"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."This elaborate statement looks suspiciously like it could be:
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 |
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|
| (5) |
| (6) |
| (7) |
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 | H1 = A ⊃ D | |
1 | True | True | True |
2 | True | False | False |
3 | False | True | True |
4 | False | False | True |
| (8) |
| (9) |
E | C | ¬C | D | H1 | H2 | 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 H1 and ¬D is the consequent in the proposition H2. How can either of H1 or H2 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) |
| (11) |
| (12) |
| (13) |
| (14) |
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H1 = (E ∧ ¬C) ⊃ D | H2 = (¬E ∧ C) ⊃ ¬D | H = H1 ∧ H2 |
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E | C | ¬C | A ⊃ D | H1 = Anything | B ⊃ ¬D | H2 = 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 | H1 | H2 | H | Reason | |
1 | True | True | False | False | False | True | True | True | Anything in H1 and H2 |
2 | False | True | False | False | False | True | True | True | Equivalent to H2 |
4 | False | False | True | False | False | True | True | True | Anything in H1 and H2 |
5 | True | True | False | False | True | True | True | True | Apparent truism |
7 | True | False | True | True | True | True | True | True | Identical to H1 |
8 | False | False | True | False | True | True | True | True | Anything in H1 and H2 |
| (17) |
This astounding proposition was the source of a huge fight between Ludwig Wittgenstein and the younger Alan Turing, who was attending Wittgenstein's lectures on the foundations of mathematics in the 1930s. Wittgenstein took the position that you can just ignore the Anything part, because p and ¬p are like two gear wheels that have 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 1950s.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) |