“This statement is false.”

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  • “This statement is false” – Propositional Logic

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Statements are very specific type of strings which are constructed recursively:

  1. If $p$ is an atomic proposition, then $p$ is a statement.
  2. If $p,q$ are statements, then: $p\land q$, $p\lor q$, $p\to q$ and $\lnot p$ are also statements.

If you try to construct $p$ which states “$p$ is true” or “$p$ is false” then you quickly realize this is not a well-formed formula.

And just like in natural language, not every string of characters has any actual meaning. The vile bile danced for a while, as the liver was guile.

It’s a paradox, meaning any interpretation of it’s truth value results in a contradiction. The statement is neither true nor false. If “this statement is false” were false, then it would be true, contradicting the assumption that it’s false. Similarly if it were true. It can be shown using the rules of logic that a contradiction $A \wedge \neg A$ implies $B$, for any proposition $B$. Hence when constructing a formal logical system we want to guarantee that contradictions are excluded, and therefore define propositions in ways that do not allow for the kind of paradox you mentioned.

You can use propositional logic to prove this statement is not a proposition:

Suppose the statement, $S$, “This statement is false”, is a proposition

That is $S\iff\lnot S$

Then $S$ is true or false, but both of those lead to a contradiction, so our supposition is false:

“This statement is false” is not a proposition.