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Logic

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.

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.

\(P\)

T

F

T

F

\(\neg P\)

F

T

F

T

\(P \leftrightarrow \neg P\)

F

F

F

F

\(P :=\) "This statement is false."

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 \(P.\)
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.

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.

Just for fun:

If I ask you this question, will you answer no? (Don't lie!)