One of the founding myths of analytic philosophy is that the predicate logic developed in the late nineteenth century was far more powerful than its predecessors. This ambitious book argues that, on the contrary, medieval philosophers developed "a system of logic that is similar to the predicate calculus in richness and power" (4)—or that, as Parsons put it in his presidential address to the American Philosophical Association, "the core of medieval logic is as accurate and as expressive as the core of contemporary logic."
The bulk of the book is devoted to identifying and reconstructing this "core theory" (1). Historians will wonder if any such thing existed across the centuries. It helps to realize that the focus is on fourteenth-century logic in the Latin West, and in particular on influential texts available in English translation. The most cited author by far is John Buridan, followed by his older contemporary, William of Ockham, with support coming mostly from the fourteenth century bookends Walter Burley and Paul of Venice and the mid-thirteenth-century logicians Peter of Spain and William Sherwood.
The first two chapters introduce the scholastic interpretation of Aristotle's logic and use natural deduction to reduce his proof system to the principles of reductio, exposition (existential instantiation), and expository syllogism (existential generalization). The Parsons project begins in earnest in chapter 3, which explores the consequences of three medieval additions to Aristotle's propositional syntax: quantified predicates, negations separated from the copula, and singular predicates. The rules of inference for this expanded system are given using a simple notation that Parsons has devised for encoding propositions into a conveniently manipulable canonical form. This proves to be a warm-up for the development in chapter 4 of "Linguish," a richer notation using grammatical role markers and allowing for verbs other than the copula, which is expanded in chapter 5 to handle other constructions like genitives and relative clauses. These two chapters constitute both the heart of the book and its pons asinorum. Only the hardiest of readers will slog through the sections on truth conditions (§4.5) and validity (§4.6) that pave the way for a completeness proof that "may be skipped without loss of continuity" (§4.7), and yet the semantic technicalities that they introduce are mercilessly relied upon in chapter 5 (and again in chapters 8–10).
One key feature of fourteenth-century logic that underpins the project is the stipulation that scope is determined by word order. For instance, the sentences Omnis homo aliquid amat and Aliquid omnis homo amat were treated as if they were unambiguous, the first saying that everyone has something they love, and the second saying that there is something such that everyone loves it. This artificial regimentation, which was largely confined to logical [End Page 348] writings, allows sentences of Latin to be treated like the formulae of a formal language, and deserves more than just scattered remarks (3, 63n., 86, 246).
The Achilles heel, though, comes in chapter 8, where Parsons expands Linguish to cover the "relatives" (anaphoric expressions) that will serve as bound variables. The trouble is that the logicians who have so far been providing us with the "core theory" had theories of relatives that were not up to scratch. What is worse, so did the late-fourteenth-century logicians who pointed this out ("Though I hope I am wrong," 244). Instead, following a suggestion of Reinhard Hülsen's, Parsons takes "singled supposition" from the standard theory of reflexive pronouns—or rather, from a version of it that he extracts from Marsilius of Inghen and diagnoses via "reverse engineering" in Burley, Ockham, and Buridan (§8.2)—and applies it to relatives in general. It is scant consolation to hear that "there is some evidence that some medieval logicians did exactly this" (244), especially when two of the three are not mentioned elsewhere. Then again, this is an under-researched area and further work might help.
In chapter 9, the resulting system is used to represent first-order logic and first-order arithmetic. It is not done...