In formal logic, "contradiction" has a very precise meaning to describe when both P and ~P are the case. This means that you can have inconsistent sets where none of the wffs are contradictions of each other. For example, consider the set of wffs {P, P→Q, ~Q}. This set is inconsistent -- there is no model in classical logic in which all three sentences are true. But there are no contradictions in this set because none of the sentences are direct negations of each other.
On the LSAT, these terms are used more colloquially, so there's not much of a meaningful difference between the two. Counterexamples are usually specific instances which are inconsistent with some general claim(s) -- e.g. "my cat Tom has no hair" is a counterexample to the general claim "all cats have hair". Contradictions are usually a bit broader and apply to any set of inconsistent sentences -- e.g. assuming there's at least one cat, "all cats have no hair" contradicts "all cats have hair". As far as I'm aware, you won't really need to understand the nuances and can treat the two terms synonymously for LSAT purposes.
A counter example is a type of contradiction. A contradiction doesn't have to be a counter example.
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2 comments
What these terms mean depend on the context.
In formal logic, "contradiction" has a very precise meaning to describe when both P and ~P are the case. This means that you can have inconsistent sets where none of the wffs are contradictions of each other. For example, consider the set of wffs {P, P→Q, ~Q}. This set is inconsistent -- there is no model in classical logic in which all three sentences are true. But there are no contradictions in this set because none of the sentences are direct negations of each other.
On the LSAT, these terms are used more colloquially, so there's not much of a meaningful difference between the two. Counterexamples are usually specific instances which are inconsistent with some general claim(s) -- e.g. "my cat Tom has no hair" is a counterexample to the general claim "all cats have hair". Contradictions are usually a bit broader and apply to any set of inconsistent sentences -- e.g. assuming there's at least one cat, "all cats have no hair" contradicts "all cats have hair". As far as I'm aware, you won't really need to understand the nuances and can treat the two terms synonymously for LSAT purposes.
A counter example is a type of contradiction. A contradiction doesn't have to be a counter example.