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In the exception paradox, the statement "every rule has an exception" leads to a contradiction. If every rule has an exception (this is the false premise), then there must be an exception to the rule that every rule has an exception.
From the logical point of view, this can be taken as a proof that the sentence "every rule has an exception" is false - a simple example of a proof technique known as reductio ad absurdum. More formally,
- Every rule has an exception. (Statement)
- "Every rule has an exception" is a rule, which has an exception. (by 1)
- There exists some rule R without exception. (by 1. and 2.)
$ P \Rightarrow Q \equiv \lnot Q \Rightarrow \lnot P $.
- Since R is a rule, by the first statement it must have an exception. But as stated in 3, it does not have an exception - an apparent contradiction. Since 3. is the negation of 1., there is a contradiction.
However, as every rule has an exception, that would include the statement "every rule has an exception" as it is it's own exception.
Yet again, the rule may be true, as well as false; in particular, if the rule R is "Every rule has an exception". Such a rule has no domain to restrict it, and so its truth and falsehood need not conflict, as they do not compete on any domain.Template:Fact
Another way of looking at The Exception paradox:
Rule of exception: Every rule must have an exception.
So there is no rule without exception. In other words: there is no exception to the rule of exception. (which is according to the rule of exception, since it states every rule 'must have'). Thus the above rule is obvious contradiction.
Further from above, There is no exception to the Rule: Every rule must have an exception, which is the exception to the rule: Every rule must have an exception.
Hence wording it finally, The only exception to the rule of exception is: there is no exception to the rule of exception.
Apparently, as per the above statement, even though it has an internal contradiction, it has the overall agreeance with the rule of exception at the same time.
Variations on the Paradox Edit
- "All generalizations are false, including this one" - Mark Twain
- The liar paradox has similar self-reference, with the added twist that rejecting it leads to another paradox.
- If everything is possible, then it is possible for anything to be impossible.
- The only rule is that there are no rules. Ignore all rules
- If everything has an opposite, then the opposite of there being an opposite to everything, is that there is not an opposite to everything.
- The only thing certain is that there is nothing certain, or "Nothing is certain.".
- "Moderation in all things, including moderation", a quotation sometimes attributed to Petronius. If everything should be taken in moderation, then moderation should itself be taken moderately, meaning that not everything would be taken in moderation.