3. Contradictions (A and not-A)

The concept of a contradiction is very important in logic. In this lecture we’ll look at the standard logical definition of a contradiction.

Here’s the standard definition. A contradiction is a conjunction of the form “A and not-A”, where not-A is the contradictory of A.

So, a contradiction is a compound claim, where you’re simultaneously asserting that a proposition is both true and false.

Given the logic of the conjunction and the contradictory that we’ve looked at in this course, we can see that the defining feature of a contradiction is that for all possible combinations of truth values, the conjunction comes out false, since a conjunction is only true when both of the conjuncts are true, but by definition, if the conjuncts are contradictories, they can never be true at the same time:

So, propositional logic requires that all contradictions be interpreted as false. It’s logically impossible for a claim to be both true and false in the same sense at the same time.ALWAYS FALSE

This is known as the “principle of non-contradiction”, and some people have argued that this is the most fundamental principle of logical reasoning, in that no argument could be rationally persuasive to anyone if they were consciously willing to embrace contradictory beliefs.

There’s a minor subtlety in the definition of a contradiction that I want to mention.

Here’s a pair of claims:

“John is at the movies.” and “John is not at the movies.”

This is clearly a contradiction, since these are contradictories of one another. John can’t be both at the movies and not at the movies at the same time.

Now, what about this pair?

“John is at the movies.” and “John is at the store.”

Recall, now, that these are contraries of one another, not contradictories. They can’t both be true at the same time, but they can both be false at the same time.

Our question is: Does this form a contradiction?

This is actually an interesting case from a formal point of view. Let’s assume that being at the store implies that you’re not at the movies (so we’re excluding the odd possibility where a movie theater might actually be in a store).

Then, it seems appropriate to say that, since they both can’t be true at the same time, it would be contradictory to assert that John is both at the movies and at the store. And that’s the way most logicians would interpret this. They’d say that the law of non-contradiction applies to this conjunction even though, strictly speaking, these aren’t logical contradictories of one another. The key property that it has, is that it’s a claim that is false for all possible truth values.

Here’s another way to look at it.

This is the truth table for the conjunction:

But in our case, the top line of the truth table doesn’t apply, since our two claims are contraries — they can’t both be true at the same time. So this case never applies.A conjunction is true only when both conjuncts are true. For all other truth values it’s false.

This kind of example raises a question that logicians might debate — whether, on the one hand, a contradiction should be defined as a conjunction of contradictory claims, or, on the other hand, whether it should be defined as any claim that is false in all logically possible worlds.The remaining three lines give you all the possible truth values for contraries, and now we see that the conjunction comes out false for all of them.

Examples like these suggest to some people that it is this latter definition which is more fundamental, that’s it more fundamental to say that a contradiction is a claim that is logically false, false in all possible worlds.

This issue isn’t something you’ll have to worry about, though. If you’re a philosopher or a logician this may be interesting, but for solving logic problems and analyzing arguments, it doesn’t make any difference.