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

Question

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. — , Aug 02 '17 at 10:39
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. — Our, Aug 02 '17 at 10:49
How did Crete become Sicily here? (I suspect Vizzini may be to blame). — hmakholm left over Monica, Aug 02 '17 at 11:37
@onurcanbektas I don't think it's the statement “On Silicy everyone lies” that is problematic since it just corresponds to define “A is Sicilian” as “for each sentence A says, it is false”. This paradox can be simplified as just “This sentence is false.”. — Dabouliplop, Aug 02 '17 at 12:26
Also, I think the OP mixed the two paradoxes “That sentence is false.” with “A Sicilian says 'I am from Silicy'". — Dabouliplop, Aug 02 '17 at 12:27
You have to define what it means to say "I lie". Does it mean that the person is lying when speaking such phrase? Or that the speaker lies frequently? Or does he/she lies all the time? — WeakestTopology, Aug 02 '17 at 12:36
@AsafKaragila: But do you think it doesn't mean what you think I think it means? — hmakholm left over Monica, Aug 02 '17 at 12:41
@cpassos: "On Silicy everyone lies", I assume this means all the time. — , Aug 02 '17 at 14:17

score 4 · Accepted Answer · 2017-08-02T12:26:17.790

4

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.

edited Aug 02 '17 at 12:26

answered Aug 02 '17 at 12:16

“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. – Dabouliplop Aug 02 '17 at 12:25
@Idéophage: there are no undecidable propositions. Propositions belong to the zero-th order logic. – Aug 02 '17 at 12:27
Ok, I confused with a statement, it seems. – Dabouliplop Aug 02 '17 at 12:34

score 1 · Answer 2 · answered Aug 02 '17 at 10:41

1

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".

answered Aug 02 '17 at 10:41

Matt

3,296

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

2 Answers2