Pinocchio Paradox Propositionally Debunked

Dear Mr. Supposed Philosopher:


My nephew taunts that were Pinocchio to say me, "Your nose will grow," a paradox would ensue.  I said, "Nuh Uh."  But, secretly, I'm not so sure.  Is he really that smart, or is he just bluffing?


Signed,
Roundly Razzed
Batavia, OH

Dear R:

I'm going with he's bluffing.  Here's why:


P =def. "Pinocchio"
R =def. "Roundly Razzed"


00. P says, "R's nose will grow."


Theorem 1:
01. If P lies, then P's nose grows. (premise)
02. If P doesn't lie, P's nose doesn't grow. (premise)
03. P lies or P doesn't (tautology)
04. So, either P's nose grows, or it don't. (1,2,3 ; CD)


Theorem 2:
05. If R's nose grows, P's does not. (premise of hypothetical effect)
06. If R's nose don't grow, then P's does. (premise of hypothetical effect)
07. R's nose grows or R's nose doesn't. (tautology)
08. So, either P's nose grows or it don't (5,6,7; CD)


Theorem 3:
09. If R is a physically maturing person, R's nose will grow. (premise)
10. If R's nose will grow, then P doesn't lie. (premise)
11. So, If R is still a physically maturing person, then P doesn't lie. (9,10 HS)


Main Argument:
12. R is still a physically maturing person. (premise)
13. Thus, P doesn't lie.  (11,12; MP)
14. Hence, P's nose doesn't grow. (2,13; MP)
15. Therefore, R's nose still does, though.(6,14; MT)

Analysis: No contradiction found. Thus, no paradox detected.

O.

REFERENCES:

Ray Wiseman Logic and Proofs (Accessed 11/29/2010) - gives a summary of the rules of propositional logic.