This is a book about applied logic. Books that popularize mathematics (and logic) often have chapters about paradoxes, or logic puzzles. If you were expecting something in that style, then it will soon become clear that you are mistaken. Eugenia Cheng who combined in her previous book *Cakes, Custard + Category Theory* her profession as a mathematician specialised in category theory and her hobby of cooking. In *Beyond Infinity* she discussed the infinitely small and the infinitely large, which is also rather mathematical, but it can also lead to paradoxes such as the Hilbert Hotel.

In the current books she discusses the roots that govern all mathematics: the rules of logic and axioms that lay at the origin. Close as this may be to the heart of a mathematician, she considers here however how logic also applies to our daily life, although in a much fuzzier way and lacking the mathematical formalism. As a consequence misunderstandings will occur. These will result in endless and unsolvable discussions, because the opponents apply different rules or different axioms and so both claim to be correct in coming to opposing conclusions.

Cheng applies this to unravel some of the currently hot topics that roam the media like (political) discussions about health care, or racism and sexism. She clearly explains why the different parties can not come to an agreement. People come to their own version of "The Truth" by making logical mistakes. For example a negation is mistaken for a contraposition, or they use a false premise which logically allows to imply anything, or people apply the rules to different classes of subjects, etc.

To be able to point out where things go wrong in many practical situations, Cheng of course needs to explain some rules and terms form logic that are much more clearly defined in a mathematical situation. Mathematicians will agree on what is true or correct because they are arguing within a much more abstract and unambiguous universe, using generally accepted rules, even though they need not make all the details of their logical deduction steps explicit. As long as their peers will be able to see how the gaps need to be filled, they will accept the result. Only if the gaps are too large, a referee will require more details.

So the first part of this book is explaining what logic actually is and how it is experienced every day by anyone. Using many examples from social discussions, political disagreement, or just parent-kid discussions, Chen introduces the different terms, using some necessary abstraction, to talk properly about the terms that are used in more formal proposition logic, including quantifiers, Venn diagrams, truth tables, negation of implications, equivalence, but also fuzzy logic (the world has many "grey zones"),... When there is a discussion about who is to blame for some unfortunate event, then one should first see how it is possible that the event came about, and those previous events are caused in turn by some other events, etc. So there can be a very complex network of causal coincidences that have eventually led to the event that is the subject of the dispute (Chen uses all these relations as a pretext to smuggle in some of her beloved category theory).

In a second part she explains the limits of logic. In practice there is no peer reviewing process of some person's argument like in a mathematical environment of publishing a paper. Which mechanisms (correct or inappropriate) are used to convince people? Perhaps (Internet) memes are assumed correct while they are not. When one comes to paradoxes, some alarm should go off to revise the system applied. In other situations, logic will not be useful like in emergencies, or when we do not have all the information to act logically, in which cases we may perhaps just follow a reflex, a gut feeling, or trust the judgement of others.

The third and last part is called beyond logic. This is where one should agree on axioms, the things that are accepted without a (logic) proof. Then there are of course the many grey zones where binary logic is not the proper tool to use. What universe is one talking about (all humans, all men, all women, all white women, all rich and white women,...?) Things may be considered equivalent (the same) for somebody, but not for the adversary. And then there are of coarse emotions that are important factors in everything we do or say.

In this book Chen is strongly engaged in social justice, minority groups, gurus, religion, climate issues, the role of science, etc. So in her last chapter she somehow summarizes how logic can help you to be a reasonable and intelligent person. There should be some framework that one believes in, and one should be sceptical towards charismatic "superstars". You should realize that there are a lot of grey zones and that you are not alone so that reaching a joint objective can be more rewarding than reaching your own. Correct and reasonable logical arguments should be used, even in a world that is not always logical.

This is an engaging book, that should be read by everyone. It will help solving disagreements, or direct discussions away from and "it is - it isn't" arguments, and help you focus on the underlying cause of the dispute. Of course real life is not mathematics, boundaries are fuzzy, and obviously, it can not prevent that people disagree, and they should if for the proper reasons and when using the correct arguments, and this is the main message of the book.