Is every statement true?

Is every statement 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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