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  Why Can't Logic Tell Us How to Resolve an Inconsistency?

Question:

In one lecture you say this:
"Logic can tell us whether a set of beliefs is consistent or inconsistent, but by itself it can’t tell us which belief to modify in order to remove the inconsistency."
I don't know why logic can't identify which belief or as I interpret it which statement or premise is inconsistent. Modify the premises or statement or belief put it back into the argument and see if this set of beliefs is consistent or not. In other words if logic can tell us that a set of beliefs is consistent than we can assume that the beliefs are all consistent. But if a set of beliefs were to be identified by logic to be inconsistent then why can't you modify each of the beliefs, one by one and substitute them back into the argument to see if it reveals a consistent result?
- Joseph

Hey Joseph,

These are good questions. It is perhaps an overstatement to say that logic alone can never tell us whether a statement is true or false. There are, for example, standard proof techniques in mathematics and logic based on the assumption that contradictions (claims of the form "p and not-p") are always false, and that any statement that logically entails a contradiction must be false -- this is "proof by contradiction".

(Of course this proof method relies on the assumption that all contradictions must be false, and there are systems of logic and mathematics that reject this assumption, but that's another matter .…)

It's helpful to be strict with the language when talking about this. Individual statements can be true or false, but not consistent or inconsistent. Consistency is a property of sets of statements, not individual statements. It's a property of the logical relationships between the members of a set of statements.

That, in short, is why the judgment that a set of set of statements is inconsistent can't tell us anything about the truth or falsity of the individual statements within the set. To say the set is inconsistent is just to say that they can't all be true at the same time.

Once a set of statements is established to be inconsistent, then we can ask which of the statements we should modify to regain consistency. My claim in the lecture was that logic, by itself, can't tell us which of the claims we should give up or modify. Your question is, why can't we test each of the statements in the set individually? Modify one, hold the others constant, and see what happens.

If I understand the suggestion properly, my answer is that it's quite easy to regain consistency -- if we're given three statements, P1, P2 and P3, and they're mutually inconsistent, then we could modify any of them to regain consistency. But that doesn't help with the problem of choosing among all the possible ways to modify the set.

In real examples of this sort of reasoning, we make modifications and compare the results with other beliefs that we happen to have and think are true, and use this to infer back to statements that we think are problematic.

This activity isn't guided by logic alone, it's also guided by our convictions about which beliefs are more or less likely to be true, and those judgments aren't a matter of logic. They aren't about judging the logical properties of sets of statements, or judging what follows from what. They depend on our substantive beliefs about what is actually true in the world.

And that's another way of saying that logic alone can't tell us how to resolve an inconsistency.