What connects Russell’s paradox, intolerance and Battenberg cake? Or Euclid’s axiomatization of geometry and sexual harassment? The definition of marriage and lasagne? They are all sets of concepts that readers will find completely sensible and illuminating by the end of this mind-expanding book about “how people construct misleading arguments, and how we can argue back”.
Eugenia Cheng, in The Art of Logic, begins by describing what logic does and doesn’t mean to a mathematician – “Maths isn’t about right and wrong, and nor are most arguments” – and goes on to demonstrate the concept of logical implication, “that ‘if’ one thing is true, ‘then’ another thing must be true, using logic”. Working backwards through a person’s logical implications, much like a child repeatedly asking “why?” can reveal their fundamental beliefs, or axioms. Two people can both be logical but still disagree, she points out, if their axioms differ.
The Art of Logic claims ‘to help achieve better mutual understanding’ through facilitating more good arguments.
The book is divided into three sections, The Power of Logic, The Limits of Logic and Beyond Logic, and it is about as difficult to summarise as a complex mathematical formula. There is a sense in which it is a complex mathematical formula – a set of logical proofs in which A implies B, and B implies C, and so on, until Z is inevitable assuming that we accept A. This makes for a very persuasive argument. Cheng also includes a chapter called ‘How to Be Right’ in which she recommends prefacing certain statements with the clarifying logical phrase, “there is a sense in which…”
Along the way, Cheng explores converse errors, grey areas, oversimplifying and levels of abstraction. She highlights the difference between false equivalence and false dichotomy. This is pure maths applied to the real world, in a surprisingly practical way. It also happily accepts that there is a limit to logic – and that logical people must use empathy and emotion if we are to stand any chance of understanding each other and getting on.
It’s a shame that not everyone can read this book, but Cheng claims it is incumbent on those of us who can to use compassion and logic to argue productively with those who can’t. In this way, advanced mathematics could make a meaningful contribution to creating a better society as well as happier conversations and relationships. There is a sense in which this book is proof it can. ( (Guardian)
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