Planet Paradox

Relevant wiki: Propositional Logic

Let's define statements $P$ and $Q$ as:

• Statement $P :=$ "If this statement is true, then our planet Earth is bigger than the planet Jupiter."

• Statement $Q :=$ "Earth is bigger than Jupiter"

We can read statement $P$ as "If $P$ , then $Q$ " which can be written as $P \implies Q$ in proposition; logic. Therefore $P$ is equivalent to $P \implies Q$ . Thus, if $P$ is true, then $P \implies Q$ must be true, and if $P$ is false, then $P \implies Q$ must be false.

When we construct a truth table, we see that this only happens when $P$ , $Q$ and $P \implies Q$ are all true.

$\begin{array}{|c|c|c|} \hline P & Q & P \implies Q \\ \hline \color{#20A900}{T} & T & \color{#20A900}{T} \\ \color{#20A900}{T} & F & \color{#D61F06}{F} \\ \color{#D61F06}{F} & T & \color{#20A900}{T} \\ \color{#D61F06}{F} & F & \color{#20A900}{T} \\ \hline \end{array}$

This leads us to believe that $Q$ is true. From the given sentence, we interpret that the statement "Earth is bigger than Jupiter" is true. This is a paradoxical result since a quick look at the facts tells us that Jupiter is in fact bigger than Earth. $_\square$

The conditional is either True or False.

If it is True, then the precedent is true, and hence the antecedent is true. Therefore, the earth is bigger than jupiter.

On the other hand, if it is false, then by definition of conditionals, the precedent is true and the antecedent is false, but this cannot be since the precedent requires the entire conditional to be true, for itself to be true.

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There is clearly something wrong here. Using this paradox, we can prove contradictions.

This is the Curry Paradox. There is no universal consensus among logicians/philosophers on how to resolve this.