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I studying graduate math (not very far into it), and I realized that some of the higher-level math texts I would like to read are hard to understand without a strong basis in logic. Now I've taken elementary courses (like general college first year) that emphasized logic.

I just started reading an introductory logic book titled *Forall X* by P.D. Magnus, in order to strengthen my skills. One of the first topics covered is *validity* and its definition:

An argument is valid if and only if it is impossible for all of the premises to be true and the conclusion false.

The author then provides an example of a valid argument, and then of an *invalid* argument, which is

London is in England.

Beijing is in China.

So: Paris is in France.

He then explains that this argument is invalid, based on his definition of *valid*

The premises and conclusion of this argument are, as a matter of fact, all true. But the argument is invalid. If Paris were to declare independence from the rest of France, then the conclusion would be false, even though both of the premises would remain true. Thus, it is possible for the premises of this argument to be true and the conclusion false. The argument is therefore invalid.

This quickly led me to think that he's circumventing any subtlety. For example, there are arguments that I could make in the same style, but where the conclusion is impossible to make false. Consider

London is in England.

Beijing is in China.

So: This is an argument.

To summarize, I do believe there's some fundamental flaw in my reasoning in regards to creating this little paradoxical-seeming statement, but at the same time, I don't think the author's logic was correct either.

5If an argument has a logical truth (like

p or not-p) as its conclusion, then the argument is indeedvalid. It is impossible for the conclusion to be false, whence it isa fortiorialso impossible for the premises to be true and the conclusion false. This is one of the limiting cases of validity - an argument with contradictory premises being another. The intuitive and technical concepts of validity part way here. Note that you’re using a demonstrative,this, in your example. Ordinary FOL can’t handle these expressions – essentially because they are context-sensitive. – MarkOxford – 2018-03-05T20:36:04.4401Logical validity should depend only on the logical form, not on the semantic meaning of any of the terms. "This is an argument" may seem obviously true in context, but it isn't true for purely formal reasons. For example, we could replace "this" with the semantically meaningless symbol x and the predicate "is an argument" with a predicate symbol like p(), so "this is an argument" becomes just p(x). In such an abstract form, p(x) is neither tautological nor would it follow from either of the previous statements if they were also "translated" into such an abstract form. – Hypnosifl – 2020-10-14T22:51:07.877