Logic: Techniques of Formal Reasoning

I used this book in a course I took in logic at the University of Virginia as an elective in the philosophy department while pursuing my degree in electrical engineering. I returned to this book in 1994 when I found myself studying math on my own. I was trying to tackle analysis and not satisfied that the proofs I was studying were sound. This book was a godsend to me. I wanted to see how far it was possible to dispense with any hand waving whatsoever when proving a theorem, and I was deeply distrustful of the typical arguments found in higher math books. Other books on logic seemed to assume you were comfortable with mathematical proof itself and proceeded to show you how to informally prove theorems about formal systems. You can imagine how little use that was to me. This book introduced the techniques of formal logic as a game with simple rules that you had to practice, which is a great way to demystify the more complex styles of argument that are commonplace in higher math. The problems grow very slowly and smoothly in complexity through the book, with no enormous gaps that readers have to fill in on their own (a rarity in math books, I think). If you are a programmer, an exercise I highly recommend is to write a program to validate proofs expressed in the system of the book, as you work your way through it. After spending six months working through this book, I proceeded to Halmos's "Naive Set Theory", another great book, BTW, that paves the way to understanding higher math. I can't imagine a better preparation for Halmos's book than KMM's Techniques of Formal Reasoning.

The first edition of this book was the textbook for my first formal logic course at UCLA in 1967, taught by Donald Kalish. Because of it, I changed my major from Physics to Mathematics with a specialty in Logic and Set Theory. Forty years later, I still pull the book out and do exercises from the later chapters, certain that such mental exercise sharpens my decision-making skills. I highly recommend this book to anyone who wishes to improve their deductive reasoning.

Logic: Techniques of Formal Reasoning, 2nd Edition, by Donald Kalish, Richard Montague, and Gary Mar 1980 (1964) I was originally introduced to the 1964 edition of Kalish and Montague's Logic: Techniques of Formal Reasoning in early 1970. As an undergraduate taking elementary logic for the first time, needless to say I found the demands of sentential and predicate calculus and theorem-proving in general to be daunting and not a little painful. It was many years later after receiving advanced degrees and teaching logic courses myself, along with researching some of the theoretical horizons in artificial intelligence, that I turned back to this most precious of textbooks. Finding that a second edition had been published, I eagerly bought a copy and set out to re-prove all those theorems. Sharpening one's logic skills can be a struggle, but it is one well worth undergoing especially with the demands for reasoned discipline imposed by Kalish, Montague, and Mar. Every so often, I go back to this text to prove the theorems once again (though I occasionally skip over a few in the first three chapters). I've found just a few suggestions I would make to the authors, if they were still around, or to whoever may edit it in the future. These pertain only to the first 5 chapters. The transition from the 125 theorems of the sentential calculus to those of the predicate calculus is a bit rough-going. Almost immediately, one is expected to engage in abbreviated theorem-proving which certainly assumes a command of all those theorems that came before. It would seem that a few more exercises would help students acquire more familiarity with those theorems and with abbreviated proofs. Moreover, one is introduced to more complex inference rules, such as separation of cases, for which few exercises have prepared one, at least up to that point. These may be minor quibbles, but they can cause a lot of confusion, especially to students introduced to logic for the first time. Additionally, well into Chapter III, it is possible to construct a proof of one of the advanced theorems with the use of hypothetical syllogism. In theorem T235 (corresponding to the Aristotelian syllogism Barbara), one can derive two pure hypothetical statements permitting the application of hypothetical syllogism (the law of transitivity) to deduce a third. Yet neither hypothetical syllogism as a specific rule of inference nor the concept of transitivity has been introduced in previous pages. In fact, hypothetical syllogism as such (including explanations of pure and mixed syllogisms) is never introduced, though principles of syllogism are. The law of transitivity is not introduced until late in Chapter V. Of course, one can derive them, but this can cause confusion for a beginner. I highly recommend this text over all others that are commonly used in basic undergraduate or even graduate courses. Though Logic: Techniques of Formal Reasoning is more demanding than, say, any of the Copi bo

With some saddness, I noted the death of Dr Kalish this month. As an undergrad at UCLA during the mid 60's, I was fortunate to take Dr Kalish's class in Symbolic Logic. Over the years, I realized that his course using this textbook was the most valuable class I took, either as an undergrad or a graduate student. The text offers disciplined procedural logic that clarifies thought processes. At various times, I worked as a mathematical programmer in several higher level languages without ever taking a formal course in any of them. I attribute my success in this area to the Techniques of Formal Reasoning.

If taught by the right person, this book will reveal all of the issues of contemporary logic. It is best supplemented on first reading with Schuman's "logic" guide. As Kalish was influenced by Russell, I found the Theorems in the Principia matched perfectly those found in this book. The latest edition of this book is much improved, in my opinion. It offers more guidance to understanding how to solve problems, and offers a great many useful hints and tips. Kalish is/was (?) the Chair of the department of UCLA. He is a modern master and authority on this subject. In my opinion, if you wish to spend time on understanding the strengths of contemporary symbolic logic, there is no better book to buy.