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Is every statement true?
Consider the following argument:
If a statement is true, then it is a member of the set of true statements.
If a statement is false, then it implies a contradiction. Since anything follows from a contradiction, it follows that the statement is true. Thus the statement is a member of the set of true statements.
Since a statement is true or false, all statements therefore belong in the set of true statements. All statements are true, with the set of false statements being a subset of the set of true statements. A statement thus is either true and true only, or both true and false.
Does this mean that all statements are true?

Is every statement true?
Consider the following argument:
If a statement is true, then it is a member of the set of true statements.
If a statement is false, then it implies a contradiction. Since anything follows from a contradiction, it follows that the statement is true. Thus the statement is a member of the set of true statements.
Since a statement is true or false, all statements therefore belong in the set of true statements. All statements are true, with the set of false statements being a subset of the set of true statements. A statement thus is either true and true only, or both true and false.
Does this mean that all statements are true?

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It is false that Cambridge is a bigger city than Oxford. But that doesn't meant that the statement that Cambridge is a bigger city than Oxford entails a contradiction. It plainly doesn't. We can all imagine a world where history went just a bit differently and Cambridge ended up the bigger city; there's no internal incoherence at all in that counterfactual story, no contradiction is entailed.

You might say, sloppily, that the claim that Cambridge is a bigger city than Oxford contradicts how things are (or some such). But contradicting how things are in this sense, i.e. being plain false, doesn't mean being necessarily false, and doesn't mean entailing a constradiction in that sense of "contradiction" in which, arguably, anything follows from a contradiction.