We are dealing just in the scope of classical (two-valued) logic and there a statement like "This statement is false" is just without a truth-value as far as I know. — Pippen
So not having a truth value is the third option. If we have the conjunction p ∧ q and if p is false and q doesn't have a truth value then the conjunction as a whole is false. — Michael
Not in classical logic. Not having a truth value is not a third option there. I think 2. of my proof is dubious, maybe nagase will check that out. — Pippen
The logicians formulate "All (S)tatements are (F)alse" as follows: All x: (Sx -> Fx). If you do that you can indeed prove that this is just plain false since if it's true it's a contradiction and so by RAA it's its negation that is consistent.
But why do I have to formulate the statement like above? Why can't I just formulate: All x: (Sx & Fx). This statement is not false, it is not well formed since it entails the liars paradox.
Both versions of the upper statement say roughly the same - that every x in the set of statements is false - but their form is different and so are their results. So who is right? Or why am I wrong? — Pippen
(All statements are false) is false, and it is equivalent to
(Some statements are true). — Owen
Because ∀x(Sx→Fx) and ∀x(Sx∧Fx) don't say the same thing. — Srap Tasmaner
They're not. Whether you want to say "all statements are false" is false or "all statements are false" is neither true nor false, it is still the case that "all statements are false" isn't true and "at least one statement is true" is true. — Michael
And it's a huge difference if a statement is false or not false (no matter if it's true or not besides that). — Pippen
In summary, my problem is why nobody interprets the statement "All sentences are false" as "All sentences are false and this very sentence is false", because in this version the whole thing wouldn't be true or false. I just don't see an error in infering one from the other. — Pippen
In formal logic directly self referential statements like the one you showed do not exist and can not be defined. — Meta
So you're thinking that since you have, in essence, "[the Liar] & P" as your conjunction, we'll be unable to construct a truth table because the first conjunct is not truth-apt. True. — Srap Tasmaner
If your point is only that this sentence can't be represented within classical logic, then duh. — Srap Tasmaner
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