Paradox extra credit sets for Math 70
Paradox: A Problem
Define “Russell’s set” R to be the set of all sets, and only those sets, which do not contain themselves:
R = { x | x Ï x }
In general: x Î R = x Ï x
and therefore: R Î R = R Ï R .
If RÎR = true, then RÎR = false.
If RÎR = false, then RÎR = true.
RÎR can be neither true nor false. A paradox!
Bertrand Russell illustrated this paradox with a story about the barber of a Spanish village. Being the only barber in town, he boasted that he shaves all those — and only those — who do not shave themselves.
Therefore barber shaves man = man does not shave man
Therefore barber shaves barber = barber does not shave barber
Therefore the barber shaves himself just as much as he does not shave himself. That’s a paradox.
Exercise, 1 point each: Derive paradoxes from these:
The Watchmen watch all those, and only those, who do not watch themselves.
The Counsellor counsels all those, and only those, who do not counsel themselves.
The Judge judges all those, and only those, who do not judge themselves.
The Jester laughs at all those, and only those, who do not laugh at themselves.
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Kleenean Logic: A Solution
To solve the problem of paradox, let there be a third logic value; call it ‘I’ for ‘intermediate’ or ‘imaginary’; and let it equal its own negation:
I = ~ I
Meantime let negation act the same as usual on the Boolean values ‘true’ and ‘false’:
~ T = F ; ~ F = T
In Kleenean logic, Ú (“or”) is the maximum operator, while Ù (“and”) is the minimum operator, on this linear order:
F < I < T
x Ù y = x iff x Ú y = y iff x < y
The Kleenean operators follow these four axioms:
Commutativity x Ùy = yÙx ; xÚy = yÚx
Identities x Ù T = x Ú F = x
Dominance x Ù F = F ; x Ú T = T
Recall x Ù x = x Ú x = x
Those identities imply these truth tables:
x: ~x Ù y: Ú y:
F I T F I T
F T F F F F I T
I I F I I I I T
T F F I T T T T
Call a function “Kleenean” if it can be defined from Kleenean “and”, “or”, “not”, and the three values F, I, T. They include:
x nor y = ~ (x Ú y)
x nand y = ~ (x Ù y)
x Þ y = (~ x) Ú y
x iff y = (x Þ y) Ù (y Þ x)
x xor y = (x Ù ~y) Ú (y Ù ~x)
Dx = x Þ x = x iff x = x Ú ~x
dx = x minus x = x xor x = x Ù ~x
x ÚB y = (x Ú y) Ù Dx Ù Dy
x ÙB y = (x Ù y) Ú dx Ú dy
x min y = (x Ù y) Ú (y Ù I) Ú (I Ù x)
Exercise, 1 point each: Make truth tables of these logic functions.
Exercise, 1 point each: find all solutions to these three systems of equations:
x = ~x
x = ~y
y = ~x
x = y nor z
y = z nor x
z = x nor y
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