# Concerning the definition of “valid”

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 access without a strong basis in logic. Now I've taken elementary courses (like general college first year) in philosophy with an emphasis in logic. So anyway, I just started reading an introductory logic bood/pdf titled forallx by P.D. Magnus, in order to strengthen myself.

One of the first things 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 lead 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.

• If an argument has a logical truth (like p or not-p) as its conclusion, then the argument is indeed valid. It is impossible for the conclusion to be false, whence it is a fortiori also 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 Mar 5 '18 at 20:36

'Validity' has different senses but the questioner has made plain which one is involved in the question - the sense given by the author.

London is in England.

Beijing is in China.

So: Paris is in France.

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 is one of the oddest explications of validity I have come across.

London is in England.

Beijing is in China.

So: Paris is in France.

The invalidity of this argument has nothing to do with the counterfactual independence of France. The point is that the premises :

London is in England

Beijing is in China

can be true without the conclusion's being true : 'Paris is in France.'

The fact that all three statements are true is irrelevant as the author knows. The truth of 'Paris is in France' is not guaranteed by the truth of 'London is in England' and 'Beijing is in China'. The truth of the premises is completely independent of the truth of the conclusion; there is no logical connexion between premises and conclusion to guarantee the truth of the conclusion.

On your own intriguing example :

London is in England.

Beijing is in China.

So: This is an argument.

Indeed, 'This is an argument' will be true whenever it features as the conclusion of two premises. But it will be true whether the premises, both or either, are true or false. The truth of the premises does nothing to guarantee the truth of the conclusion. The conclusion follows only because there are premises - any premises. Their truth-value does no work.

Nice intellectual testing of the author, though.

Logic does have more than one context of the term validity.

In one case validity used to refer to a type of argument that was truth preserving. That is, once we start with all true premises and there is a proper relationship between statements then the conclusion would also have to be true as well. It is impossible for the premises to be true and the conclusion to be false by definition of the term valid. When you have no relationship or the wrong relationship we can see blatantly false conclusions from true premises. So the study of mood and form were introduced in classical logic. The logician can evaluate the relationship of any argument of any subject matter without mastering the subject at hand. So I don't have to be a economist to evaluate an argument in the field of Economics. I don't have to know Biology to evaluate an argument about evolution, etc. This deductive reasoning can apply to any subject whatsoever --- it is universal.

On the other hand if a pattern of reasoning is shown to have a flaw that argument is also invalid. That is, I would be able to change the content of the argument to some other subject and get a false conclusion from true premises. In other words I can present a counter example to the argument form presented and state we can't have an argument form that is both true and false at the same time. This is mentioned because many people reason from worldly knowledge so they know some statements are true but they don't really understand logic. So they can make a fallacious argument with true premises and true and a false conclusion. Because the argument works when they conviently want it to work they conclude the argument is valid. This is not the case in logic. Deductive reasoning is about absolutes or non absolutes. There is no middle ground. An argument form that cannot be made to have a false conclusion is valid. If I can trust your argument into a form of statements TRUE TRUE False then the form is invalid.

According to the explanation of validity you yourself quoted, the argument with the conclusion "This is an argument" is not formally valid.

Thus, it is possible for the premises of this argument to be true and the conclusion false. The argument is therefore invalid.

It is not formally valid because the conclusion can be in fact interpreted as referring to anything at all, including therefore things that are not even an argument, let alone this one.

Each one of these interpretations (and there is an infinity of them) makes the conclusion false.

This is so because the premises of your argument do not formally compel the only interpretation that makes the conclusion true.

The fact that we do read the conclusion as obviously true, and therefore necessarily true, comes from a premise which is not made explicit here in your argument. This premise is broadly the totality of the semantic necessary to interpret the conclusion as we do, including the definition of "this" as referring to the most proximate item of the kind mentioned. I believe this premise is too complicated and ill-defined to be possibly formalised properly.