can an argument containing a contradiction be valid argument

Question

I know that validity has nothing with truth of the conclusion or with how good argument is in general, and an argument is valid iff the truth of its premises guarantees the truth of its conclusion.

However, my teacher gave us an argument such that

Grass is green.

Grass is not green.

Therefore Cows bark.

and he said that it is a valid argument, because it is impossible for this example when all premises are true ,the conclusion is false. However, I think that this example contradicts with the main idea of validity. Because the premises don't guarantee the truth of the conclusion.

Can you enlighten me about whether it is a valid or not.

Answer 1

Well, in what way do you think the truth of the premises does not guarantee the truth of the conclusion? In which situations is the promise "If the premises are true, the conclusion will be true" broken?

The definition of validity is:

For all interpretations it holds that if all premises are true under that interpretation, then the conclusion is true under that interpretation as well.

The negation of this is

Not for all interpretations it holds that if all premises are true under that interpretation, then the conclusion is true under that interpretation as well.

which is equivalent to

There is an interpretation for which it does not hold that if all premises are true under that interpretation, then the conclusion is true under that interpretation as well.

which is in turn equivalent to

There is an interpretation such that all premises are true but the conclusion is false under that interpretation.

That is, an argument being invalid amounts to saying that there is a concrete counter interpretation which makes all premises true but the conclusion false. If there is no interpretation which can make all premises true to begin with, then in particular there can be no such counter interpretation. If there arises no situation in which the condition on the truth preservance guarantee (the truth of the premises) takes effect, then there is no situation in which this promise can be broken.

So yes, the argument is valid, precisely for the reason cited by your teacher. An argument that is valid because the premises are contradictory is called vacuously valid.

You may be interested in the notion of a sound argument: A sound argument is one which is valid and where in addition the premises are true in the real world. Premises which are contradictory can obviously not be true in the real world, so the above argument is unsound. This may be closer to your intuition of a "correct" argument than the notion of validity.

Answer 2

You state (slightly paraphrased):

the truth of the premises doesn't guarentee the truth of the conclusion.

But the stance of classical logic (re: this, see below) is that in fact the truth of the premises does guarantee the truth of the conclusion. There is no conceivable situation where the premises are true but the conclusion is false, since there is no conceivable situation where the premises are true, full stop. The principle at play here is called "ex falso quodlibet" or "the principle of explosion."

And in fact this isn't special to classical logic. Intuitionistic and modal logics also have this principle by default (there are e.g. such things as "relevant modal logics" - see below - but the usual modal logics are simpler than that). Basically, this situation occurs in any logical system in which (i) a deduction is thought of as valid iff it has no countermodel and (ii) the semantics for the system does not permit "impossible" models.

That said, not every logical framework accepts this sort of reasoning. In particular, we might want to use a framework where "From A we can deduce p" only happens if A somehow "causes" p - the deduction above would intuitively not be acceptable in such a situation since impossible grass has no bearing on cow-nature. The terms "relevance logic" and (more generally) "paraconsistent logic" are key here.

Answer 3

The example argument is not valid.

(1) The terms of the conclusion (cows, bark) appear nowhere in the premises. I believe this fallacy is irrelevancy.

(2) A syllogism has exactly three terms; all three appear in the premises. The premises in the example have only two terms (grass, green). In effect, there is no middle term.

(3) A syllogism having a negative premise must have a negative conclusion. Here, one premise is negative but the conclusion is positive.

(4) If a term is distributed in the conclusion, it must be distributed in the premises. Here, "cows" in "Cows bark" is distributed. The term "cow" is not distributed in either premise.