Gillian Russell’s new book Barriers to Entailment (hereafter BE) takes on the ambitious project of formulating and proving five barriers of entailment in a unified framework. Each barrier states that a certain class of sentences does not follow from another class of sentences:The particular/universal barrier: no universal claims from particular ones.The past/future barrier: no claims about the future from those about the past.The is/must barrier: no claims about how things must be from those about how things are.The indexical barrier: no indexical claims from nonindexical ones.The is/ought barrier: no normative claims from descriptive ones. The last barrier is known as Hume’s law, a controversial thesis in logic and metaethics. BE makes a compelling case for studying these barriers together and developing a general account of them. It begins with an extensive survey of formal and informal counterexamples to Hume’s law found in the literature and convincingly demonstrates that they can be reconstructed to challenge the other barriers (chap. 1).The primary technical feature of this book is Russell’s model-theoretic approach to sentence classification for each barrier, developed and expanded from her previous work (Restall and Russell 2010). This original approach allows a systematic and generalizable sentence classification method, which is an important novel alternative to syntactic or lexical methods that classify sentences based on the mere presence or absence of particular logical operators or linguistic items. To illustrate the central idea of the model-theoretic approach, consider a pair of particular and universal sentences, Fa and ∀xFx, in first-order logic (FOL). Suppose that both Fa and ∀xFx are true in a certain model. When adding a new object, such as b where ¬Fb, in the domain of the model, the truth value of Fa does not change, while that of ∀xFx does. Thus, the truth value of a particular sentence is not sensitive to model-extension, whereas that of a universal sentence is. This method is easily applicable to other classes of sentences with respect to different types of changes in models. Roughly, the truth value of a future sentence is sensitive to changing what will hold in the future in a model of tense logic, but that of a past sentence is not. Unlike contingent sentences, the truth value of □P would change when adding a ¬P-world to a model. Indexical sentences are sensitive to context-change. Unlike descriptive sentences, normative sentences are sensitive to norm-shifting. Each kind of change in models is represented as a binary relation R over the set of admissible models.Russell introduces two model-theoretic concepts, fragility and breakability, that capture a sentence’s sensitivity with respect to R in two different ways (chap. 2). So there are two ways of classifying sentences. Using fragility, for example, FOL-sentences are classified as universal if they are fragile with respect to model-extension and as particular if they are antifragile with respect to model-extension. We get a breakability-based taxonomy for particular/universal sentences using breakability in the place of fragility. One important difference between the two is that a breakability-based taxonomy classifies sentences in two kinds, R-breakable and R-unbreakable sentences, while a fragility-based taxonomy is not dichotomous. For example, a mixed sentence like ∀xGx∨Fa is neither particular nor universal (neither model-extension fragile nor model-extension antifragile) in the fragility-based taxonomy for particular/universal sentences.Russell formulates and proves the first three barriers using respective fragility-based taxonomies and then generalizes them to the following theorem.The GBT states that an argument is invalid when the set of its premises (premise set) is of one class and the set of its premises and the conclusion (argument set) is of another class. Note that the class of the conclusion itself is not at issue here. As Russell herself admits, this is not a naturally expected form of a barrier (78), but this particular formulation is adopted to deal with some of the prominent counterexamples in the literature. Take the particular/universal version of Prior’s dilemma (1960) as an example:(1) Fa⊧Fa∨∀xGx(2) Fa∨∀xGx, ¬Fa⊧∀xGxIf the particular/universal barrier is formulated as the thesis that an argument from particular premises to a universal conclusion is invalid, then one of the above arguments is a counterexample to the barrier regardless of how we classify Fa∨∀xGx. If Fa∨∀xGx is universal, then (1) is a counterexample. If it is particular, then (2) is a counterexample. So either classification would result in disproving the particular/universal barrier. However, Prior’s dilemma does not pose a threat to the particular/universal barrier when it is formulated in terms of a particular premise set and a universal argument set as in the GBT. The argument set {Fa,Fa∨∀xGx} in (1) is particular. The premise set {Fa∨∀xGx,¬Fa} in (2) is not particular. Therefore, neither argument is a counterexample to the particular/universal barrier according to the GBT.However, Russell rejects the GBT as her defense of all five barriers because of its limitations in handling arguments involving modal necessity claims:(3) □P⊧OP(It is necessarily that φ. Therefore, it is obligatory that P.)(4) □P⊧FP(It is necessarily that φ. Therefore, it will be the case that P.)These arguments threaten to be counterexamples to the is/ought and the past/future barriers respectively because intuitively □P is neither an ought-sentence nor a future sentence. Russell puts forward the following tetralemma for the defenders of the past/future barrier (chap. 6.5):1. Accept that □P is a future sentence.2. Accept that FP isn’t a future sentence.3. Accept that □P⊧FP isn’t valid.4. Rework the theorem to allow a limited class of exceptions.Russell argues for option 4 through a process of elimination (chap. 6.5): clearly, option 3 is not desirable, and options 1 and 2 would result in unsatisfactory sentence classifications. Avoiding options 1 and 2 turns out to be somewhat challenging because the fragility-based taxonomy does not allow us to classify □P and FP into different sentence classes (chap. 6.4). Analogous tetralemmas hold for the indexical and is/ought barriers (chaps. 7 and 8). So Russell concludes that the barriers must not only allow exceptions but also be formulated using breakability.Here is Russell’s final general formulation of the barriers:Russell shows that all five barriers are instances of the LBT proved in one big quantified-deontic-modal-temporal-indexical logic (chap. 9).I believe it would have been worth discussing further whether the LBT is a better defense of the five barriers than the GBT. The major issue Russell takes with the GBT is its unsatisfactory taxonomical consequences. On the other hand, the LBT, formulated using breakability, intuitively classifies sentences but allows exceptions. I think proponents of the barriers, particularly those of Hume’s law, have good reason to want both: getting the taxonomy right and formulating a barrier without exceptions. Unfortunately, they cannot have both in Russell’s model-theoretic framework. So there is a dilemma between two options: (1) a barrier theorem without exception using a fragility-based taxonomy which has some unintuitive results in sentence classification and (2) a limited barrier theorem with exceptions using a breakability-based taxonomy which results in more intuitive sentence classification. Russell advocates (2) using the tetralemmas, but it’s not obvious to me that the taxonomical cost is greater than allowing exceptions as Russell assumes. Is it really implausible to classify □φ as a future sentence? And normativism about □φ is not an undefendable position. For example, Amie Thomasson (2020) argues that □φ is normative in that it instructs us what to infer from what. The indexical tetralemma discussed in chapter 7.5 sounds more compelling than others because it seems harder to defend classifying ∀xFx as indexical. Consequently, some might wonder if formulating and proving the barriers in a limited form is the best defense of all five barriers. At least, two horns of the dilemma may have to be weighed more carefully.In the last two chapters, Russell proves informal versions of the barriers by defending them against informal counterexamples, such as arguments involving speech acts, propositional attitude verbs, truth predicates, or thick normative expressions. The key idea of Russell’s informal approach is combinations. Combinations are the different ways the world could have been (different possible worlds) and the different ways our language could have been (different interpretations). This ingenious informal counterpart of a model in a formal system allows Russell to apply her model-theoretic approach to informal counterexamples without formally representing them and to sidestep controversial formal or theoretical issues. For example, regardless of which theory about thick normative terms is accepted, the limited informal is/ought barrier holds because any natural language argument from descriptive premises to a normative conclusion is either invalid or satisfies the unless-clause of the barrier (chap. 11.4).Despite BE’s formal and informal defense of the five barriers in a unified framework, readers might find it unsatisfactory regarding Hume’s law. Consider the following arguments:(3) □P⊧OP(5) P⊧P∨OP1(6) ∀x(Ax→Bx)⊧∀x(Bx→OCx)→∀x(Ax→OCx)2These are just a few examples of valid arguments from descriptive premises to a normative conclusion in quantified deontic modal logic but satisfy the unless-clause of the limited is/ought barrier, Russell’s final formulation of Hume’s law (chap. 8). Therefore, they are exceptions. I think it is natural to wonder why they are exceptions, not counterexamples, to Hume’s law. One person’s modus ponens is another person’s modus tollens. The contraposition of the limited is/ought barrier says: if an argument from descriptive premises to a normative conclusion is valid, then it satisfies the unless-clause. For skeptics, thus, proving the limited is/ought barrier is nothing but identifying a specific model-theoretic property of working counterexamples to Hume’s law. And, in fact, many proposed counterexamples much discussed in the literature share this property. However, I believe it is meaningful progress to shift the focus of the debates on Hume’s law to a critical examination of the model-theoretic aspects of the exception class of the barrier.Overall, BE is an excellent showcase of how careful formal work can shed new light on classic problems and how philosophical questions can be sharpened through formalism. BE also carefully explores classification issues in tense logic (chap. 3), clarifies different versions of the modal barriers (chap. 4) and the is/ought barriers (chaps. 8.1 and 9.7), and develops indexical logic through synthesizing important insights on indexicals in the literature (chap. 7). Despite its innovative formal approach and broad coverage of various logics, anyone familiar with elementary FOL and some modal logic should be able to follow and appreciate Russell’s thorough and rigorous work in BE. BE might leave us wondering how effective it is to prove the limited version of the barriers to defend them since they are free of counterexamples at the cost of allowing exceptions. But because it provides a rigorous recasting of some issues surrounding proposed barriers, including Hume’s law, and promotes the study of barriers to entailment in general, BE is an important contribution.