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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.

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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

  • Is not the validity of an argument is about the connection between premises and the conclusion? I can see that the reasoning behind this argument is ill-formed, but still it being valid nags me, so if the conclusion is true then the argument is valid? – RaGa__M Aug 10 at 20:11
  • @Explorer_N If the conclusion is always true the conclusion would be a valid argument by itself, because one could use any premises one wanted, or no premises, to show the conclusion is true. It fits the definition of validity that Wikipedia describes. Perhaps a different definition of validity would require more interesting arguments, those that connect actually useful premises to the conclusion. However, in that case since this second sentence is always true, the two sentences would not be an argument under such a definition and so the question of validity would not arise. – Frank Hubeny Aug 10 at 20:21
  • Your answer is misleading people. First you are specifically speaking of only Mathematical logic. The definition from wiki is wrong & poorly worded. This nonsense is evident in you & others believe the premises don't even have to be related. So I can just write any three random premises & some humans will now think "well it is valid". You need to address this problem of any three true random premises DO NOT present an argument. What you wrote can lead to any nonsense being valid & logical. Mathematical logic doesn't have middle terms. This is why you think the way you do – Logikal Aug 11 at 14:30
  • @Logikal I prefer relevance logics. From an informal logical perspective people in an argument commit to the premises as true and then argue from there. The issue of false premises doesn't appear. Reasoning from such is meaningless, not invalid. However, for classical truth-functional logic any sentence that can be assigned a truth-value is in the domain of sentences that can form an argument. That classical logic wants to assign all such arguments a "valid" or "invalid" value. That is where the problem comes in. There should also be a "meaningless" value for arguments. – Frank Hubeny Aug 11 at 14:45
  • The are indeed meaningless statements. However when it comes to arguments there a strict rules that many people don't seem to realize or honor. Propositions are NOT sentences in philosophy. Declarative Sentences express a proposition. If there is nothing relating the premises together, there is no argument --so validity doesn't come up at all. Many people use false premises in philosophy probably not Mathematical logic. A person can be unaware a premise is false. What then? Rational folk would start over but some folk decide to be irrational. Psychological issues arise in the real world. – Logikal Aug 11 at 16:21

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