# Is a paradox a counterexample to $p \lor \neg p$ always being true?

Consider the following paradox:

On Silicy everyone lies all the time, and if you are not from Silicy, you speak the truth all the times. Call the proposition "I am from Silicy" $p$.

Then $p$ is not true, because then he is from Silicy, thus lies.

But $p$ cannot be false either because then he is not from Silicy, which means he always speaks true, which contradicts that $p$ is false.

Can someone explain to me where I go wrong here? My knowledge of logic is very limited: I know truth tables, the logical symbols and some tautologies such as de Morgan's laws, and that is pretty much it.

• Such self-referential phrases ($p$ is about $p$) do not fit in the framework of Boolean algebra, they are not acceptable propositions. – Yves Daoust Aug 2 '17 at 10:39
• @YvesDaoust I think your comment should be an answer – TZakrevskiy Aug 2 '17 at 10:47
• As @YvesDaoust points out, you cannot assume something that contradicts with your assumptions, so statement "On silicy everyone lies" is not a valid nor acceptable statement.I mean of course you can assume such thing, but if you do that, you can derive almost anything from it. – onurcanbektas Aug 2 '17 at 10:49
• How did Crete become Sicily here? (I suspect Vizzini may be to blame). – Henning Makholm Aug 2 '17 at 11:37
• @HenningMakholm: in Lieland, Sicily is called Silicy. – Yves Daoust Aug 2 '17 at 12:15

## 2 Answers

Such self-referential phrases ($p$ is about $p$) do not fit in the framework of Boolean algebra/propositional logic, they are not acceptable propositions.

A proposition must have a well-defined truth value, either $\text{true}$ or $\text{false}$, and $\text{true}\lor\lnot\text{true}$, $\text{false}\lor\lnot\text{false}$ do hold.

• “A proposition must have a well-defined truth value.” I think it is questionable, there are undecidable propositions. But every mathematical object must be well defined, of course, and in general a definition cannot refer to the object being defined. – Idéophage Aug 2 '17 at 12:25
• @Idéophage: there are no undecidable propositions. Propositions belong to the zero-th order logic. – Yves Daoust Aug 2 '17 at 12:27
• Ok, I confused with a statement, it seems. – Idéophage Aug 2 '17 at 12:34

Proposition $p$: Someone from Sicily says "I lie". Your first assertion is correct, this cannot hold if we assume that everyone on Sicily lies.

Therefore, $p$ is false. Someone from Sicily, in this case, did not say "I lie".