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# Logic and its Limits by Patrick Shaw

### Edward Ingram enjoys a surprisingly lively introduction to logic by Patrick Shaw.

Logic is the branch of philosophy and mathematics that addresses the structure of good arguments – arguments the conclusions of which follow from their premisses. Knowledge of logic is not only indispensable to students of philosophy but is also useful in all aspects of intelligent discourse. Unfortunately, the teaching of logic suffers from a problem. In order for the student to appreciate the delights of higher level logic (and they are delights) he or she has to wade through mountains of not so delightful stuff. This problem is not alleviated by the fact that both logic texts and the teaching of the discipline are often dull, and that the relevance of logic to practical affairs can be obscure. Shaw’s aim in Logic and its Limits is to address these problems of presentation, and to raise some of the broader issues that arise from practical logic. In my view, he is successful. Within its set limits, the book is both accessible and entertaining.

Shaw begins with an exposition of what constitutes an argument, noting that the premisses of arguments can be either explicit or (more often, as it happens, in everyday speech) implicit. From this starting point he proceeds to propositional logic (sometimes called sentential calculus) This is the branch of logic concerned with arguments comprising inferences from simple sentences to simple sentences. He continues to consider syllogisms which comprise strings of the type ‘All X is Y’, ‘Some X is Y’, ‘No Y is Z’ etc., and which lead to conclusions, such as ‘Therefore All X are Z’. A classic example of a syllogism is: All men are mortal, Socrates is a man, therefore Socrates is mortal.

Having provided an exposition of formal elementary logic, Shaw continues to discuss the more practical aspects of logic: common fallacies in statistical reasoning, the cheap tricks used in political debate, how far one should trust appeal to authority, and so forth. All this is admirable, and it is made all the more so by the inclusion of copious exercises for the reader with (mercifully) solutions at the end of the book. I was particularly pleased by the discussion on statistical inference.

I do, however, have some caveats.

In his coverage of sentential calculus, Shaw describes truth tables (if you do not know what truth tables are, don’t worry, truth tables are simply rows and columns of ‘truth values’ by reference to which one can easily determine the validity of sentential arguments). Shaw explains the easy way of determining validity by reference to truth tables; however, as well as the easy way there is the very easy way, which Shaw fails to delineate. For the record, the very easy way involves arranging truth values such that if, and only if, an argument is valid, all the truth values in the rightmost column of the table come out ‘true’.

There are also some more significant omissions. There is little mention of systems other than propositional and syllogistic logic, and no mention at all of either modal logic or fuzzy logic – modal logic involves considerations of necessity and probability, fuzzy logic permits propositions with partial, as opposed to absolute, truth values. For a book that aims to specify the limits of logic (mathematical logic as opposed to practical logic) some mention should have been made of attempts to rectify its shortcomings. More importantly, logic only becomes philosophically interesting at its higher levels, and although comprehensive treatment of advanced logic is ruled out in a book of this sort, something of its flavour could have been provided – Smullyan provides just such a taste in The Lady or the Tiger? (OUP, 1991).

For all the above, Logic and its Limits contains many pleasures. One of these is certainly Shaw’s analysis of a 1950s radio debate between Bertrand Russell and Malcolm Muggeridge, noting the various logical errors and distortions of meaning of the protagonists (if so fine a logician as Russell could fall prey to such lapses, one feels, what hope is there for the rest of us?). Moreover, in an appendix, Shaw presents as clear an exposition of the shortcomings of syllogistic logic as I have yet seen.

So, if you want a simple, easily-digested introductory text on logic, one that will help you analyse both your arguments and those of others, and one that will at least tell you something of what logic is about, you can do a lot worse than buy this book.

I have no hesitation in recommending Logic and its Limits to my students.