The definition of a logically valid argument in most logic books is that an argument is valid iff there is no possible world where the premises are true and the conclusion is false. But this relies on the existence of possible worlds. For a person who does not believe in possible worlds, this would force them to accept that any argument with a true conclusion is valid. Now, perhaps they can bite the bullet, and accept, contrary to most logic books, that any argument with a true premise is valid. But perhaps we can salvage our intuition about validity by using another definition of validity which does not rely on possible worlds. Has anyone defined logical validity without using possible worlds? This question is somewhat similar to one I have asked before, but in the previous question, I was talking about actualism, whereas here I am asking for a definition of validity.
The short answer is, yes, there are lots. There are at least a dozen different accounts of validity, or logical consequence, and there is no need to refer to possible worlds. It is worth noting though, that even if you don't believe in the existence of possible worlds, it is still feasible to refer to them simply as sets of descriptions of sets of states of affairs, without any commitment to their existence.
Stewart Shapiro, in a paper called "Logical Consequence: Models and Modality" lists the following definitions: Φ is the logical consequence of premises Γ iff…
Modal, or metaphysical definitions:
- (1) It is not possible for every member of Γ to be true and Φ false.
- (2) Φ holds in every possible world in which every member of the premises hold.
- (3) There is a deduction of Φ from Γ by a chain of legitimate, gap-free (self-evident) rules of inference.
- (4) It is irrational to maintain that every member of Γ is true and Φ is false.
Linguistic, or semantic, characterizations:
- (5) Φ holds in every interpretation of the language in which every member of Γ holds.
- (6) The truth of the members of Γ guarantees the truth of Φ in virtue of the meanings of the terms.
- (7) The truth of the members of Γ guarantees the truth of Φ in virtue of the meanings of a special collection of the terms, the “logical terminology”.
- (8) There is no uniform substitution of the non-logical terminology that would render every member of Γ true and Φ false.
- (9) The truth of the members of Γ guarantees the truth of Φ in virtue of the forms of the sentences (or propositions).
Shapiro says that the list is not exhaustive, and indeed it is not. Some others that could be added are:
- (10) The conclusion holds under all permutations of the domain of quantification.
- (11) The information present in, or represented by, Φ is entirely contained within the information present in, or represented by, Γ.
- (12) Necessarily, if Γ holds, Φ holds. Note that while this is often conflated with (1) and is equivalent to it in classical logic, it is not equivalent in all logics. A relevance logician might agree with (12) and disagree with (1) since (1) is committed to explosion, but (12) is not if we use a relevant conditional.
To make matters more complicated, some logicians hold that there is such as thing as ‘material validity’ which is a kind of logical validity, but distinct from formal validity. For a more detailed treatment, there is an interesting three-part paper by Ladislav Koreň called Quantificational Accounts of Logical Consequence.
"There is no possible world where the premises are true and the conclusion is false" does not rely on the existence of possible worlds, it's merely saying that it's impossible for the all premises to be true while the conclusion is false.
Using the concept of possible worlds separates validity of the logic itself from correctness of claims about the reality, allowing you to reason about them independently and avoid having one interfere with the other. This may, for example, help to prevent someone from saying/asserting "but the conclusion is true in our world" (to perhaps rebut a logically valid argument that demonstrates otherwise). Whether the conclusion is true in our world is irrelevant as far as logical validity is concerned, because that deals with premises and conclusions in the abstract. If I say "All men are mortal, Socrates is a man, therefore Socrates is mortal", this is a logically valid argument regardless of whether any of those things are true. And then, if you accept that the argument is valid and the premises are true, then you must logically accept the conclusion.