by Brian Bosse, Copyright March 13, 2008, all rights reserved. 36 views
The following is taken from Logic - Techniques of Formal Reasoning, pg. 103.
According to ancient legends the Oracle of Delphi was infallible. The Oracle was once asked whether any of the triumvirate of Pompey, Caesar, and Crassus would be assassinated. The Oracle gave a two-sentence reply:
More than one of the triumvirate will be assassinated. Crassus will be assassinated and either Caesar will not be assassinated or Pompey will be assassinated, if and only if, Pompey will not be assassinated and either Crassus or Caesar will be assassintated.
Question: According to the Oracle, who among the triumvirate will be assassinated?
Answer:
P = "Pompey will be assassinated."
Q = "Caesar will be assassinated."
R = "Crassus will be assassinated."
The second sentence of the Oracle can be symbolized as follows:
(R ∧ ((¬Q ∨ P) ↔ (¬P ∧ (R ∨ Q))))
Since we are assuming that this statement is true and because the statement is a bi-conditional, then we know each part of the bi-conditional has the same truth value. Let's consider the case where each part of the bi-conditional is true.
Case 1:
((R ∧ (¬Q ∨ P)) ↔ (¬P ∧ (R ∨ Q)))
T T T T F T T T T T F
This yields that only Crassus will be assassinated. However, we know from the Oracle's first sentence that there will be at least two assassinations. As such, case 1 fails. We now turn to case 2 where each part of the bi-conditional is false.
Case 2:
((R ∧ (¬Q ∨ P)) ↔ (¬P ∧ (R ∨ Q)))
F T F
At this point we need to consider further cases. Let's assume that R is true, and call this case 2a.
Case 2a:
((R ∧ (¬Q ∨ P)) ↔ (¬P ∧ (R ∨ Q)))
T F F F T T F F T T T
We end up with a contradiction. (¬Q ∨ P) is false, but 'P' is true. The only time a disjunction is false is when both parts of the disjunct are false. As such, case 2a fails. This means 'R' must be false. 'R' being false simply means that Crassus will not be assassinated. Since the Oracle told us at least two would be assassignated and we know that Crassus is not one of the two, then we can conclude that Caesar and Pompey will be assassignated. Q.E.D.
-----No comments yet.