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.