Paradox

Although we say that every statement is either true or false, there are stange statements that are either both or neither. A paradox is a statement that makes a tautology false, or a contradiction true.

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As a first example, we make a contradiction true. The following statement is a contradiction: \[P \leftrightarrow \neg P\] This can be verified with a truth table.

The following is a paradox: If \(P\) is true, then \(P\) is false. So \(P \rightarrow \neg P.\) On the other hand, if \(P\) is false then \(P\) is true. So, \(\neg P \rightarrow P.\) Combining the two statements, \[(P \rightarrow \neg P) \wedge (\neg P \rightarrow P) = P \leftrightarrow \neg P\] So, the statement \(P \leftrightarrow \neg P\) is true for \(P.\)

Definition of \(\neg\)

Definition of \(\rightarrow, \leftrightarrow\)

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In this example, we will not break down the logic. Suzy is taking a multiple choice test, and she closes her eyes and guesses answers at random. One of the question on the test is the following:

"What is the probability that Suzy gets this question right?"
a. \(25\)%
b. \(50\)%
c. \(25\)%
d. \(0\)%

The answer cannot be \(25\)%, because \(25\)% appears as 2 answer choices. That would give Suzy a \(50\)% chance of being right. But, the answer cannot be \(50\)% because then she would only have a \(25\)% chance of being right. So, maybe there is no way to get it right. But, the answer can't be \(0\)%, because then she has a \(25\)% chance of being correct.

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The takeaway is that statements of truth cannot always be determined to be true or false. In fact, there are questions in any system of logic that cannot be answered.