A nonsensical argument

Question

I am reading An Introduction to Formal Logic by Peter Smith. I was checking if the following argument ( Question no-8, Exercise-1 ) is valid or not:

All the slithy toves did gyre and gimble in the wabe. Some mome raths are slithy toves. Hence some mome raths did gyre and gimble in the wabe.

I concluded that it is a valid argument based on its inference pattern. But in the answer sheet, the author has written following:

This has the look of an inference of the shape: All Ss are G. Some Ms are S. So, some Ms are G.

And a genuine inference of that form will of course be valid. However, our Jabberwockian example is not a genuine inference, since the premisses and conclusion are nonsense, and make no contentful claims. And if it isn’t a genuine inference, where we infer a contentful claim from two other contentful claims, it can’t be a valid genuine inference in particular!

Only assuming that the words (slithy toves, gyre, gimble, wabe and mome raths) are meaningless makes this argument deductively invalid.The problem is that English is not hard and fast language, it cannot be that all meaningful words are written in dictionary. So, shouldn't this argument be valid?

Answer 1

I wonder whether the author is trying to make the point that certain argument forms do not apply when one of the objects is vacuous.

However, in this case it is an $AII$ of the first figure, where this restriction does not apply.

(See https://proofwiki.org/wiki/Definition:Figure_of_Categorical_Syllogism for a link to where such things are discussed.)

It is stated that "some mome raths are slithy toves". Hence the existence of mome raths is asserted. From that assertion it follows that slithy toves exist, and it is further asserted that all such slithy toves also in fact gyre and gimble.

Therefore it is completely valid that:

• given the fact that there exist some mome raths which are slithy toves, and
• given that all slithy toves gyre and gimble

it is the case that some mome raths gyre and gimble.

It is important that we are not scared off by arguments making deductions about entities that may or may not exist, because in mathematics it is often the case that we don't yet know whether such entities exist -- but if they do exist, then we can deduce things about them which, in turn, may allow us to deduce whether or not such entities can exist or can be proven to exist.

Further considerations, which may or may not shed further light:

The validity of an argument is independent of the truth of its premises. An argument with false premises may well be valid, but you may deduce only that the conclusion is true if the premises are true.

Hence the difference between a valid argument and a theorem.

I have a battered old copy of E.J. Lemmon's "Beginning Logic" on my shelf which I used to teach myself this stuff. It contains an entire section on the syllogism, which is analysed in some breadth, if not depth.

Answer 2

The definition of a logical sentence requires it to only use terms of some predefined language (in this case, English) and possibly some free variables. Even if English is naturally a flexible language, when you talk logic, you must treat it as a formal language, and therefore you're not allow to freely add new terms.

The expressions that you're given look like they are logical sentences, they seem to have the correct grammar, but since they use terms that don't exist in the accepted language, they are not. And you cannot apply logic to expressions that are not logical sentences.

You could however define a new language, let's call it an expanded English that does include these terms and then it will be a logical sentence in that language, and the reasoning will be valid. The thing is, it should be done beforehand, and if it is not stated, it is presumed that the language you're working in is normal English.