A proof that penguins rule the universe

This is a version of Curry's Paradox.

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• We start with the following sub-argument.
• Premise, just for this sub-argument: "If A is true, penguins rule the universe". Call this premise A.
• Suppose A is true. Then:
  • If A is true, penguins rule the universe. (This is just A with the quotation marks removed, which is OK because for the moment we're assuming A.)
  • So penguins rule the universe (still supposing A). This is the conclusion of the sub-argument (still supposing A).
• That's the end of the sub-argument.
• In the sub-argument, we've shown that if A is true then penguins rule the universe.
• But that's what A says.
• So A is true.
• But if A is true, penguins rule the universe.
• So penguins rule the universe.

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There are no special tricks involved in this. The question of how best to avoid this paradox is still an open question among logicians.

An obvious solution would be to disallow premise A. But then it's hard to say what's wrong with it. One option is to say that logic has to conform to Bertrand Russell's theory of types (or simplifications of that, of which the most famous is Quine's system called New Foundations), but that solution is contentious. After all, if we can get from a premise that doesn't claim anything definite about penguins to a conclusion that does, using only the laws of logic, you'd think we would blame the laws of logic rather than the premise.

But solutions to Curry's paradox that revise the laws of logic (instead of disallowing premise A) are very hard to find, and even more contentious than the theory of types.

See the Stanford Encyclopaedia of Philosophy entry by J.C. Beall (http://plato.stanford.edu/entries/curry-paradox) for more details.