Logical Methods

This book is intended as an introduction to logic and mathematical reasoning, with a view to applications in pure mathematics, analytical philosophy, and computer science, particularly in the current context where we are witnessing the emergence and rapid development of technologies with semantic features.

 

In terms of form, the philosophy behind this book is as follows: a profound understanding of a subject is not acquired all at once, because discerning deep structures and making sense of them takes time, as does abstraction and generalization. Therefore, in the hope of facilitating the intellectual digestion of the material covered, it is presented in stages. Chapters presenting some of the basic principles of set theory are intertwined with those introducing propositional logic and the different types of proof, in a coherent and rational order. While the choice of topics covered in the first two-thirds of the 24 short chapters that make up this book is standard for a work of this type, some of the topics covered in the last third point to the fact that it was developed from material used by the author as part of an introductory course in logical methods offered to students enrolled in an undergraduate program in a computer science department. For instance, elementary combinatorial techniques are covered before introducing certain notions of abstract algebra and then presenting the rudiments of graph theory. After this interlude (which will likely be somewhat less appropriate in the context of a course offered exclusively to undergraduate students in pure mathematics who will no doubt cover this content in a course entirely dedicated to discrete mathematics), we make a final return to logic proper to deal with formal languages, deterministic finite automata, and natural deduction.

 

The author maintains in the preface that he has sought to plot a course midway between a bottom-up approach (i.e., by proceeding from the axioms towards the more complex structures by successive operations) and a top-down approach (i.e., by mobilizing one’s intuition and the study of examples to progress incrementally from the concrete to the abstract). Ultimately, the approach used by the author is closer to the latter. One would search in vain for theorems, corollaries, and lemmas in the 300 pages that make up this book, the author having deemed it preferable to adopt an informal and refined style, where it is the definitions–and not the assertions and their demonstrations—that constitute the framework of this work. Each of the 137 definitions presented is highlighted, both in terms of content and form. To promote understanding and clear up misunderstandings, most of the definitions are followed by examples to help explain their meaning, along with counterexamples or non-examples, to point out, through the use of contrast, what they do not mean. Therefore, very little basic knowledge is required for this introduction to logical methods produced by Roger Antonsen, which is written in an accessible style that aims to offer an overview without skimping on the details. There are nearly 60 exercises embedded in the text, some of which are accompanied by a sketch of a solution.

 

However, none of the approximately 350 problems presented at the end of the various chapters is accompanied by a solution or an answer. Of note, the material is also enriched by 38 boxes titled “Digression.” This supplementary material provides an eloquent overview of topics that could very well be part of an introductory course in logical methods intended for an audience composed entirely of undergraduate mathematics students (e.g., Russell’s paradox, the continuum hypothesis, etc.). Some aim to inspire the reader to explore the subject in more depth beyond an introductory course (e.g., the P vs. NP problem, modal logic, second-order logic). Finally, others mainly aim to introduce interesting elements outside the regular curriculum (e.g., Lucas numbers, Bell numbers, Graham’s number, the Four color theorem).

 

In the past, Roger Antonsen has demonstrated his vibrant interest in the visual representation of mathematics by collaborating on the book Illustrating Mathematics. The same sensibility is expressed in this book. Indeed, contained in the book are several hundred small figures; arrow, Venn, and Hasse diagrams; and simplified visual representations intended to illustrate and arouse the reader’s intuition around certain ideas or abstract concepts that are more difficult for the mind to grasp (e.g., the reflexive, symmetric, or transitive closure of a relation). The author has also elected to use color to draw the reader’s attention to certain subtleties worthy of attention. This strategy, used wisely, does not encumber the presentation unnecessarily, and the desired effect is obtained.  Interestingly, the author has assembled, in a section just before the index and symbols index titled The Road Ahead, about 30 suggestions for further reading. Grouped by type (e.g., the Classic, introductory books on logic, popular science & recreational mathematics, etc.), each title is accompanied by a brief description of what makes them unique or appealing.

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Frederic Morneau-Guerin is a professor in the Department of Education at Universite TELUQ. He holds a Ph.D. in abstract harmonic analysis.