Consequences of a closed, token-based semantics: the case of John Buridan

Abstract

This paper argues for two principal conclusions about natural language semantics based on John Buridan's considerations concerning the notion of formal consequence, that is, formally valid inference. (1) Natural languages are essentially semantically closed, yet they do not have to be on that account inconsistent. (2) Natural language semantics has to be token based, as a matter of principle. The paper investigates the Buridanian considerations leading to these conclusions, and considers some obviously emerging objections to the Buridanian approach.

Acknowledgements

Earlier drafts of this paper were presented at the Moody Conference in Medieval Philosophy on ‘Truth and Proof in the Middle Ages’ at UCLA, on 1 February 2003 and in the Philosophy Colloquium Series of Boise State University, Idaho, 3 May 2003. I am grateful to both audiences for many insightful comments, in particular to Andrew Cortens, David Kaplan, Elizabeth Karger and Tony Roark. But I want to thank especially Calvin Normore for the numerous inspiring conversations we had on the subject during my visit at UCLA's Philosophy Department during the academic year 2002–2003. Research for this paper was supported by an ACLS Fellowship and a Fordham Faculty Fellowship for the same period.

Notes

Some of the material contained in this paper will form part of the author's monograph The Logic and Metaphysics of John Buridan, forthcoming in the Past Masters series of Oxford University Press.

Obviously, this consequence is only valid on the medieval analysis of universal affirmatives, attributing ‘existential import’ to them. For a more detailed discussion of the issue see Klima ( 2001 , pp. 197–226). Modern readers who still do not like this analysis may substitute their favorite example of a valid consequence here; Buridan's point remains the same.

Antecedens autem et consequens relatiue dicuntur ad inuicem; ideo per inuicem describi debent. Dicunt ergo multi quod propositionum duarum illa est antecedens ad aliam quam impossibile est esse ueram illa alia non existente uera et illa est consequens ad reliquam quam impossibile est non esse ueram reliqua existente uera, ita quod omnis propositio ad omnem aliam propositionem est antecedens quam impossibile est esse ueram illa alia non existente uera. Sed haec descriptio deficit uel est incompleta, quia hic est bona consequentia omnis homo currit; ergo aliquis homo currit, et tamen possibile est primam esse ueram secunda non existente uera, immo secunda non existente. Buridan 1976 (henceforth TC) (pp. 21–22)

In the context of Buridan's nominalist philosophy, we can safely disregard the issue of real propositions espoused by Burleigh and other realists. Cf. Nuchelmans 1980 (pp. 9–13).

Et ideo aliqui dicunt dictam descriptionem debere suppleri sic: illa propositio est antecedens ad aliam propositionem quam impossibile est esse ueram illa alia non existente uera istis simul formatis. TC (pp. 21–22).

Sed adhuc dico quod haec descriptio non est bona, quia hic non est bona consequentia nulla propositio est negatiua; ergo nullus asinus currit, et tamen secundum dictam descriptionem oporteret eam concedere esse bonam; ergo etc. Primam praemissam probo. Quia ex opposito consequentis non sequitur oppositum antecedentis; non enim sequitur ‘quidam asinus currit; ergo quaedam propositio est negatiua’. Secunda autem praemissa manifesta est. Quia primam, scilicet quae designatur esse antecedens, impossibile est esse ueram; ergo impossibile est ipsam esse ueram alia non existente uera. TC (pp. 21–22)

To be sure, given Buridan's token-based conception of propositions, the phrase ‘the proposition “p”’ does not always manage to single out a single proposition-token (although in some contexts, when the uniqueness of reference is secured by the context itself, it can). For brevity and naturalness of expression, when no confusion arises, we can still use this phrase as an abbreviation of the expression ‘any proposition of the form “p”’.

Buridan 2001 (henceforth SD) (p. 953).

I owe the gist of this objection to David Kaplan.

I owe this point to Elizabeth Karger.

See SD (9.8, 8th sophism, pp. 971–974). I have modified Buridan's sophism, since his treatment of it relies on a rather dubious ‘parity of reasoning’-style argument attempting to establish the equivalence of ‘Plato says something false’ and ‘Socrates says something false’ uttered by Socrates and Plato respectively. I believe the modified version serves to illustrate the points I want it to illustrate without having to rely on this type of reasoning.

I owe this anecdotal remark to Peter King. As in almost all cases of such anecdotal remarks, the Italian saying applies here too: se non è vero è ben trovato.

The terms ‘the proposition I am uttering’ and ‘false’ do indeed stand for the same thing, namely, the proposition I am uttering, given that it is in fact false.

Et licet haec concedantur, tamen non sequitur quod omnis propositio affirmatiua de inesse et de praesenti sit uera cuius termini supponunt pro eodem, quia in propositione asserente se esse falsam potest esse quod termini supponant pro eodem, et tamen ipsa est falsa; uerbi gratia, si aliquis dicat ‘propositio quam ego profero est falsa’. Et causa est quia, quamuis illa propositio de sua forma designet idem esse pro quo termini supponunt et ita sit, tamen cum hoc, propter significationem praedicati, designat quod non sit idem. Quamcumque enim propositionem dicimus esse falsam designamus non esse idem pro quo etc. Ideo talis propositio designat esse idem et non esse idem, et ideo, licet qualiter significat ita sit, tamen non qualitercumque significat ita est, et ideo est falsa. TC (c. 5, pp. 25–26)

J. Buridan, Quaestiones in primum librum Analyticorum Posteriorum, q. 10 (unpublished edition by H. Hubien). Buridan here apparently took over Thomas Bradwardine's solution. Cf. Read 2002 (pp. 189–218). For more detailed accounts of Buridan's parallel passages see Pironet 1993 (pp. 293–300); Buridan 1994 (Introduction §3).

SD (pp. 967–968).

The point of the argument is that the proposition ‘the proposition “p” signifies itself to be true’ cannot be true. For according to Buridan's theory of sentential nominalisations, the phrase ‘itself to be true’ can be taken to stand either materially, for the proposition ‘the proposition “p” is true’ (but if ‘p’ is a proposition about things that are not propositions, then it certainly does not signify this proposition, despite what the original proposition claims in this sense), or personally, for whatever the terms of the phrase ‘that p is true’ are jointly true of, which in the case of an impossible proposition cannot be anything, and thus cannot be signified by the proposition ‘p’. Cf. SD (p. 969, n. 183). So, Buridan actually had a very good reason to reject Bradwardine's (and his own earlier) solution: the solution provided in terms of signification cannot be expressed in a true sentence.

SD (p. 857).

SD (p. 969).

Cf. SD (9.8, p. 970).

This is the principle G. E. Hughes called Buridan's ‘entailment principle’, which he ingeniously defended against an apparently obvious objection in the Introduction of his John Buridan on Self-reference (Hughes 1982 , pp. 23–27). (Hughes' defense of this principle, by the way, actually relies on considerations similar to those concerning the situation of the ‘Reciprocal Liar’, pointing out that two tokens of the same proposition-type may have opposite truth-values. So, I think Hughes is absolutely correct in remarking that Buridan had everything at his disposal to reply to the objection along the lines Hughes does.) As in a stimulating discussion of an earlier draft of this paper at Boise State University Tony Roark has insightfully pointed out, a similar objection raised by Andrew Cortens, attempting to refer to the proposition in the antecedent of this principle, can be handled by relying on Buridan's conception according to which propositional components of complex propositions are not themselves propositions. For more on Buridan's conception, see Klima 2004 .

A similar, yet slightly different, account is provided by Hughes 1982 (pp. 22–23). Indeed, perhaps surprisingly, I also agree here with Stephen Read's ( 2002 , p. 202) conclusion: ‘Buridan ends up with no theory of truth at all’. Yet, pace Read, I do not find this to be a fatal flaw of Buridan's approach. For on my understanding (TB) does not even attempt to be a theory of truth, and Buridan does not even need such a theory. He has a theory of correspondence, which is all he needs for his logic, and he provides the trivial assertibility conditions for the predicate ‘true’ in his token-based semantics, whereby he can show why, despite possible appearances to the contrary, what Tarski would call the ‘semantic closure’ of his theory does not entail paradoxical results.

Cf. Gaifman 2000 (pp. 79–121). It is quite revealing that Gaifman's main motivation for a token-based semantics is the same sort of semantic paradox that I dubbed the ‘Reciprocal Liar’ in Buridan.

Ideo alii aliter diffiniunt, dicentes quod illa propositio est antecedens ad aliam quae sic se habet ad illam quod impossibile est qualitercumque ipsa significat sic esse quin qualitercumque illa alia significat sic sit ipsis simul propositis. TC (c. 3, p. 22) We should also keep in mind the crucial distinction concerning the correct interpretation of this rule Buridan introduced in his Sophismata: ‘… a consequence is valid if it is impossible for things to be as the antecedent signifies without their being as the consequent signifies. And this rule can be understood in two ways: first, that it is one proposition about impossibility in the composite sense, in the way that this is commonly used, and its sense then is that this is impossible: ‘When it is formed, things are as the antecedent signifies and not as the consequent signifies’. And taken in this way the rule is not valid […]. Taken in the other way, the rule is understood as a proposition about impossibility in the divided sense, so that its sense is: ‘a consequence is valid if in whatever way the antecedent signifies [things to be], it is impossible for things to be in that way without their being in the way the consequent signifies [them to be]’ SD (9.8. Second sophism, pp. 957 – 958).

Tamen adhuc illa descriptio non est uera de uirtute sermonis, quia supponit quod omnis propositio uera ex eo sit uera quia qualitercumque significat ita est, quod prius negatum est. Tamen dictum fuit quod hoc modo loquendi uteremur ad sensum prius datum; ideo sic illam descriptionem concedemus. TC (p. 21)

See TC (c. 1, pp. 18–19). Cf. SD (9. 2, 14th conclusion, pp. 858–859).

Immo etiam saepe utemur modo loquendi secundum primam descriptionem prius manifeste improbatam, quia ipsa in paucis consequentiis habet instantiam. Tamen quocumque modo loquendi utemur nos intendemus sensum praetactum. TC (c. 3, p. 22).

Concerning Buridan's treatment of consequences as a special type of hypothetical propositions in which the clauses are asserted, see Klima 2004 .