I'm not sure if this will come up as a "verified purchase", but I've read this book in its entirety through Kindle Unlimited.
The Art of the Argument purports to present an effective method of argument (The Argument) which is the means of saving civilisation, but it won't help civilisation and it won't help you. The language is muddled and confusing enough that nobody will learn logic from this book. The philosophical grounding of The Argument is incoherent to the point where nobody will come away with a good idea of what makes a statement true or known. The advice to ostracise those who won't play by the rules of The Argument is likely to, if followed, result in you losing friends and colleagues for no good reason.
Some justification for these points:
1. Molyneux confuses his reader with misleading terminology and convoluted analogies. As several other reviewers have commented, he mixes up sound and valid arguments, and refers to true premises as "valid". What's more, inductive reasoning is given three distinct meanings, none of them getting to the bottom of what inductive reasoning is. Truth is conceived of as both a relation on statements and a relation on concepts, without any explanation of how one conception of truth is supposed to help us understand the other. "Synonym logic" is conceived of as being two terms presented as having different meanings when they are the same, and in the next moment as being two terms presented as having same meanings when they are different.
I conclude that no naive reader will come away from this book with an understanding of the basic elements of logic.
2. For Molyneux, an understanding of arguments requires an understanding of truth. For one might either argue the truth or value of a position, but an argument concerning value (for him) must be grounded in truth. Let's go by Molyneux's definition of truth on statements. According to him, a statement is true if it is "a rational statement about empirical reality that can be objectively verified." This can't be the case, as we would presume that "all truths are empirically verifiable" is true, but by his own definition it can't be, because that's not a statement that can be empirically verified.
If we concede that not all true statements are empirically verifiable, but they're either "obvious" or empirically verifiable, there are at least two issues left in being sure of what we know. First, it's not clear what exactly is obvious; second, there is no account here of when we can discard empirical evidence in the face of reason. As Molyneux mentions, we can produce prima facie empirical evidence that the earth is flat, but how do we discount that evidence? When we acquire knowledge, how do we balance what is known through reason with what is obtained by sense experience? Molyneux's incoherent hardline empiricism betrays a lack of understanding of the extent of this problem.
I conclude that no naive reader will come away from this book with an understanding of what makes a statement true or known.
3. The Argument, such as it is, is meant to be a debate between at least two interlocutors who first agree to precise definitions for all of their terms, then proceed to persuade each-other of the truth or value of their position using only reason and evidence. It is meant to be the form by which all nonviolent disagreements are resolved. I've already argued that this book does not provide a good idea of when a statement is true, or whether reason or evidence should be prioritised in a given situation.
While agreeing on definitions beforehand is nice, the demand to reach agreement beforehand is unreasonable. After all, the meanings of terms like "dark matter", "bipolar disorder", "race", and "intelligence" are underdetermined by what we know; not only might our mutual understanding of these terms change over time, but because we don't really know what they represent in reality, there are reasonable competing hypotheses for what they are.
Molyneux thinks that debates can be resolved "nine times out of ten" by demanding everyone defines their terms beforehand, but this is being disingenuous. A wide variety of reasonable debates of substance concern objects or ideas we don't know enough about to spell out exactly what they are. Furthermore, he advocates that those who don't follow the doctrine of The Argument should be ostracised, a harsh penalty for simply not agreeing to an unreasonable condition to define all terms.
Therefore, a naive reader who takes this book's doctrine to heart will end productive discussions before they start (by quibbling over definitions) and is likely to lose friends and colleagues for no good reason (because they won't follow the letter of The Argument).
I don't recommend that you purchase this book. A good introduction to logic is Irving Copi and Carl Cohen's Introduction to Logic. A sensible guide for writing a persuasive essay (a more important skill IMO than making a succinct point at debate club) is found in Anthony Weston's Rulebook for Arguments. A good introduction to the study of argumentation is Stephen Toulmin, Richard Rieke, and Allan Janik's Introduction to Reasoning.
I've elaborated on these points in a lengthy review on medium.com, which I won't link here because I'm only supposed to link to other amazon products, but if you're curious you can google my name and "art of the argument".