Can the above argument be valid?
If it can, I want to know why, as far as I see, even though the conclusion (no squares are circles.) is true, it doesn't seem to be following the premises (not a fallacy?)
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I am contributing an answer to this question because although I agree with Frank's answer, there is more that could be said about why the argument in question seems unsatisfactory, even though it is indeed valid.
Traditionally, logic arose as a way of codifying the rules of what differentiates a good argument from a defective one. Back in Aristotle's day, and even right up until the early 19th century, there was no attempt to distinguish between logic and reasoning. Logic was conceived of as the 'laws of thought' that expressed normative rules telling us how to reason well and avoid errors. With the development of modern logic in the late 19th century, logicians came to treat logic as being concerned with the relationship of consequence between propositions. As a result, today the term 'logic' is used (by logicians at least) to refer to the relationships between truths and falsehoods, and the term 'valid' is used to describe arguments where it is impossible for the premises to be true and the conclusion false. Logic is not concerned with whether the argument is persuasive, i.e. with whether the premises of the argument provide a reason to believe the conclusion.
This division between logic and reasoning leads to some counterintuitive consequences. One is that an argument with contradictory premises is always valid, whatever the conclusion. This is called the principle of explosion. It seems odd because if one tried to apply it as a principle of reasoning it would be absurd. If I discover that I have inconsistent beliefs, this does not provide me with a rational warrant to believe absolutely anything. But as a purely logical relation, it is a part of the classical system of logic and can be proved from other rules. The same goes for the fact that an argument with a tautologous conclusion is always valid whatever the premises. It seems odd, because the premises may be entirely unrelated to the conclusion. But since the conclusion cannot be false under any circumstances, there is no way for the premises to be true and the conclusion false, and so the argument is valid. As a principle of reasoning, we might say that such an argument is defective because the premises do not offer us a reason to believe the conclusion, but we cannot fault the logic itself.
So far, I have been referring to classical logic, which is the most commonly used kind. There are other logics that attempt to describe the relationship of logical consequence using properties other than just truth and falsehood. In particular, there is a family of logics called relevance logics, under which the argument in your question would indeed come out as invalid. The issue with using a relevance logic, or any non-classical logic, is that it only makes sense if we can provide a satisfactory semantics, i.e. a way of understanding what the symbols in the logic mean. Relevance logics can be interpreted in terms of channels of information, so that a valid argument is one that correctly channels information from a site where the premises hold true to a site where the conclusion holds true. There is more information about this in the Stanford Encyclopedia entry on relevance logic.
Having something that is a circle and not a circle (square) at the same time is contradictory, that is, it is false. However, saying something is not that would be true.
So, the second sentence is true, but I came to that conclusion without referencing the first sentence of the argument: All philosophers are nerds. If the argument's conclusion is the second sentence and the second sentence is true regardless of any other premises, then the conclusion can be derived without referencing any other premise. The argument is valid.
Does it matter that I stated or did not state the first sentence? No. The truth of the second sentence does not depend on what other sentences I might state. It is always true.
Does it matter that the first sentence might be true or that it might be false? No. The truth of the second sentence does not depend on the truth value of the first sentence. It is always true.
Here is Wikipedia's description of validity in logic:
In logic, an argument is valid if and only if it takes a form that makes it impossible for the premises to be true and the conclusion nevertheless to be false.
For the argument we are considering, the second sentence, that is, the conclusion, is always true. Nothing can make it false. It is impossible for the conclusion to be false. So it is impossible for the premises, the first sentence, to be true and the conclusion to be false.
The OP asks:
Can the above argument be valid?
The argument is valid because the conclusion, the second sentence, is always true.
Wikipedia contributors. (2019, August 4). Validity (logic). In Wikipedia, The Free Encyclopedia. Retrieved 19:18, August 10, 2019, from https://en.wikipedia.org/w/index.php?title=Validity_(logic)&oldid=909360709