Deductively valid arguments and situations

Question

In book 'Logic: A Very Short Introduction', Graham Priest has quote about deductively valid arguments.

Here is one problem. Assuming that the account is correct, to know that an inference is deductively valid is to know that there are no situations in which the premisses are true and the conclusion is not. Now, on any reasonable understanding of what it is to be a situation, there are an awful lot of them: situations about things on the planets of distant stars; situations about events before there were any living beings in the cosmos; situations described in works of fiction; situations imagined by visionaries. How can one know what holds in all situations? Worse, there would appear to be an infinite number of situations (situations one year hence, situations two years hence, situations three years hence, . . .). It is therefore impossible, even in principle, to survey all situations. So if this account of validity is correct, and given that we can recognize inferences as valid or invalid (at least in many cases) we must have some insight into this, from some special source. What source?

But I don't understand what problem he is referring to? If a argument is deductively valid we don't care if premises are true or not right? That is we can evaluate validity of argument without looking at if premises are true, isn't it?

So why does he list and mention all these situations? I am not following what point he is trying to make. I am relatively beginner in philosophy, so can someone enlighten me?

Answer 1

"If a argument is deductively valid we don't care if premises are true or not right?" Well, valid means that if the premises are true, the conclusion is guaranteed to be true as well (one way of describing logical validity is that if you have some premises P and a conclusion Q, then the statement P -> Q is a tautology). Note his comment "to know that an inference is deductively valid is to know that there are no situations in which the premisses are true and the conclusion is not". So he's asking rhetorically how it is that we can be sure of this--if we name some premises, how can we be so confident there's no "situation" anywhere in space and time where those premises hold true but the conclusion does not, given we can't actually survey all situations or even imagine all possible situations in detail? Presumably he goes on to discuss the answer(s) to this rhetorical question after the section you quoted.

I should add that when he talks about propositions that can apply to multiple different "situations", I imagine he means some kind of fill-in-the-blank propositions where the blanks can be filled by different specific objects, like the following:

A. ___ is the fourth planet from the star it orbits.

B. ___ has two moons.

C. ___ is the fourth planet from the star it orbits, AND ___ has two moons.

All three would be true of Mars, so they apply to the "situation" represented by our own solar system, but they could apply to many other planetary systems in the universe as well, i.e. many other situations. So I think Priest would be asking how we know that in any situation where A and B are true, C is true as well. C seems like a pretty trivial logical deduction from A and B, but there are more complicated examples of logical deductions from premises, and even in the simple case one might see it as an interesting philosophical question to ask how we know there are no weird logic-violating planetary systems out there in the universe where A and B are true but C is not.

Answer 2

It seems he's questioning the classic formal logic's source of validity when it comes to any kind of logical forms having a universal quantified physical premise with its conclusion, such as the universal proposition Everyday the sun rises in the east on earth. Just look out of your window, there's no such logic printed in the sky. So from what source can we be sure it was the case long ago when there were no living beings? And from what source can we be sure it'll continue to be so in future without exceptions? Maybe the author is even skeptical about modus tollens or do not see deductive inference rules as mere formal rules. Quoting from your above texts "How can one know what holds in all situations?", so once the author starts to doubt certain logical entailment validity applied elsewhere under deductive formal logic, eventually one will doubt any universal logical consequence therein which may lead to outright skepticism about deductive reasoning and formalism of math.

Graham Priest is known for his defense of dialetheism and non-classical logics such as paraconsistent logic and the non-being Asian metaphysical logic. So maybe he's stimulating his readers to think about these assumed undisputable formal logic and thus hints there're other types of logics in application. However without access to his full text, these are just what I can conceive of...

Answer 3

Assuming that the account is correct, to know that an inference is deductively valid is to know that there are no situations in which the premisses are true and the conclusion is not.

This is an excellent point brought by Priest. Consider the modus ponens:

(A → B) ∧ A ⊢ B

The implication (A → B) ∧ A ⊢ B is obviously true and we don't need to scan the whole universe, or the past and the future, to see that it is true. We only need to look at it and exercise our supreme wits. However, consider now that we may want to apply the modus ponens to real-world situations. Obviously, we understand that the modus ponens will be true of all real-world situations. In fact, it is true even of all imaginary world situations. No worries here.

Now consider the implication A → B which is nestled within the modus ponens. It may be true and it may be false, and we don't know of any a priori why it should be true. Whether it is true or not is not going to affect the truth of the modus ponens, but it remains that we have there, nestled within the modus ponens, an implication which may be true or false. We don't need to know whether it is true or false to decide on the truth of the modus ponens, but we sure need to know whenever we will want to apply the modus ponens to real cases.

When we do that, we are not interested in proving the truth of the modus ponens, we already know it is true. We are only interested in applying it to a concrete situation. However, what we do need to know in this case is whether B is true, and to be able to decide on that, we need to know if the two terms A → B and A are true. Concerning A, this is possibly trivial given that we are supposed to be considering a concrete situation. If A for example means "Trump lost the election", we will probably assume that A is true. In any case, we will not need to scan the entire universe and beyond to decide. We just look at the situation we are considering.

However, the implication A → B is something else entirely. Suppose A means "x is a man". This seems easy enough to handle. For instance, if x is Trump, we obtain "Trump is a man", and we can easily decide that A is true, even those of us who will want to add some disparaging qualifications. And again, no need to scour the entire creation. However, we have a problem, with A → B. If A → B means "If x is a man, then x is mortal", then do we know that? Even if we assume that only the Earth harbours humans, we don't really know if all humans in the past were mortals and if all humans in the future will still be mortal. I guess this is what Priest is after.

To be clear, this isn't a logical problem at all. This is 100% an empirical problem. In fact, the problem already appears with things like "Trump is a man" because we don't actually know that. All we can do is feel confident that this is true, or even just true on balance of probabilities. However, once we admit doing it for "Trump is a man", there is not justification for not also doing it for "If x is a man, then x is mortal". Maybe this is false, but we will nonetheless trust that this is true. And logic does not require that we commit no mistake in this respect. All be need to do is reason logically, trust our perception and our common sense, and hope for the best. This seems to work well. So Priest made a good point, but this is no logical problem. This is yet again the eternal problem that we don't actually know anything about material world.

Answer 4

You are confusing 2 things

(1) knowing what is the actual truth value of the premises , that is what situation corresponds to the actual world

(2) knowing the truth value of the premises in all situations.

The method of testing a reasoning for validity is :

• consider frst all logically possible situations

• consider , among these situations, those in which the premises are true

• check that , in all these situations , the conclusion is also true.