Understanding deductive validity

Question

I read the following:

An argument is deductively valid iff it is impossible for all of the premises to be true and the conclusion false.

Definitions like these strike me as describing what something CANNOT BE rather than telling me WHAT SOMETHING IS. Is it correct to understand that a definition structured like this is effectively "exempting" something explicit while "tolerating" anything else?

For example, based on my understanding of this definition, the following pairs are all "deductively valid":

1. (all false premises, false conclusion)
2. (some false premises, false conclusion)
3. (all false premises, true conclusion)
4. (some false premises, true conclusion)

The only pair that is NOT deductively valid is:

5. (all true premises, false conclusion)

Is that correct?

Answer 1

That's sort-of correct, but it's missing a key point: validity doesn't depend on what happens to be true or false at the time. In essence, validity needs to take into account all possibilities. So it would not be allowing some false premises, but allowing always false premises.

Another way to state the definition would be that whenever the premises are true, the conclusion is true as well. For an analogy, consider the definition "a joke is funny iff it's impossible to hear the joke and not laugh". To evaluate whether a given joke is funny, we only care about what happens when the joke is told (whether people laugh). Similarly, to tell whether a conclusion logically follows from premises, we need to consider what happens when the premises are true.

Consider the premises $P$ and $P\rightarrow Q$, and the conclusion $Q$. Whether or not $P$ is actually true or false is irrelevant for whether the deduction is valid. The only way truth or falsity comes into play is when something is necessarily true or necessarily false. For example, the premise $P$ and $\neg P$ (i.e. a contradiction), you can conclude anything, and still have the reasoning be valid (it's impossible for all the premises to be true and the conclusion false, because it's impossible for the premise to be true).

Answer 2

An argument is deductively valid iff it is impossible for all of the premises to be true and the conclusion false.

Definitions like these strike me as describing what something CANNOT BE rather than telling me WHAT SOMETHING IS. Is it correct to understand that a definition structured like this is effectively "exempting" something explicit while "tolerating" anything else?

Yes, exactly.

For example, the following pairs are all "deductively valid":

1. (all false premises, false conclusion)
2. (some false premise, false conclusion)
3. (all false premises, true conclusion)
4. (some false premise, true conclusion)

Is that correct?

Actually, none of your four cases need be deductively valid. For example, referring to the real numbers, the argument $$\forall x \forall y\; x=y;\;\text{therefore, }\;3≠3$$ has a false premise and a false conclusion, but is nonetheless invalid. This is because it is not impossible for the all of the premises to be true and the conclusion false: we just have to restrict the universe to $\{3\}.$

Compare the quoted definition

An argument is deductively valid iff it is impossible for all of the premises to be true and the conclusion false

with this wrong definition:

• An argument is deductively valid iff it is not the case that all of the premises are true and the conclusion is false.

Here, “it is impossible” particularly doesn't just mean “it is not the case”; the former refers to verifying that <premises are all true, conclusion is false> applies to no possible interpretation/context.

Answer 3

You're close. If you don't like the negative definition, an argument is valid IFF the argument is such that IF true premises THEN necessarily true conclusion.

1. False premises, False conclusion: Valid/Invalid Argument
2. False premises, True conclusion: Valid/Invalid Argument
3. True premises, True conclusion: Valid/Invalid Argument
4. True premises, False conclusion: Invalid Argument

Proving validity is, last I checked, problematic. Most of the time we're advised to stick to known argument forms, that are proven to work (their validity has been proven). Try anything else and you'll have more than the point you're trying to make to worry about.