If we define a lie, L, as any false claim contrary to the claimant's belief then:http://en.wikipedia.org/wiki/Eubulides#Paradoxes_of_Eubulides wrote:A man says: "What I am saying now is a lie." If the statement is true, then he is lying, even though the statement is true. If the statement is a lie, then he is not actually lying, even though the statement is a lie. Thus, if the speaker is lying, he tells the truth, and vice versa.
∀x: L := ¬x ∧ C(x) ∧ B(¬x)
If the claim, x, is "what I am saying now is a lie" then the application of the above definition is:
L := ¬L ∧ C(L) ∧ B(¬L)
A lie is being partly defined as not a lie: this is incoherent. Therefore, if the claim, x, is "what I am saying now is a lie" then it cannot be a lie, and if it is not a lie then it is false. This then shows that a lie cannot be defined as any false claim contrary to the claimant's belief as there is at least one false claim that, even if contrary to the claimant's belief, necessarily is not a lie:
∃x: ¬(L := ¬x ∧ C(x) ∧ B(¬x))
Not every falsehood is a lie.
A possible counter-claim is that a lie need not be false; that a lie is simply defined as any claim contrary the claimant's belief (if I believe that there is another beer but tell you that there isn't then I am lying even if I am wrong and there is another beer):
∀x: L := C(x) ∧ B(¬x)
If the claim, x, is "what I am saying now is a lie" then the application of the above definition is:
L := C(L) ∧ B(¬L)
If the claimant believes that what he is saying now is not a lie then the above definition is satisfied. Therefore, if the claim, x, is "what I am saying now is a lie", then it can be a lie, and if it is a lie then it is true.
Not every lie is a falsehood.
A similar line of reasoning can be used to solve the Pinocchio paradox:
Premise 1. Pinocchio's nose grows if and only if he claims any falsehood
Premise 2. Pinocchio claims "my nose grows now"
If his claim is true then his nose will not grow as it only grows if he claims any falsehood; but his claim is that his nose will grow and so if true his nose will grow. This is a contradiction. If his claim is false then his nose will grow as it grows if he claims any falsehood; but his claim is that his nose will grow and so if false his nose will not grow. This is a contradiction.
If Pinocchio's nose grows, G, if and only if he claims any falsehood then:
∀x: C(x) ∧ ¬x ↔ G
If Pinocchio's claim, x, is "my nose grows now" then the application of the above premise is:
C(G) ∧ ¬G ↔ G
Pinocchio's nose grows if and only if his nose doesn't grow: this is a contradiction. Therefore, if Pinocchio's claim, x, is "my nose grows now", then his nose doesn't grow, and if his nose doesn't grow then it is false. This then shows that Pinocchio's nose does not grow if and only if he claims any falsehood as there is at least one falsehood that, even if claimed, necessarily will not cause his nose to grow:
∃x: C(x) ∧ ¬x → ¬G
The first premise is shown to be false; proof by contradiction.
