"A decision was wise, even though it led to disastrous consequences, if the evidence at hand indicated it was the best one to make, and a decision was foolish, even though it led to the happiest possible consequences, if it was unreasonable to expect those consequences."We would like to evaluate its truth functionality. To make that task easier, let's simplify it slightly as: Proposition 1 A decision was good even though the consequences were bad, if the evidence for the decision was good, and a decision was bad even though the consequences were good, if the evidence for taking that decision was bad.
Symbol  Name  Meaning 
∧  'and'  Conjunction 
∨  'or'  Disjunction 
¬  'not'  Negation 
⊃  'hook'  Implication 
C  good consequences 
¬C  bad consequences 
D  good decision 
¬D  bad decision 
E  good evidence 
¬E  bad evidence 

 (1)  

 (2) 

 (3)  

 (4) 
 (5) 
 (6) 
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E  C  ¬C  A = E ∧ ¬C  
1  True  True  False  False 
2  True  False  True  True 
3  False  True  False  False 
4  False  False  True  False 
p  q  p ⊃ q  
1  True  True  True 
2  True  False  False 
3  False  True  True 
4  False  False  True 
A  D  H_{1} = A ⊃ D  
1  True  True  True 
2  True  False  False 
3  False  True  True 
4  False  False  True 
 (8) 
 (9) 
E  C  ¬C  D  H_{1}  H_{2}  H  
1  True  True  False  False  True  True  True 
2  False  True  False  False  True  True  True 
3  True  False  True  False  False  True  False 
4  False  False  True  False  True  True  True 
5  True  True  False  True  True  True  True 
6  False  True  False  True  True  False  False 
7  True  False  True  True  True  True  True 
8  False  False  True  True  True  True  True 
1. The evidence is good, the consequences are good but the decision is bad.Notice in case 1 that the decision was bad, in spite of E and C being good. In Table 6, that state corresponds to D being False. Formally, D is the consequent in the proposition H_{1} and ¬D is the consequent in the proposition H_{2}. How can either of H_{1} or H_{2} be True if the consequent is False? To resolve this paradox, we need to examine how the truth functionality of implication in Table 4 works.
8. The evidence is bad, the consequences are bad yet the decision is good.
 (10) 
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H_{1} = (E ∧ ¬C) ⊃ D  H_{2} = (¬E ∧ C) ⊃ ¬D  H = H_{1} ∧ H_{2} 

 (15)  

 (16) 
E  C  ¬C  A ⊃ D  H_{1} = Anything  B ⊃ ¬D  H_{2} = Anything  
1  True  True  False  False False  True  False True  True 
4  False  False  True  False False  True  False True  True 
8  False  False  True  False True  True  False False  True 
E  C  ¬C  A  D  H_{1}  H_{2}  H  Reason  
1  True  True  False  False  False  True  True  True  Anything in H_{1} and H_{2} 
2  False  True  False  False  False  True  True  True  Equivalent to H_{2} 
4  False  False  True  False  False  True  True  True  Anything in H_{1} and H_{2} 
5  True  True  False  False  True  True  True  True  Apparent truism 
7  True  False  True  True  True  True  True  True  Identical to H_{1} 
8  False  False  True  False  True  True  True  True  Anything in H_{1} and H_{2} 
 (17) 
This astounding proposition was the source of a huge fight between Ludwig Wittgenstein and the younger Alan Turing, who was attending the former's lectures on the foundations of mathematics in the 1930's. Wittgenstein took the position that you can just ignore the Anything part, because p and ¬p are like two gear wheels that become stuck. Today we might say, the computer program becomes "wedged." Turing, on the other hand, found this intolerable. He complained that if you have the possibility of a contradiction (e.g., in arithmetic) and you just ignore it, how do you know that the bridge you are building won't fall down? Today we might say, the computer program produces an error or has a "bug" which goes unnoticed (until it's too late). Ultimately, Turing voted with his feet and quit attending the remainder of Wittgenstein's lectures [3]. Looked at from the standpoint of modern computers, it seems that both were right. Computer programs exhibit both these problems all the time. Moreover, in my view, this bone of contention about the implication of a contradiction may be at the root of why modern computers (the electronic circuits) are too brittle to represent "intelligence" in the way promised by AI advocates, including Turing, since the 1950's.Anything follows from a contradiction.
If George Bush is elected to a third Presidential term, I'll be a monkey's uncle, otherwise there is life on Mars.As with Herodotus, we assign the propositions as shown in Table 10.
p  George Bush will be elected to a third Presidential term 
q  I'll be a monkey's uncle 
A  there is life on Mars 
 (18) 