Tuesday, 23 October 2012

The best medieval solution to the Liar ever

It's been much too long since I last posted here at M-Phi! (I've been unusually busy with all kinds of things.) What follows here is still not a proper post: it is in fact a review of Stephen Read's edition and translation of Bradwardine's treatise on insolubles, which I just wrote for Speculum. But I figured that it may be of general interest -- after all, any M-Phi'er worthy of the title should be familiar with Bradwardine on the Liar.

Thomas Bradwardine (first half of the 14th century) is well known for his decisive contributions to physics (he was one of the founders of the Merton School of Calculators) as well as for his theological work, in particular his defense of Augustinianism in De Causa Dei. He also led an eventful life, accompanying Edward III to the battlefield as his confessor, and dying of the Black Death in 1349 one week after a hasty return to England to take up his new appointment as the Archbishop of Canterbury.

What is thus far less well known about Bradwardine is that, prior to these adventures, in the early to mid-1320s, he worked extensively on logical topics. In this period, he composed his logical tour de force: his treatise on insolubles. Insolubles were logical puzzles to which Latin medieval authors devoted a considerable amount of attention (Spade & Read 2009). What is special about insolubles is that they often involve some kind of self-reference or self-reflection. The paradigmatic insoluble is what is now known (not a term used by the medieval authors themselves) as the liar paradox: ‘This sentence is not true’. If it is true, then it is not true; but if it is not true, then what it says about itself is correct, namely that it is not true, and thus it is true after all. Hence, we are forced to conclude that the sentence is both true and false, which violates the principle of bivalence. It is interesting to note that, in the hands of Tarski, Kripke and other towering figures, the liar and similar paradoxes re-emerged in the 20th century as one of the main topics within philosophy of logic and philosophical logic, and remain to this day a much discussed topic.

Bradwardine’s De insolubilibus has been recently given its first critical edition, accompanied by an English translation and an extensive introduction, by Stephen Read. One cannot overestimate the importance of the publication of this volume for the study of the history of logic as a whole; prior to this edition, Bradwardine’s text was available in print only in an unreliable edition by M.L. Roure in 1970. Moreover, Bradwardine’s treatise is arguably the most important medieval treatise on the topic. So far, the general philosophical audience is mostly familiar with John Buridan's approach to insolubles; the relevant passages from chapter 8 of his Sophismata have received multiple English translations and been extensively discussed. But Buridan's text pales in comparison to Bradwardine’s treatise; Bradwardine not only offers a detailed account and refutation of previously held positions (chapters 2 to 5), but he also presents his own novel, revolutionary solution (chapters 6 to 12).

The backbone of Bradwardine’s solution is the idea that sentences typically signify several things, not only their most apparent signification. In particular, they signify everything they entail. Moreover, Bradwardine postulates that, for a sentence to be true, everything it signifies must be the case; in other words, he associates the notion of truth to universal quantification over what a sentence says. Accordingly, a sentence is false if at least one of the things it signifies is not the case (existential quantification). He then goes on to prove that insoluble sentences say of themselves not only that they are not true, but also that they are true. Hence, such sentences say two contradictory things, which can never both obtain; so at least one of them is not the case, and thus such sentences are simply false.

Unlike Buridan, who merely postulates without further argumentation that every sentence implies that it is true, Bradwardine makes no such assumption, and instead proves (through a rather subtle argument, reconstructed in section 5 of Read’s introduction) that specific sentences, namely insolubles, say of themselves that they are true. In this sense, Bradwardine’s analysis can rightly be said to be more sophisticated and compelling than Buridan’s.

Bradwardine’s solution to insolubles is not only of interest to the historian of logic, and indeed Read and others have written extensively on its significance for contemporary debates on paradoxes of self-reference. In fact, a whole volume was published on the philosophical significance of Bradwardine’s analysis (Rahman et al. 2008). According to Read, the Bradwardinian framework allows for the treatment of a wide range of paradoxes as well as for the development of a conceptually motivated, paradox-resistant theory of truth in terms of quantification over what a sentence says. Alas, the latter project was not to succeed, for the following reason. As pointed out by Read himself in his critique of Buridan (Read 2002), a theory that says that every sentence signifies (implies) its own truth cannot offer an effective definition of truth, as every sentence becomes what is known as a truth-teller: one necessary condition for its truth is that be true (as it is one of the things it says), ensuing a fatal form of circularity. Now, as it turns out, while Bradwardine does not postulate that every sentence signifies its own truth, this does follow as a corollary from his general principles (Dutilh Novaes 2011). Thus, Read’s own criticism against Buridan’s approach applies to Bradwardine as well. This does not affect the Bradwardine/Read solutions to the paradoxes because all of them (paradoxes) come out as false, but ultimately Bradwardine cannot deliver a satisfactory theory of truth.

However, this observation should in no way be construed as a criticism of Read’s work in general and of his edition and translation of Bradwardine’s treatise in particular. It is indeed the job of a reviewer to spot shortcomings in a volume, even if only minor ones, but this reviewer failed miserably at this endeavor. Read’s volume is an absolutely exemplary combination of historical and textual rigor (for the edition and translation of the text) with philosophical insight into the conceptual intricacies of the material; it is both accessible and sophisticated. As such, it is to be emphatically recommended to anyone interested in the history of logic as well as in modern discussions on paradoxes and self-reference.


  1. Interesting! Can you tell us a few of your favorite examples of the 'insolubles' Bradwardine studied, other than the Liar? And by the way, what did he call the Liar?

    1. Hi John, I left the book in my office, so a proper reply will have to wait until tomorrow, ok? But the Liar was just one insoluble among others for the medievals, albeit the one most extensively discussed by them; it did not have a special name. In fact, I wonder when the Liar was first referred to as the 'Liar'. I think Russell used the exact term, and there is also a German book called Der Lugner. Theorie, Geschichte und Auflosung (by A. Rustow, 1910). Coming to think of it, it was probably Russell, but I'd have to look it up.

    2. Hi again. So some of the other examples discussed by Bradwardine are what we now refer to as epistemic paradoxes, such as 'This sentence is not known by you' (very much like the knower paradox). He discusses those in chap. 9 of the treatise.

  2. I'll have to go read Dutilh Novaes 2011: it's not obvious to me what exactly the problem about being a truth-teller is. For your account of Read, I'd need to know what the conditions on giving a definition of truth are. For your account of your objection, I'd need to know how "a fatal form of circularity" is generated. It looks like being a truth-teller is "light" and so might not burden any story with enough substance to generate problems; might be light enough that it's just automatically satisfied if everything else is.

    1. Unfortunately, the review is supposed to be around 900 words (it's already longer than that), so I can't really spell out the details in it. My goal has been to entice people's curiosity so that they go look for themselves :) (Seems to have worked with you...)
      The argument is on pp 68-71 of my 2011 article. The main point is the following:
      "But it [the proposed definition of truth] can never show in a definitive way that a sentence is true, since one of the necessary conditions for the truth of a sentence is precisely that it is true: the definition corresponds to a looping, non-terminating procedure." In other words, you end up with the same thing on the definiens AND the definiendum side of the definition (to be sure, more stuff in the definiens side).

  3. Hi Catarina,

    Thanks for adding a bit of activity to M-Phi's recent, hopefully temporary, dormancy! (I'm so busy with the new job.)

    Something I'm not sure about - are Bradwardine's truth bearers sentences or propositions?


    1. Hi Jeff! I can imagine how busy you must be at your new job, constantly changing from togas to gowns and having copious dinners at colleges.

      But anyway, the issue of what truth bearers are for Bradwardine and other medieval authors is complicated. It's certainly not propositions (in the modern sense), but there is still the debate between types and tokens. Stephen and I don't entirely agree on this one; I think Bradwardine individuates sentential meaning in terms of tokens, while Stephen thinks it is in terms of types. My main argument is that the very core of Bradwardine's shield against revenge presupposes tokens: Socrates' assertion 'Socrates says something false' (and nothing else) says of itself that it is true (as per the argument referred to but not explained above), whereas my assertion 'Socrates says something false' does no such thing, and thus it is not an insoluble (paradoxical). I discuss these matters in my 2009 PQ paper, 'Lessons on sentential meaning...'

  4. No togas; but some amusing dinners, yes.

    So not propositions in the modern sense, but either sentence types or sentence tokens? In your example, the types (i.e., the string "S"^"o"^"c"...^"l"^"s"^"e") of the two speech acts are identical (because linguistic strings are individuated by syntax alone) whereas the context has changed. That truth for sentence types is context dependent is the standard view: the semantic primitive is "string S expresses (in L) in context C the proposition A".
    In this case, if we call the sentence S, then since the context is different, S is self-referential in the first context, but S is not self-referential in the second. I guess that "S expresses (In L) in context C the proposition A" is something like "S signifies the proposition A".

    So, whether this is a viable approach to the liar paradox will turn on the properties of this semantic primitive of "signifying" or "expressing". It seems likely to me that a revenge issue may arise because, possibly, one can show that no sufficiently rich interpreted language L can contain its own "signifies" predicate.


    1. This is a complicated bit of the theory (just as the bit about types/tokens): what are the 'things' that a sentence signifies? At some point Stephen seemed to suggest that these should be sentences too, but I argued against that in my 2009 PQ paper. it has to be something else, perhaps close to the modern notion of propositions.

      Here's a possible revenge sentence: "This sentence signifies other than is the case'. If it signifies as is the case, then it doesn't; if it does not, then it does. So while the truth predicate seems to be brought under control by means of the quantificational definition, now all the paradoxical burden can be transfered to the notion of signifying.

  5. Hi Catarina,

    What I'm guessing is that the theory has the following axioms

    (1) $T(x) \leftrightarrow \Pi p(Sig(x, \langle p \rangle) \rightarrow p)$.
    (2) $Sig(\ulcorner \phi \urcorner, \langle \phi \rangle)$.
    (3) $\Sigma p(p \leftrightarrow \phi)$.

    where $\Pi$ and $\Sigma$ are substitutional/second-order quantifiers (for sentence variables), $Sig(x,y)$ is the primitive "signifies" predicate, and (3) is comprehension for sentence variables.
    Then I think we can prove:

    (T-Out) $T(\ulcorner \phi \urcorner) \rightarrow \phi)$.

    But we can't prove (T-In). To prove that we should need,

    (4) $Sig(x, \langle \phi \rangle) \wedge Sig(x, \langle \theta \rangle) \rightarrow (\phi \leftrightarrow \theta)$.

    But (4) requires that all things a sentence signifies be materially equivalent, which is not so for the sentence which says of itself it is not true.

    So, the theory avoids inconsistency by avoiding (4).

    Does that sound right?


    1. Yes, that's exactly right, the theory does not have T-In as an axiom: from Phi you don't get T('Phi'). I discuss this in more detail in my paper in the Rahman et al. volume, called 'Tarski's hidden theory of meaning'.