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Books

Barriers to Entailment by Gillian Russell

Christopher John Searle recommends a study of which moves are allowed in logical arguments.

Gillian Russell is Professor of Philosophy at the Australian National University. In her new book Barriers to Entailment: Hume’s Law and Other Limits on Logical Consequence, she ventures into the intricate terrain of obstacles to logical entailment, particularly focusing on Hume’s Law (‘You can’t derive an ought from an is’). According to Russell, a barrier to entailment exists when it is logically illicit to derive a conclusion of one logical type from premises of a different logical type. Philosophy is replete with such violations, and in this intense and ground-breaking book, Russell makes it her aim to expose the principal felons. Her exploration is not just academically rigorous but also timely, addressing core issues in contemporary philosophical discourse and inviting readers to reconsider fundamental aspects of logic and reasoning. Her conclusions affect every aspect of rational thought, from the foundations of ethics to the limits of metaphysics. This is an extremely important work that ought not be ignored.

Russell’s academic journey, including her PhD in philosophy from Princeton in 2004, has been marked by notable contributions in the philosophy of logic and language, uniquely positioning her to map out this relatively under explored and extremely challenging landscape. This book critically examines a variety of problems, including those contingent on moving between the normative and descriptive statements, particulars and universals, and between past and future propositions.

In her introduction, Russell briefly touches on the work of Charles Pigden, which offers a crucial insight into the limitations and scope of logical inference. Russell situates her discussion within the broader narrative by examining Pigden’s thesis that logic, by its nature, is conservative. This means that, barring the introduction of ideas through addition or conjunction (ie, ‘and X’), a valid inference hasn’t been made if the conclusion introduces expressions not found in the premises. Pigden illustrates this with a memorable example: we cannot infer hedgehog conclusions from non-hedgehog premises. This principle (not the hedgehogs) was known to medieval scholastics as quaternio terminorum, or ‘the fallacy of the fourth term’. Russell refines Pigden’s definition by stating that “a predicate or propositional variable cannot occur non-vacuously in the conclusion of a valid inference unless it also appears among the premises” (p.4). However, Pigden’s stance causes some problems in logic, notably how we can admit logical operators into conclusions that are absent from the premises. For example, in the logical equivalence of double negation, p = ¬¬p; but ¬ (‘not’) is not in the original proposition p. Pigden resolves this by allowing logical operators in first-order logic (such as ‘not’, ‘and’, or ‘or’) as the only exceptions to the conservative rule.

Russell, however, thinks Pigden’s conservatism is overly restrictive, and opens up a field of objections to it. Her overarching aim in Barriers to Entailment is to address these objections without confining entailment barriers to first-order logic. However, for logical purists like Pigden, who adhere to W.V.O. Quine’s view that first-order logic is the only true logic, Russell’s broader approach might seem irrelevant.

The significance of the principle of the conservativeness of logic for philosophy cannot be overstated, though it is often overlooked, by philosophers and logicians alike. Violations of this principle are arguably at the root of much metaphysical confusion. Recognising and adhering to this principle as a fundamental rule of inference, coupled with our standard definitions of validity, could prevent a myriad of logical errors and philosophical missteps, making it a pivotal concept in the discipline.

Barriers to Entailment is divided into three main parts, each focusing on different aspects of the barriers. Part I, ‘Getting Started’, begins with an in-depth survey of both formal and informal counterexamples to logical entailment. This section thoroughly examines first-order logic and its implications for the scope of entailment barriers, including a detailed discussion on model extension and the taxonomy of particular and universal sentences. It then explores the concept of time in logic, discussing Hume’s other law (‘All inferences from experience presuppose that the future will resemble the past’), and delving into the complexities of past and future sentences in logical arguments. This section includes a discussion of the General Barrier Theorem – “a thesis about invalidity that is both general and true” – before moving on to a detailed exposition of entailment barriers in modal (or probability) logic.

“...we cannot infer hedgehog conclusions from non-hedgehog premises.”
hedgehog © George Chernilevsky 2020 Public Domain

Logic Puzzles

Russell’s survey dissects various counterexamples to Hume’s Law in the literature that challenge the presumed unidirectionality of barriers such as the is/ought divide (can you get an is from an ought?). She meticulously examines ‘contraposition strategies’, which involve flipping the premise and conclusion while changing their truth values, to reveal potential crossings between normative (ought) and descriptive (is) realms through supposed logical manipulations with negation and modal operators. For example, if we assume the Kantian principle that ought implies can, then from the conditional statement “If Ali ought to donate blood, then Ali can donate blood”, we can derive the contrapositive: “If it is not the case that Ali can donate blood, then it is not the case that Ali ought to donate blood” (pp.33-34). This demonstrates how the is/ought gap is vulnerable to contraposition: if normative statements imply descriptive ones, then through logical equivalence, descriptive statements can imply normative ones. Russell resolves this by questioning whether the normative status of propositions is preserved under negation and by classifying statements on their behaviour across models rather than relying on syntax alone. The discussion extends to the contentious debate concerning change of subject matter, over whether valid inferences permits a shift from one topical domain (for example, the factual) to another (for example, the ethical or normative) – which debate lies at the heart of the is/ought barrier. Russell critiques the notion that employing non-classical logic (where contradictions are allowed, for instance) to prevent such shifts in subject matter, could simplify entailment barriers. She argues that this approach fails to capture the complexity of logical entailment. Instead she advocates a nuanced understanding that encompasses logical subtleties without diluting the classical form of logical reasoning.

According to Russell, a sentence is ‘fragile’ if it is true in a particular model but there exists at least one extension to the model where changing some condition makes it false. For example, “All objects in this set are red” is fragile because adding a non-red object to the set (extending the model) would falsify the sentence. Anti-fragility, in contrast, is when the premises of an argument are true in a certain model, and they remain true in all models that are related to the original by a certain relation R. Anti-fragility, in contrast, is when the premises of an argument are true in a certain model, and they remain true in all models that are related to the original by a certain relation R. For example, “This object is red” is anti-fragile because adding new objects to the model does not affect the truth of the sentence about this specific object. This concept is crucial for understanding the stability and robustness of argument premises under different but related conditions, and it plays a significant role in determining whether a set of premises can legitimately entail a particular conclusion. In Russell’s terminology, a set of sentences or premises is said to be satisfiable if there exists at least one use where they are all true together. These definitions together allow us to state Russell’s General Barrier Theorem: If a set of sentences is both satisfiable and antifragile in respect to some relation, then that set cannot entail a conclusion unless the union of the set with the conclusion is itself antifragile. In simpler terms, if we start with a set of sentences that are both true and stable, we can only reach a conclusion that keeps this set stable. If adding a new sentence to the original group makes the whole set unstable, then the conclusion does not logically follow from the premises. Russell applies this theorem to the premises of generalised arguments to show that the putative conclusions do not follow. For instance: we can’t deduce an ought from an is; we can’t infer universal conclusions from particular premises (‘This cow is brown, so cows are brown’); we can’t infer future conclusions from past premises (‘It rained yesterday, so it will rain tomorrow’).

Part II, ‘Getting Complex’, deepens the discussion by examining the intricate interplay between modal (probability) logic and tense, as well as the complexities of context-sensitivity in logical reasoning. This part covers various logics, such as tense-modal logic and deontic-alethic modal logic, exploring complex philosophical principles like ‘ought implies can’ and ‘will implies can’. It also addresses the challenges of context-shifting in logic and how this gives rise to various logical dilemmas.

The final part of the book, ‘Getting Informal’, shifts the focus back to the realm of informal logic. This section emphasises the practical implications of Russell’s theories, exploring how they apply in lower order and classical logics, and in philosophical discourse more generally. It covers topics such as domain extension, norm-switching, and the implications of Hume’s Law to informal contexts. The section concludes with a comprehensive analysis of informal counterexamples, bringing the discussion back to more surface-level logical considerations.

“In every system of morality, which I have hitherto met with, I have always remark’d, that the author proceeds for some time in the ordinary way of reasoning, […] when of a sudden I am surpriz’d to find, that instead of the usual copulations of propositions, is, and is not, I meet with no proposition that is not connected with an ought, or an ought not.” David Hume, Treatise of Human Nature (1740)

Logical Landmarks

This book’s proof of the Strong General Barrier Theorem is a landmark achievement in twenty-first century philosophy. Not since Ludwig Wittgenstein’s Tractatus (1921) has such an important contribution been made to philosophical logic. Russell has shown the limits of deductive inference, which means, the limits to the very expression of thought. The proof she presents is integral to understanding how and why logical inferences can or cannot be legitimately made based on the properties of the premises involved. If ever there was a litmus test for philosophers to gauge the validity of their arguments, this ought to be it. My only hope is that this work does not go unnoticed.

However, whilst Russell’s treatment of such sophisticated topics is highly commendable, the book also presents certain challenges. The highly nuanced nature of technical concepts such as R-anti-fragility and satisfiability, though well-explained, might pose difficulties for readers not well versed in higher-order logic. Contrary to the claim made in the blurb that the work is “suitable for any reader with some background in first-order model theory (of the kind that might be taught in a first class in logic)”, the book is heavily reliant on a logical formalism that even some postgraduate philosophy students might struggle with. This is a shame, as the importance of the work could be missed by a general philosophical audience, who would almost certainly benefit from it. And while Russell does an admirable job of clarifying her terms, the book could also benefit from more straightforward examples and practical applications. This would help to bridge the gap between theoretical exposition and philosophical practice, making the work more accessible to a broader readership. Maybe Russell ought to write a popular version for a general audience. Or perhaps we will see the thrust of Barriers to Entailment condensed into a short, punchy paper, with clear and simple examples, that will rock the establishment.

Overall, Russell’s work in Barriers to Entailment stands as a testament to her scholarly rigor and her innovative approach to philosophical logic. The book not only advances the discourse on logical entailment, but also encourages deeper engagement with the underlying principles governing logical reasoning.

Perhaps the most exciting, but unstated, corollary of Russell’s proof is the implications it may have for certain projects in metaphysics, ethics, and the philosophy of mind. Given the General Barrier Theorem, it is clear that from empirical premises nothing metaphysical can be legitimately inferred, and vice versa. It is incumbent on the metaphysical realist to demonstrate from what premises their metaphysical conclusions are drawn. For they cannot be empirical; and if they’re metaphysical, how were they established? Unfortunately, Russell barely alludes to this in her book.

In terms of normative ethics, the book conclusively proves that Hume was right: you cannot derive an ought from an is, and that debate should now be closed.

Barriers to Entailment stands out for its rigorous analysis and innovative approach to understanding the conditions for and limitations of logical reasoning. In this, it significantly contributes to contemporary philosophical logic in the analytic tradition, and also provides a clear framework for future research.

Christopher John Searle is a postgraduate research student taking a PhD in Philosophy at the Open University.

• Barriers to Entailment: Hume’s Law and other limits on logical consequence, Gillian Russell, OUP, 2023, 320pages, £60.00 hb, ISBN-13: 978-0192874733