# Validity of an argument with a contradictory conclusion [duplicate]

I have an argument with the conclusion: 'The sky is blue and it is not the case that the sky is blue'.

Without knowing the premises of this argument, am I right in saying this argument is invalid since the this conclusion is necessarily false?

Given that your conclusion is contradictory, it cannot be true in classical logic, but that doesn't necessarily mean that any argument that has it as its conlusion is invalid. It is possible for an argument to have inconsistent premises. So, for example, the following argument is valid:

1. The sky is blue.
2. It is not the case that the sky is blue.
3. Therefore, the sky is blue and it is not the case that the sky is blue.

This is valid syntactically because 3 follows from 1 and 2 by rule of conjunction, and it is valid semantically because every model of the premises is also a model of the conclusion (because there are none of either).

The important thing to remember about validity in logic is that it is not an evaluative term meaning that an argument is good or strong or convincing. When assessing an argument it is important to ask whether it is sound (the premises are true) and whether it is cogent (the premises support the conclusion).

In classical propositional logic, an argument is invalid iff. there is a situation in which the premises can be true and the conclusion false. So by that measure (assuming that you have premises that can possibly be all true at one time) your argument that concludes in a contradiction would be invalid.

However, what is worth noting is that you didn't state was what kind of logic is being used to analyze this argument - Paraconsistent logics such as dialetheism accept that there can be true contradictions, which can throw a spanner in the works here.

• Not necessarily; if the premises are unsatisfiable, i.e. "always false", then the argument is valid, because "if FALSE, then FALSE" is TRUE. Nov 10, 2015 at 8:04

No, if you are asking: for any set of propositions P, is any argument from P to 'The sky is blue and it is not the case that the sky is blue' invalid?

For example consider P = the two premises The sky is blue and It is not the case that the sky is blue; then you conclude validly that The sky is blue and it is not the case that the sky is blue (by the valid inference rule that given or assumed both A and B, then (A and B)).

You may have valid arguments whose conclusion is false.

The sentence in your post is not an argument but a proposition. Because you do not make a deduction but you state a proposition.

I assume that you suppose the validity of 2-valued propositional logic. Then your proposition has a truth-value, which is either true or false. With A := "The sky is blue" your proposition has the form

"A and not A".

According to the law of non-contradiction your proposition is false - just by formal reasons, independently from the content of the special A.