Consistent vs Inconsistent Sets of Claims

Consistent vs Inconsistent Sets of Claims

Like the terms “valid” and “invalid”, the most common use of “consistency” and “inconsistency” in everyday language is different from its use in logic.

In everyday language, something is “consistent” of it’s predictable or reliable. So, a “consistent A student” is a student who regularly and predictably gets As. A consistent athlete is one who reliably performs at a certain level regardless of the circumstances. An inconsistent athlete performs well sometimes and not so-well other times, and their performance is hard to predict.

This isn’t how we use the terms “consistent” and “inconsistent” in logic.

In logic, “consistency” is a property of sets of claims. We say that a set of claims is consistent if it’s logically possible for all of them to be true at the same time.

What does “logically possible” mean here? Logically possible means that the set of claims doesn’t entail a logical contradiction.

A contradiction is a claim that is false in all logically possible worlds, and we usually write the general form of a contradiction as a claim of the form “A and not-A”.

“not-A” is usually interpreted as the contradictory of A, but as we saw in the last tutorial, this is can also be the contrary of A.

So, if it’s logically impossible for a set of claims to be true at the same time, then we say that the set is logically inconsistent.

Examples

Let’s look at some examples:

“All humans are mortal.”

“Some humans are not mortal.”

These clearly form an inconsistent set, since these are logical contradictories of one another. “Mortal” means you will some day die. “Not mortal” means you’ll never die, you’re “immortal”. If one is true then the other must be false, and vice versa.

Now what about this set?

“All humans are mortal.”

“Simon is immortal.”

Can both of these be true at the same time?

In this case, the answer is “yes”, both of these can be true. They’re only inconsistent if you assume that Simon is human —if that were true, then these would be inconsistent. But “Simon” is just a name for an individual, so Simon could be a robot or an angel or an alien, and if so then the claim about all humans being mortal wouldn’t apply.

Now, if you added the assumption about Simon being human as a claim to this set, like so ...

“All humans are mortal.”

“Simon is immortal.”

“Simon is human.”

... then you’d have an inconsistent set. Here you have three claims where, if any two of them are true, the third has to be false.

Let’s take a moment and look at this at little closer.

If all humans are mortal, and if Simon is immortal, then it logically follows that Simon can’t be human.

By “logically follows” I mean that you can construct a valid argument from these premises for the conclusion that Simon is not human, like so:

1. All humans are mortal.
2. Simon is immortal.
Therefore, Simon is not human.

This is what it means to say that the set entails a logical contradiction. From the set one can deduce a claim that is either the contradictory or the contrary of one of the other claims in the set.

Here’s another way to represent this. The first two claims entail a claim that is the contradictory of the third claim. And from this it becomes evident that, to assert that all the claims in the set are true is to assert a formal contradiction.

Now, we can run this with any pair of claims in the set. If we set the second and third claims as true, for example, then we can infer that the first must be false. If Simon is immortal and if Simon is human, then it must be the case that not all humans are mortal, which contradicts the first claim.

The only remaining pair to check is the first and the third claims. If it’s true that all humans are mortal, and it’s true that Simon is human, we can validly infer that Simon is mortal, which contradicts the second claim.

An Important Fact About Inconsistent Claims

This example helps to illustrate another important fact about inconsistent sets of claims. IF we’re given a set of claims that we know is inconsistent, then we know that at least one of the claims in the set must be FALSE.

So, if we want re-establish consistency, we need to abandon or modify at least one of these claims.

We used logic to establish that the set is inconsistent, but it’s important to understand that logic alone can’t tell us which of these claims to modify.

Logic tells us is that you can’t consistently believe all of these claims at the same time, but it doesn’t tell us how in any particular case to resolve the inconsistency.

Nevertheless, it can be very helpful in argumentation to have a group of people come to agree that a set of claims is inconsistent. In the end they may disagree about how to resolve the inconsistency, but it’s still an achievement to get everyone to realize that they can’t accept everything on the table, that something has to go.