# 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
• Logical 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 Oct 14 at 22:51

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.

If you are studying an introductory text on logic, it is almost certainly introducing you to classical logic, which is the most commonly used kind. Non-classical logics are usually a more advanced topic. Naively and pre-theoretically, an argument is valid iff the following is true:

1. If the premises of the argument are all true, the truth of the conclusion follows by necessity.

This is equivalent in classical logic (and many other common systems of logic) to:

1. It is impossible for all of the premises to be true and the conclusion false.

There are many different accounts that explicate validity in more detail, but let's stick with these. The consequence of the second way of characterising validity is that two unintuitive edge cases emerge. One is that if an argument has inconsistent premises then it is always valid, no matter what the conclusion. This is because it is impossible for all the premises to be true and hence a fortiori impossible for all the premises to be true and the conclusion false. The other case is that if an argument has a logical truth for a conclusion then it is always valid. This is because it is impossible for the conclusion to be false and hence a fortiori impossible for the premises to be true and the conclusion false.

Your example with the conclusion, "This is an argument" is an attempt to exemplify the latter case, but it doesn't really work, because "This is an argument" is merely obviously true, not a logical truth, or even a necessary truth in some broader sense. All kinds of things are obviously true without being necessarily true, e.g. it sometimes rains in New York. If we substitute your conclusion for a logical truth, such as something that looks like "P or not P" then it would indeed be valid.

If this seems strange and unintuitive, maybe the following line of reasoning can help. Classical logic is monotonic, which is to say that if a given argument is valid, adding extra premises does not make it invalid. So, take the following valid argument:

``````Either Mary or Jane won the race.
Mary didn't win the race.
Therefore, Jane won the race.
``````

``````Either Mary or Jane won the race.
Mary didn't win the race.
Some people like peanut butter.
The moon is made of cheese.
Therefore, Jane won the race.
``````

This argument is still valid. It doesn't matter that two of the premises are redundant and one of those is false. Now if our conclusion is a logical truth, such as the classical tautology P or not P, this can be proved using logic alone and we don't need any premises at all. So if we start from no premises and add a couple we can get:

``````Some people like peanut butter.
The moon is made of cheese.
Therefore, P or not P.
``````

This is still valid. The premises are not needed to make the conclusion valid, but by the property of monotonicity they do not make a valid argument invalid.

It follows from this that classical logic does not require the premises of an argument to be relevant to the conclusion. There are other logics, such as the family of logics called relevance logics, in which relevance is required for validity. Going back to the two ways of characterising validity that I gave at the start, a relevance logician would accept 1 but reject 2.

Quite simply validity is an ideal by which we can quantify degrees of validity. It is a concept such as zero or infinity, we are tricked into seeing them as different by our desperate attempts to experience an external reality. Infinity and zero are useful but not tangible or even nearly possible to experience and validity is the same. Think the past is gone future unmade and now no time long so when is anything valid?