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The Boundary Stones of Thought, by Ian Rumfitt. Oxford The Boundary Stones of Thought,by Ian Rumfitt. Oxford: Oxford University Press, 2015. Pp. xiv + 345.⇤ Peter Fritz Final Draft 1 Introduction In his book The Boundary Stones of Thought, Ian Rumfitt considers five argu- ments in favor of intuitionistic logic over classical logic. Two of these arguments are based on reflections concerning the meaning of statements in general, due to Michael Dummett and John McDowell. The remaining three are more spe- cific, concerning statements about the infinite and the infinitesimal, statements involving vague terms, and statements about sets. Rumfitt is sympathetic to the premises of many of these arguments, and takes some of them to be e↵ective challenges to Bivalence, the following princi- ple: (Bivalence) Each statement is either true or false. However, he argues that counterexamples to Bivalence do not immediately lead to counterexamples to (the Law of) Excluded Middle, and so do not immediately refute classical logic; here, Excluded Middle is taken to be the following principle: (Excluded Middle) For each statement A, A A is true. p _¬ q Much of the book is devoted to developing and assessing the most compelling versions of the five arguments Rumfitt considers. Overall, Rumfitt argues that each of these challenges is ine↵ective against classical logic. As a preliminary to his exploration of the five arguments against classical logic, Rumfitt discusses a general problem for advancing a debate between pro- ponents of competing logics. The problem is this: If in arguing for the logic you favor, you appeal to inferences which you accept but your opponent rejects, you will fail to convince them, even if you succeed in justifying your position. Rumfitt’s solution to this difficulty consists of two steps: The first, which is in- dependent of the particular dispute at hand, is an account of logic in terms of truth; this is developed in Part I of the book. The second step is specific to a given dispute: Consider a particular case which an opponent considers to con- stitute a counterexample to the logic you favor. Develop a theory of the truth of ⇤Forthcoming in Mind.Thefinalpublicationisavailableat https://doi.org/10.1093/mind/fzx016. 1 statements which covers the alleged counterexample (although it need not cover all statements) which you and the opponent agree on. Using the term loosely, call such a theory a semantic theory. Then show, using only inferences accept- able to both parties in the dispute, that this semantic theory, together with the account of logic developed previously, entails that the alleged counterexample to your favored logic is not a genuine one. Assuming that the opponent agrees with the proposed account of logic, this allows a rational discussion between proponents of competing logics to advance. Rumfitt carries out this second step for several disputes in his discussion of the five arguments against classical logic; this is done in Part II of the book. The general methodological proposal just sketched is set out in chapter 1 of the book. In the remainder of this review, I will go through each of following chapters of the book, summarizing and commenting on its main contributions. Rumfitt’s development of the di↵erent arguments against classical logic, the corresponding semantic theories, and their assessment, is complex and subtle. I will therefore comment in more detail on those parts of the book which are general, namely Part I on the nature of logic and a concluding chapter on Bivalence, and less on those chapters of Part II which are specific to particular challenges against classical logic. 2 The nature of logic The terms ‘statement’ and ‘true/false (statement)’ are central to Rumfitt’s dis- cussion of the nature of logic. Since the details will matter, let me quote the passage in which he introduces these technical terms: Let us consider those ordered pairs whose first element is a meaning- ful, indeed disambiguated, declarative type-sentence, and whose sec- ond element is a possible context of utterance; by a possible context of utterance, I mean a determination of all the contextual features which can bear upon the truth or falsity of a declarative utterance. Some of these ordered pairs will be such that, were the declarative type-sentence that is the first element uttered in the context that is the second element, a single complete thought would then be ex- pressed: the resulting utterance would say that such-and-such is the case. As I shall use the term, a statement is an ordered pair that meets this condition. [. ] When a statement expresses the thought that such-and-such is the case—or more briefly, says that such-and- such—the statement is true if and only if such-and-such really is the case, and false if and only if such-and-such is not the case. (pp. 20– 21) A footnote adds that this account of will have to be refined for certain statements expressing multiple thoughts; this becomes important in the final chapter on Bivalence. Rumfitt sets out his views on logic in chapter 2. He argues that in success- ful deductive reasoning, the conclusion does not always follow logically from the premises. His main example concerns electrical circuits; he argues that in contexts in which such circuits are the relevant subject matter, sound deduc- tions may take the laws of electrical circuits for granted, without taking these 2 to be tacit – sometimes called enthymematic – premises. He calls the relations which are operative in the application of deductive capacities ‘implication re- lations’, and argues that they satisfy the familiar constraints on consequence relations often attributed to Tarski (reflexivity, monotonicity and cut) and truth- preservation. Thinking of arguments as pairs of sets of statements and state- ments, all of which share the same context, one can think of implication relations as sets of arguments; the members of such a relation are the arguments which are sound according to it. A thinker’s logical competence is now identified as a capacity to obtain, from arguments which are sound according to a given implication relation, further arguments which are sound according to the same relation. Such a capacity is underwritten by a corresponding law. E.g., a thinker’s capacity to reason dilemmatically is underwritten by the Law of Dilemma, which Rumfitt states as follows, where X and Y are variables for sets of statements, and A, B and C are variables for statements: Whatever implication relation R may be, if X, A stand in R to C, and Y,B stand in R to C,thenX and Y together with any disjunction of A with B also stand in R to C. (p. 54, (2)) What are the logical laws in general? Rumfitt answers this question (p. 67) by stating that they come in two kinds: the structural principles, such as the one which states that every implication relation is monotonic, and the laws for particular logical notions, leaving it somewhat open what exactly the logical notions are. As a by-product of the development so far, one can give an account of the more familiar relation of logical consequence: let this be the set of arguments which are sound according to all implication relations. As Rumfitt notes (p. 56), the situation is reminiscent of the di↵erence between conceiving of (a) logic as constituted by the logical truths and conceiving of it as a relation of logical consequence. Just as in the latter case, the logical truths can be seen as by- product of an underlying relation of logical consequence, so Rumfitt considers the relation of logical consequence to be a by-product of a relation of soundness- preservation. Rumfitt’s discussion is in many ways compelling, and full of careful and subtle argumentation, such as the distinction between deduction and (deductive) inference. Nevertheless, it leaves a number of crucial matters open. One of them concerns the logical laws: Rumfitt says virtually nothing about them apart from their falling into the two categories mentioned above. Another surprising gap in Rumfitt’s discussion of the nature of logic concerns the status of formal languages. Any mathematical study of logics such as classical first-order predicate logic operates not with statements in Rumfitt’s sense but with formulas of a formal language. The details of these syntactic matters are often left somewhat underspecified, but it is usually presupposed that a rigorous syntactic theory could be developed within a standard background theory such as ZFC set theory. The availability of such a mathematical treatment of syntax is important for many theorems about various logics, such as the L¨owenheim- Skolem theorem of classical first-order predicate logic, in which set-theoretic reasoning about infinite cardinalities is applied to the syntax of the language. Rumfitt’s discussion of logic makes no mention of such formulas, being couched solely terms of statements. This is partly obscured by his use of letters A, B, . 3 and notation adopted from sequent calculi, but these letters are clearly intended to stand for statements (although Rumfitt himself slips occasionally into taking them to stand for formulas, as on p. 53). Any connections between Rumfitt’s notions, such as his relation of logical consequence among statements, and familiar notions from the mathematical study of logic, such as the consequence relation of classical first-order predi- cate logic among formulas, are left to the reader to draw. Doing so is far from straightforward. In fact, some of the ways Rumfitt uses logical symbols to talk about statements, which might suggest such a connection, are even hard to un- derstand on their own. For example, Rumfitt uses strings like ‘ A’ or ‘A B’, where ‘A’ and ‘B’ are taken to stand for statements.
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