does not cite Buridan, but notes the connection of the ancient paradoxes with Cervantes: “if I mistake not, a similarly ludicrous instance may be found in Don Quixote”.Pseudo-Heytesbury’s

*Insolubilia Padua*also presents the penny version (Pironnet, 2008, p. 294), namely “Socrates non habebit denarium”. See also Wyclif’s*Summa insolubilium*(ca. 1360), Book III, chapter 13 [261], which refers to it as “casu[s] de pertransitione pontis” (Spade & Wilson, 1986). I am indebted to an anonymous referee for these references.Jones (1986) conjectures that Cervantes might have relied on treatises by the Spanish philosophers Domingo de Soto (

*Introductiones Dialectice*, p. 1529) and Juan de Celaya (*Insolubilia et Obligationes*, p. 1517), as well as on Paul of Venice’s*Logica Magna*, first published in 1499. Surprisingly, Jones does not cite Buridan or earlier sources. On the analyses of the Bridge given by Paul of Venice and by Domingo de Soto, see Ashworth (1976).Jacquette (1991) considers that Cervantes’s version makes the paradox easy to solve, because Cervantes’s version refers to the bystander’s first-person

*intention*to cross the bridge.Bradwardine’s argument rests on a principle explained in Chapter 6 of his

*Insolubles*, namely “every proposition signifies or means as a matter of fact or absolutely respectively everything which follows from it as a matter of fact or absolutely”. I leave aside a formalization of Bradwardine’s theory in what follows, to focus on the other three accounts, which subscribe to transparency.Buridan contrasts two analyses of conditionals in his discussion of the second question raised by the puzzle, one we may call strict (a conditional cannot be true if it is possible for the antecedent to be true and the consequent false), and one we can call material, based on the fact that it obeys the principle of conjunctive sufficiency (a conditional is true when antecedent and consequent are both true). Buridan argues that the latter suffices for the argument. On the tradition to treat promissive conditionals as material conditionals until the Renaissance period, see in particular Ashworth (1972a).

In other words, I handle “will” throughout in a predictive sense, and not in the deontic sense of “ought” in which it is sometimes interpreted. Throughout this paper I therefore talk of the decree as being true or false, instead of fulfilled vs. unfulfilled. One may, in principle, introduce a predicate of fulfillment distinct from the truth predicate, in order to give a more elaborate account. For more on the distinction between truth and fulfillment, see Bonevac (1990). For a defense of the view that legal statements are evaluable as true and false, see Priest (2006, p. 186).

Jacquette (1991, p. 459) writes: “The analysis jumbles several things together. Again there are apparent semantic errors from the standpoint of modern logic”. Ulatowski (2003, p. 87) hammers the point: “...Buridan’s solutions to the bridge paradox are inadequate: the logico-semantic issue jumbles several things together...”. However, both Jacquette and Ulatowski actually rely on Scott’s edition and translation of Buridan. Hughes’ edition explicitly differs from Scott’s in a number of places, and Hughes correctly explains how Plato’s conditional must be false (Hughes 1982, pp. 160–161), yet without formalizing the argument.

Wyclif’s treatment of the Bridge is much closer to Buridan’s than it is to Bradwardine’s in that regard. See (Spade and Wilson, 1986, pp. 261–262), where Wyclif writes that even if a tyrant makes a universal order as expressed in the bridge decree, “non ex hoc quod ipse sic ordinat sequitur universalem istam esse veram” (from the fact that he orders things that way, it does not follow that this universal statement is true).

Church’s treatment in Exercise 15.10 runs as follows:

*P*represents the fact that the man crosses the bridge,*R*the fact that the man’s utterance is true,*Q*the fact that the man is hanged, and*S*the fact that the law is obeyed. His assumptions are that \(R\leftrightarrow P \wedge Q\),*P*, and \(S\rightarrow (Q\leftrightarrow (P \wedge \lnot R))\). Church’s exercise is to show that \(\lnot S\) follows from those assumptions, by modus ponens and given the tautology \((R\leftrightarrow P \wedge Q) \rightarrow (P\rightarrow ((S\rightarrow (Q\leftrightarrow P \wedge \lnot R))\rightarrow \lnot S))\).This point is acknowledged by Jacquette, see (Jacquette, 1991, p. 462). Although Jacquette sees this, he does not appear to see it as the conclusion of a classical reductio argument.

See Égré et al. (2021) for a presentation. Whether the Cooper conditional is suitable to deal with other self-referential paradoxes beside the Bridge is not obvious, however, but I leave this issue aside here.

Read (1995, p. 149) comments on Sancho’s decision to let the man pass free as follows: “this is

*in effect*to set the law aside, and declare it inoperable in this case. In other words, the law should have been more carefully framed in the first place, and was unsatisfactory” (my emphasis). That is, Read looks at the decision from the viewpoint of classical logic. But note that this does not represent Sancho’s point of view. Sancho himself does not claim that the law is inoperable.This claim is controversial, as highlighted to me by David Ripley, depending on how “just true” is formalized. In Beall (2009)’s extension of LP with transparent truth, \(T\langle A\rangle \) and \(T\langle A\rangle \wedge \lnot T\lnot \langle A \rangle \) are equivalent. If the latter is taken to paraphrase “just true”, then “true” and “just true” collapse.

Moreover, (T\(^{\star }\)) and (F\(^{\star }\)) come out equivalent to tolerant readings of (T) and (F) in three-valued logic if we assume that for

*Sxy*and \(\textsf{F}Pa\) the distinction between tolerant and strict collapses.Throughout we read a natural deduction proof of

*A*with set of open assumptions \(\Gamma \) as a proof of \(\Gamma , \Delta \models A\), for any \(\Delta \). The metainference from \(\Gamma , A\models ^{ss} \bot \) to \(\Gamma \models ^{ss} \lnot A\) is not*ss*-valid. (Thanks to D. Ripley for a related discussion).See Read (2002) for a detailed account of Bradwardine’s treatment of the Liar, and its supposed influence on Buridan.

As in the case of the Bridge, I believe that the principle of charity should be rigorously applied. Presently, however, I find harder to accept Buridan’s (and Bradwardine’s) denial that the Liar’s falsity should entail is truth. Despite that, two abstract elements of convergence may actually be underlined concerning Buridan’s treatment of the Liar and the Bridge, and the strict-tolerant account in particular: both accounts accept that those sentences are truth-apt, and both accept that truth is not a simple notion. In Buridan, it involves distinct dimensions, see Herzberger (1973); in ST, it comes in a weak vs a strong form.

For a discussion of this passage, in particular in relation to Bradwardine’s treatment of the Liar, see Dutilh Novaes and Read (2008).

## References

Ashworth, E. J. (1972). Strict and material implication in the early Sixteenth Century.

*Notre Dame Journal of Formal Logic,**13*(4), 556–560.Ashworth, E. J. (1972). The treatment of semantic paradoxes from 1400 to 1700.

*Notre Dame Journal of Formal Logic,**13*(1), 34–52.Ashworth, E. J. (1976). Will Socrates cross the bridge? A problem in medieval logic.

*Franciscan Studies,**36*(1), 75–84.Barrio, E., Rosenblatt, L., & Tajer, D. (2015). The logics of strict-tolerant logic.

*Journal of Philosophical Logic,**44*(5), 551–571.Beall, J. (2009).

*Spandrels of truth*. OUP.Beaver, D. (2001).

*Presupposition and assertion in dynamic semantics*(Vol. 29). CSLI Publications.Biard, J., (Ed.). (1993).

*Sophismes, de Jean Buridan*. Jacques Vrin: Paris. French edition and translation, based on a first translation by Fabienne Pironnet.Bochvar, D. A. (1937). [On a three-valued calculus and its applications to the paradoxes of the classical extended functional calculus].

*Mathematicheskii Sbornik, 4*(46), 287–308. English translation by M. Bergmann in*History and Philosophy of Logic*2 (1981), 87–112.Bonevac, D. (1990). Paradoxes of fulfillment.

*Journal of Philosophical Logic,**19*(3), 229–252.Burge, T. (1978). Buridan and epistemic paradox.

*Philosophical Studies,**34*(1), 21–35.Cervantes, M. (1615).

*Don Quixote*. Gutenberg Project, 2019 edition. English translation by John Ormsby.Church, A. (1956).

*Introduction to mathematical logic*(Vol. 13). Princeton University Press. 1996 Reissue.Cobreros, P. (2016). Supervaluationism and the timeless solution to the foreknowledge problem.

*Scientia et Fides,**4*(1), 61–75.Cobreros, P., Egré, P., Ripley, D., & van Rooij, R. (2012). Tolerant, classical, strict.

*The Journal of Philosophical Logic,**41*(2), 347–385.Cobreros, P., Égré, P., Ripley, D., & van Rooij, R. (2013). Reaching transparent truth.

*Mind,**122*(488), 841–866.Cobreros, P., Egré, P., Ripley, D., & van Rooij, R. (2015). Vagueness, truth and permissive consequence. In D. Achouriotti, H. Galinon, & J. M. Fernández (Eds.),

*Unifying the philosophy of truth*(pp. 409–430). Springer.Cobreros, P., Egré, P., Ripley, D., & van Rooij, R. (2020). Inferences and metainferences in ST.

*Journal of Philosophical Logic,**49*(6), 1057–1077.Cooper, W. (1968). The propositional logic of ordinary discourse.

*Inquiry,**11*(1–4), 295–320.Dicher, B., & Paoli, F. (2019). ST, LP and tolerant metainferences. In C. Baskent & T. Ferguson (Eds.),

*Graham Priest on dialetheism and paraconsistency*(pp. 383–407). Springer.Dutilh Novaes, C. (2009). Lessons on sentential meaning from mediaeval solutions to the Liar paradox.

*The Philosophical Quarterly,**59*(237), 682–704.Dutilh Novaes, C., & Read, S. (2008). Insolubilia and the fallacy secundum quid et simpliciter.

*Vivarium,**46*(2), 175–191.Égré, P. (2019). Respects for contradictions. In C. Baskent & T. Ferguson (Eds.),

*Graham Priest on Dialetheism and Paraconsistency*(pp. 39–57). Springer.Égré, P. (2021). Half-truths and the Liar. In C. Nicolai & J. Stern (Eds.),

*Modes of truth*(pp. 18–40). Taylor and Francis.Égré, P., Rossi, L., & Sprenger, J. (2021). De Finettian logics of indicative conditionals. Part I: Trivalent semantics and validity.

*Journal of Philosophical Logic,**50*, 187–213.Field, H. (2008).

*Saving truth from paradox*. OUP.Glanzberg, M. (2004). A contextual-hierarchical approach to truth and the Liar paradox.

*Journal of Philosophical Logic,**33*(1), 27–88.Herzberger, H. G. (1973). Dimensions of truth.

*Journal of Philosophical Logic,**2*(4), 535–556.Hughes, G. E. (1982).

*John Buridan on self-reference: Chapter Eight of Buridan’s Sophismata, with a translation, an introduction, and a philosophical commentary*. Cambridge University Press.Hyde, D. (1997). From heaps and gaps to heaps of gluts.

*Mind,**106*(424), 641–660.Jacquette, D. (1991). Buridan’s bridge.

*Philosophy,**66*(258), 455–471.Jones, J. R. (1986). The Liar Paradox in Don Quixote II, 51.

*Hispanic Review,**54*(2), 183–193.Klima, G. (Ed.). (2001).

*Summulae de dialectica, by Jean Buridan*. Yale University Press.Klima, G. (2008). Logic without truth. In S. Rahman, T. Tulenheimo, & E. Genot (Eds.),

*Unity, truth and the liar: The modern relevance of medieval solutions to the liar paradox*(pp. 87–112). Springer.Kretzmann, N., & Kretzmann, B. E. (Eds.). (1990).

*Sophismata, by Richard Kilvington*. Cambridge University Press.McGee, V. (1990).

*Truth, vagueness, and paradox: An essay on the logic of truth*. Hackett Publishing.Øhrstrøm, P., & Hasle, P. (2007).

*Temporal logic: From ancient ideas to artificial intelligence*(Vol. 57). Springer.Perini-Santos, E. (2011). John Buridan’s theory of truth and the paradox of the Liar.

*Vivarium,**49*(1–3), 184–213.Pironnet, F., (Ed.). (2004).

*Summulae de practica sophismatum, by Jean Buridan*. Aristarium 10-9: Brepols.Pironnet, F. (2008). William Heytesbury and the treatment of Insolubilia in fourteenth-century England followed by a critical edition of three anonymous treatises De insolubilibus inspired by Heytesbury. In S. Rahman, T. Tulenheimo, & E. Genot (Eds.),

*Unity, truth and the liar: The modern relevance of medieval solutions to the liar paradox*(pp. 255–333). Springer.Priest, G. (1979). The logic of paradox.

*Journal of Philosophical logic,**8*(1), 219–241.Priest, G. (2006).

*In Contradiction*(2nd ed.). Oxford University Press.Priest, G. (2019). Respectfully yours. In C. Baskent & T. Ferguson (Eds.),

*Graham priest on dialetheism and paraconsistency.*Springer.Priest, G. (2021). Substructural solutions to the semantic paradoxes: Dialetheism in sheepÕs clothing? Manuscript.

Prior, A. (1962).

*Some problems of self-reference in John Buridan*(pp. 281–296). Dawes Hicks Lecture on Philosophy.Quine, W. V. O. (1953). On a so-called paradox.

*Mind,**62*(245), 65–67.Read, S. (1995).

*Thinking about logic*. Oxford University Press.Read, S. (2002). The Liar paradox from John Buridan back to Thomas Bradwardine.

*Vivarium,**40*(2), 189–218.Read, S., (Ed.). (2010).

*Insolubilia, by Thomas Bradwardine*. Dallas Medieval Texts and Translations 10: Peeters. Edition, translation and introduction by Stephen Read.Ripley, D. (2012). Conservatively extending classical logic with transparent truth.

*The Review of Symbolic Logic,**5*(2), 354–378.Ripley, D. (2013). Paradoxes and failures of cut.

*Australasian Journal of Philosophy,**91*(1), 139–164.Rossi, L. (2019). A unified theory of truth and paradox.

*The Review of Symbolic Logic,**12*(2), 209–254.Scott, T. K. (Ed.). (1966).

*Sophisms on meaning and truth, by Jean Buridan*. Appleton-Century-Crofts.Spade, P. V., & Wilson, G. A. (Eds.). (1986).

*Johannis Wyclif Summa insolubilium*. Medieval and Renaissance Texts and Studies.Spector, B. (2016). Multivalent semantics for vagueness and presupposition.

*Topoi,**35*(1), 45–55.Thomason, R. H. (1970). Indeterminist time and truth-value gaps.

*Theoria,**36*(3), 264–281.Ulatowski, J. W. (2003). A conscientious resolution of the action paradox on Buridan’s bridge.

*Southwest philosophical studies*, (pp. 85–93).Van Fraassen, B. (1968). Presupposition, implication, and self-reference.

*The Journal of Philosophy,**65*(5), 136–152.Walker, J. (Ed.). (1847).

*Murray’s compendium of logic*. Longman, Brown, Green, Longmans.Zehr, J. (2014).

*Vagueness, presupposition and truth-value judgments*. PhD Thesis, ENS, PSL University.

## Acknowledgements

I wish to thank Jean-Pascal Anfray, Timothée Bernard, Pablo Cobreros, Dimitri El Murr, Julie Goncharova, Louis Guichard, James Hampton, Klaus von Heusinger, Andrea Iacona, Benjamin Icard, Hitoshi Omori, Paloma Pérez Ilzarbe, Simone Picenni, Graham Priest, Lorenzo Rossi, David Ripley, Johannes Stern, and Denis Thouard for very helpful exchanges, and audiences in Göttingen (AIL1), Bristol, Torino, Pamplona, and Paris. Special thanks go to two anonymous referees and to the editors of this special issue for comments. I am particularly grateful to Paloma Pérez Ilzarbe and to David Ripley for detailed feedback and generous comments on specific aspects of the paper, to Jean-Pascal Anfray for his help with a Latin translation, and to an anonymous referee for pointing out various sources and exegetical aspects that were missing in the first version of this work. Any errors, whether historical or logical, are my own. This research was partly supported by the programs FRONTCOG (ANR-17-EURE-0017) and AMBISENSE (ANR-19-CE28-0019-01).

## Author information

### Authors and Affiliations

### Corresponding author

## Ethics declarations

### Conflicts of interest

The author declares no conflict of interest.

## Additional information

### Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

T.C.: Non-Classical Approaches to Paradox.

## Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

## About this article

### Cite this article

Égré, P. Truth and Falsity in Buridan’s Bridge.
*Synthese* **201**, 17 (2023). https://doi.org/10.1007/s11229-022-03907-4

Received:

Accepted:

Published:

DOI: https://doi.org/10.1007/s11229-022-03907-4