John Buridan on Self-Reference: Chapter Eight of Buridan's Sophismata, with a translation, an introduction and a philosophical commentary by G.E. Hughes (New York: Cambridge University Press, 1982), xi + 233 pp.

Alfred J. Freddoso
University of Notre Dame

This is an excellent book, a paradigm of its genre. In Chapter Eight of the Sophismata Buridan proposes his highly sophisticated solutions to a wide variety of alethic (Liar-type), epistemic and pragmatic paradoxes involving self-reference. Hughes' almost equally impressive contribution consists of no less than (i) a clear and penetrating introduction (pp. 1-37), (ii) a new (albeit non-critical) edition of the Latin text along with a facing English translation that is unfailingly accurate and smooth (pp. 38-129), (iii) notes to the Latin text (pp. 131-139), and (iv) a painstaking and philosophically illuminating commentary (pp. 141-227). The work as a whole has been fashioned with great care and obviously represents a labor of love as well as of scholarship on Hughes' part. (For confirmation of this last point, consult the final note on p. 139.)

The epistemic paradoxes are probably the deepest, while the pragmatic paradoxes are undoubtedly the most entertaining. Here, however, I will confine my brief remarks to Buridan's distinctive resolution of the alethic paradoxes. Since the issues raised by these paradoxes are, in Hughes' words, "profound, complicated and ramified," I can do little more in a short review than scratch the surface. Accordingly, I will be content to describe Buridan's strategy in rough outline and then to focus on a limited, though significant, problem which both Buridan and Hughes have, I think, neglected to face up to. (Unless the context indicates otherwise, I will be using the term 'proposition' in the way that Buridan uses the Latin 'propositio', viz., to refer to contingently existing sentence-tokens.1)

Almost all the gifted 20th century philosophers who have thought deeply about problematic propositions (sophisms) such as

    (A) (A) is false
    (B) (C) is false


    (C) (B) is true

/78/ have held that they are neither true nor false. That is to say, these philosophers have accepted the prima facie compelling claim that such sophisms are true or false only if they are both true and false. Buridan demurs. (A), (B), (C) and their ilk are, he contends, one and all false and not true.2

To buttress this contention, he must refute the seemingly powerful arguments for the claim that if the sophisms in question are false, then they are true as well. I will look at two such arguments, each centering about (A).

Consider, first, the following chain of reasoning purporting to take us from (A)'s falsity to its truth:

    (1) (A) is false. (assumption)
    (2) If (A) is false, then there is something for which (A)'s subject-term ('(A)') and its predicate-term ('false') both supposit (or, on a Fregean account: then the thing denoted by (A)'s subject-term satisfies the concept expressed by its predicate-term).3 (premise)
    (3) So there is something for which (A)'s subject-term and predicate-term both supposit. (1,2)
    (4) But a (singular) proposition is true if and only if there is something for which its subject-term and predicate-term both supposit. (premise)
    (5) So (A) is true. (3,4)

Buridan retorts that, contrary to appearances as well as to the common opinion of philosophers, (4) is false. The reason is that its right-hand side (or the Fregean equivalent thereof) embodies only a necessary and not a sufficient condition for the truth of a singular proposition.

This claim, needless to say, cries out for elaboration and defense. Let 'N is P' be a schema representing singular propositions, with 'N' and 'P' serving to represent the subject- and predicate-terms, respectively. To oversimplify just a bit, on Buridan's showing a complete account of the semantic truth conditions of singular propositions will look like this:

    (T) 'N is P' is true if and only if

      (i) there is something for which both 'N' and 'P' supposit in the proposition 'N is P'; and
      (ii) '"N is P" is true' is true.

('"N is P" is true' constitutes what Hughes calls the 'implied proposition'.4)

Ordinarily, conditions (i) and (ii) are both satisfied if either is. For instance, given that the propositions 'Socrates is sitting' and '"Socrates is sitting" is true' both exist, the former is true if and only if the latter is also true. That is why it is so easy for us to slip into thinking that condition (i) is sufficient by itself. According to Buridan, however, it is precisely in the case of certain self-referential propositions (e.g., (A)) /79/ that condition (i) is satisfied without condition (ii) also being satisfied. For even though there is something, viz. (A) itself, for which '(A)' and 'false' both supposit, the implied proposition, viz. '"(A) is false" is true', is nonetheless false--as is shown, Buridan asserts, by the standard arguments for (A)'s falsity. (Hughes presents two such arguments on p. 24.)

Now this appeal to the standard arguments for (A)'s falsity has all the appearances of being question-begging. After all, the champion of the argument expressed by (1)-(5) will reject the standard arguments for (A)'s falsity precisely because he has what he takes to be a sound argument showing that (A) is false only if it is also true. That is, the argument captured by (1)-(5) is meant to be at least an indirect response to the standard arguments for (A)'s falsity. So Buridan's invocation of the latter to undermine the former seems clearly to be dialectically improper. Hughes, however, is evidently not sensitive to this particular criticism of Buridan. (See pp. 24-25 for the relevant discussion.)

The very same criticism applies to Buridan's treatment of the second argument from (A)'s falsity to its truth. This argument is found in the Eleventh Sophism (see p. 89) and is almost identical to the argument which Hughes considers on pp. 25-27. To understand this argument we must have a decent grasp of what Hughes calls the 'principle of truth-entailment'. I will try to state it as clearly and accurately as I can. Let S be a schematic letter which takes propositions as substitutions, and let [S] represent a proper name of the proposition substituted for S. Then, according to Buridan, the following is true for any such substitution:

    (PTE) Necessarily, if S and [the proposition] [S] exists, then [the proposition] [S] is true.

So, for instance, it follows from (PTE) that if Socrates is sitting and the proposition 'Socrates is sitting' exists, then the proposition 'Socrates is sitting' is true.

(Since on Buridan's view propositions are contingently existing sentence-tokens, it is entirely possible that the first conjunct of the antecedent of a substitution instance of (PTE) should be true without the second conjunct being true. Suppose, for instance, that Socrates were sitting but that no propositions existed. In that case Socrates would be sitting even though the proposition 'Socrates is sitting' would not exist and hence would not be true. Indeed, some propositions are such that it is impossible for them to satisfy the antecedent of (PTE). Consider the proposition 'There are no negative propositions'. It is impossible that there should be no negative propositions and yet that the proposition 'There are no negative propositions' should exist. In such a case, the relevant substitution instance of (PTE) is true by virtue of having an impossible antecedent.)

(PTE), I think we can agree, has an aura of truth about it. But now consider the following argument:

    (6) (A) is false. (assumption)
    (7) If (A) is false, then (A) exists. (premise) /80/
    (8) So (A) exists. (6,7)
    (9) But necessarily, if (A) is false and (A) exists, then (A) is true. (PTE)
    (10) So (A) is true. (6,8,9)

Buridan responds here by rejecting the claim that (10) follows from (6), (8) and (9), on the grounds that far from being a true and innocuous non-self-referential statement about the sophism (as is (6)), the first conjunct of the antecedent of (9) in fact just is the sophism--and the sophism, of course, is false and not true. Hughes gives a more complicated, disjunctive response to the argument discussed on pp. 25-27, a response calculated to work regardless of whether or not the first conjunct of the antecedent of (9) is taken to be the sophism. But, as before, both Buridan and Hughes presuppose (i) that the sophism is false and (ii) that this claim about the sophism can justifiably be used to undermine the attempt to derive the sophism's truth from its falsity. And, again as before, neither Buridan nor Hughes seems to exhibit any qualms about the dialectical propriety of this tactic.

How might Buridan and his followers respond here? Perhaps they could plausibly claim that they are not obliged to provide a non-question-begging answer to the two arguments laid out above. More concretely, they might insist that because their proposal for handling the sophisms in question stands alone in preserving the principle of bivalence, it automatically wins out over its competitors as long as it can be shown to be merely consistent. So, they might continue, the objection pressed above rests on a mistaken picture of the dialectical milieu in which the debate between Buridan and Hughes, on the one hand, and their opponents, on the other hand, is taking place.

I am willing to concede that this line of response shows some promise, though it obviously raises further questions that must be addressed forthrightly. For instance, those who, unlike Buridan, distinguish sharply between sentence-tokens and propositions might well hesitate to attribute such overriding significance to the preservation of sentential (as opposed to propositional) bivalence. Indeed, one who reflects upon the matter carefully might find it intuitively more evident that neither (B) nor (C) above expresses a proposition than that bivalence holds for all syntactically well-formed sentence-tokens. It would be interesting to have Hughes' thoughts on this matter.

In any case, it is fitting that Buridan, properly packaged and interpreted, should be able to contribute, in propria persona as it were, to the lively contemporary discussion of self-reference. Hughes tells us that his main concern has been "to make Buridan's ideas accessible to present day philosophical readers for the sake of their inherent importance." In this he has succeeded admirably.


1. To be exact, Buridan takes a proposition to be a meaningful sentence-token that is spoken or written or (in the case of the mental language) thought with assertive intent.

2. Philosophers who hold that the sophisms in question are neither true nor false /81/ fall into two broad classes. Included in the first class are those who deny that the sophisms are (or express) propositions and hence deny that they have truth-values at all. The second class comprises those who affirm that the sophisms are propositional and thus that they have truth-values, but who deny that they have classical truth-values. The latter group of philosophers (at least) must, of course, still contend with the so-called 'strengthened liar', e.g.,

    (D) (D) is either false or neither true nor false.

(D), it seems, is true if false, false if true, and true if neither true nor false. On Buridan's theory, by contrast, (D) is simply false, and so it poses no new problems not already posed by (A), (B) and (C).

3. I will not bother to restate (3), (4) or (T) below in Fregean terms. But it is important to see that Buridan's resolution of the alethic paradoxes in no way depends upon his preference for a two-name account of predication over a function-argument account.

4. Condition (ii) could also be stated as follows: There is something for which both '"N is P"' and 'true' supposit in the proposition '"N is P" is true'.