Studies and Exercises in Formal Logic — Inside the Classic

Modality and the Import of Propositions

Keynes devotes considerable attention to distinctions of modality, treating assertoric, problematic, and apodeictic propositions as more than mere variations in force. In the excerpts, he contrasts the assertoric interpretation of hypotheticals—where “If A then C” merely negatives the conjunction AC′—with the modal interpretation, which denies the possibility of that conjunction. This dual analysis recurs throughout: for example, the contrapositive of “If patience is a virtue, there are painful virtues” is given as “If there are no painful virtues, patience is not a virtue,” and Keynes notes that the equivalence holds on either interpretation. He also warns that converting “If A then C” to “If C then A” is a fallacy analogous to simply converting a categorical A proposition, calling it “perhaps the most dangerous fallacy to be guarded against in the use of hypotheticals.” The treatment of problematic hypotheticals extends the analogy with categoricals, showing that conversion and contraposition apply mutatis mutandis.

The Interplay of Categorical and Hypothetical Forms

A central tension in Keynes’s exposition is the relation between hypothetical and categorical propositions. He defines a true hypothetical as expressing a relation between two propositions of independent import, not between two terms, and insists that it is not easily reducible to categorical form. Yet he acknowledges that an equivalent categorical can sometimes be obtained, with subject and predicate drawn from the original clauses. This duality is explored through immediate inferences: obversion, conversion, contraposition, and inversion are applied to hypotheticals, often yielding results that mirror categorical logic but with important caveats. For instance, Keynes notes that from “If A then C” one may infer by inversion “It is possible for both A and C not to be true,” but only if the falsity of C is assumed possible. Such theoretical niceties, he admits, have little practical importance but raise problems of considerable theoretical interest.

Exercises and the Pedagogy of Formal Logic

The book’s subtitle promises “Studies and Exercises,” and the preface to the first edition (1884) states that the pages contain “a number of problems worked out in detail and unsolved problems, by means of which the student may test his command over logical processes.” The fourth edition retains this practical orientation while updating the theoretical framework. Keynes employs Euler’s diagrams for illustration, and the excerpts show worked examples of contraposition and conversion with both categorical and hypothetical propositions. The exercises are not mere drills; they are designed to expose common fallacies, such as the illicit conversion of hypotheticals or the inference from “If A then C” to “If A′ then C′.” By embedding exercises within a rigorous theoretical discussion, Keynes aims to cultivate not just technical skill but a deeper understanding of logical structure.

Readers approaching this edition should note that Keynes assumes familiarity with traditional syllogistic logic and uses it as a springboard for more nuanced analysis. The appendices—on fundamental laws of thought and complex propositions—are integral to his project, not afterthoughts. Pay close attention to the footnotes and cross-references, which often qualify or extend the main text. The exercises are best tackled after absorbing the corresponding theoretical sections, as they frequently test the very distinctions Keynes has just drawn.