red75 comments on Unsolved Problems in Philosophy Part 1: The Liar’s Paradox

[red75]

Yablo’s version looks like unrolled infinite loop of function

s :: Bool
s=not s

[cousin_it]

Not to me it doesn’t. Yablo’s version has a “forall” that your translation misses. So in Yablo’s version there’s no consistent way to assign truth values to S(n), but in your version we could make S(n) = “n is odd” or something.

[red75]

Not exactly. My version is incorrect, yes. But there is, uhm, controversial way of consistent assignment of truth values to Yablo’s statements.

In my version n-th step of loop unrolling is

S'(n) = not not ... {n times} ... S

or

S'(n)=not S'(n+1)

Yablo’s version

S(n)=not exists m>n such that S(m)=true

or

S(n)=(not S(n+1)) && (not exists m>n+1 such that S(m)=true)

If we extend set of natural numbers by element omega such that

forall n in N : (omega>n),
not exists n in N : (n+1=omega),
omega=omega+1

Than we can assign S(n)=false for all n in N, and S(omega)=true.

Edit: Oops, second version of Yablo’s statement, which I included to demonstrate why I had an idea of loop unrolling, is not consistent when n equals omega. Original Yablo’s statement is consistent although.

Edit: Meta. The thing I always hated about my mind is that it completely refuses to form intuitions about statements which aren’t directly connected to object level (but then what is object level?).

Edit: Meta Meta. On introspection I don’t feel anything about previous statement. Pretty damn consistent...