This is a pretty good introduction to logic. It explains the subject clearly and concisely. And it uses some interesting examples. The explanation of the fallacy in St. Anselm's subtle, but ultimately sophistical, "proof" of God's existence, is a model of careful reasoning and explanation. But the book has some flaws.
First, it focuses on modern symbolic logic, which would be valuable for technical specialists, but not for the intelligent laymen the "Very Short Introduction" series is aimed at. The reader who just wants to think more clearly would typically prefer a verbal and intuitive approach, as in Aristotle's original works on the subject or "Logic Made Easy" by Deborah Bennett.
Second, the book is simply wrong or misleading in certain particulars. It's attack on the cosmological argument for God's existence misses the point entirely. Unlike the ontological proofs offered by Anselm, Descartes and Godel, cosmological arguments are essentially inductive and probable, and thus are not examples of purely logical reasoning. Moreover, in order to defeat this argument, the author has to put it in the most unflattering form.*
As another example, the book attributes to Aristotle the view that everything is predetermined, that all things proceed from necessity and cannot be changed. But this was not Aristotle's view. He wrote that things happen in nature "always or for the most part", so that when quantum theorists like Heisenberg were trying to understand the inherently indeterminate world they were exploring, a world that violated the strict determinism of Newtonian mechanics, they turned to Aristotle, as someone who recognized intuitively that nature could be indeterminate.
And, on a related note, when the author gives as an example of what is strictly impossible the sentence "I will jump out of bed and hover 2m above the ground," due to the violation of gravity implied, he seems unaware of the inherent indeterminacy in modern quantum mechanics which describes such things technically possible (though extremely improbable).
Regardless, this book is worth reading. Because it is a short and lucid account, the reader can get through it quickly. And logic is, sadly, a kind of forgotten subject. If more people were trained in logical thought, more of them would see through the sophistical arguments offered by politicians and pundits who, unlike Anselm, make fairly obvious logical errors and ground their arguments more on emotions than reason.
*Note that when the author deals with a variant on the cosmological argument (which he calls the argument from design) in the context of probability, he does so sophistically. He applies Bayes' Theorem to the argument, in an effort to debunk it. But his application is highly suspect. And, at any rate, an introductory account on logic seems like a strange place to espouse a dogmatic atheism.