Decisions, Decisions

Decisions, Decisions

Neil J. Gunther

Created Jan 13, 2009
Updated Oct 7, 2010

Contents

1  The Proposition
2  Logic Notation
3  Truth Evaluation
    3.1  Antecedent
    3.2  Consequent
    3.3  Implication
    3.4  Generalizing
4  Interpretation
5  Implication Redux
6  Conclusion
7  Postscripts

1  The Proposition

Consider the following aphorism attributed to Herodotus in 500 B.C. [1]:
"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 is good even though the consequences are bad, if the evidence for the deciusion was good and conversely, a decision is bad even though the consequences be good, if the evidence for taking that decision was bad.

2  Logic Notation

Using the conventional notation for logical connectives in Table 1,
Table 1: Logic notation
 Symbol   Name   Meaning 
∧ 'and'  Conjunction 
∨ 'or'  Disjunction 
¬ 'not'  Negation 
⊃  'hook'   Implication 
and the symbols in Table 2
Table 2: Logic statements
  C   the consequences are good
 ¬C   the consequences are bad
  D   the decision was good
 ¬D   the decision was bad
  E   the evidence was good 
 ¬E   the evidence was bad 
Then Proposition 1 contains two pieces, which can be reexpressed as:
Proposition 2
  1. If the evidence was good and the consequences bad, then the decision is good

  2. If the evidence was bad and the consequences are good, then the decision is bad

Using more of the symbols in Tables 1 and 2, we can convert each piece to:
H1
=  if (E ∧¬C) then D
(1)
H2
=  if (¬E ∧C) then ¬D
(2)
but the if-then combination corresponds to implication in Table 1, so we can write:
H1
=  (E ∧¬C) ⊃ D
(3)
H2
=  (¬E ∧C) ⊃ ¬D
(4)
The original aphorism by Herodotus is the conjunction of these two propositions:
H = H1 ∧ H2
(5)
The conjunction H is true only when both pieces are true. Therefore, we can examine the truth functionality of either H1 or H2 separately in order to understand the truth functionality of (5).

3  Truth Evaluation

Consider just H1 and associate "good" with logically True and "bad" with logically False.

3.1  Antecedent

Let's denote the antecedent in (3) by A = E ∧¬C and examine its truth table (TT).
Table 3: Truth table for the antecendent A
E C ¬C E ∧¬C
1 True True False False
2 True False True True
3 False True False False
4 False False True False
We see from Table 3 that A is only True in one case (row 2), viz., when E and ¬C are both True. That, of course, was the intention behind the original aphorism.

3.2  Consequent

H1 in (3) is a logical implication A ⊃ D where D is the consequent.
The general truth functionality for implication is shown in Table 4.
Table 4: Truth table for implication
p q p ⊃ q
1 True True True
2 True False False
3 False True True
4 False False True
The implication is true in every case except for row 2.
Since the antecendent A can be either True or False, we compare it with all possible combinations of truth values for the consequent D. The result is a TT for H1 that is identical to Table 4.
Table 5: Truth table for H1
A D A ⊃ D
1 True True True
2 True False False
3 False True True
4 False False True
Row 1 of Table 5 corresponds to (1) rewritten as:
H1 = If A then D
(6)
which is the first part of Proposition 2.1. In English:
Proposition 3 If A is good, then the decision is good
H1 is True when both the antecendent A and the consequent D are True.
But H1 is also True in rows 3 and 4, where the antecendent A is False and independent of the truth of D. What does this mean?

3.3  Implication

Technically, H1 only pertains to the case where D is good; the case of a bad decision (row 4) being covered by H2. So, row 3 could be written in English as:
Proposition 4 If A is bad, then the decision is good
which does not seem to correspond to what Herodotus intended.
Let's expand the case of the false antecendent. A false antecendent ¬A = ¬( E ∧¬C ), is logically equivalent to ¬A = ¬E ∨C. Notice that distributing the negation causes the and to become an or. In words, row 3 says:
Proposition 5 If either the evidence is bad or the consequences are good, the decision is good.
Only one of either the evidence or the consequences has to be good for the decision to be good.
In summary, both of these statements: are identically true, even though it may not seem like it when they are expressed in English.
In Section 5 we attempt to clarify this situation by reconsidering the logic of an implication as a ternary operator. Jumping ahead, we can rewrite (6) as:
H1 = if A then D else X
(7)
where X stands for any (undefined) proposition that can be either true of false. If the antecendent A is True, then H1 gets the logical value of D (True or False). This is equivalent to rows 1 and 2 of Table 5. Herodotus was referring to the case where D is True.
If the antecendent A is False, then H1 gets the logical value of X. This is equivalent to rows 3 and 4 of Table 5. As we explain in Section 5, the intent is to allow anything to be possible, so X is always assigned a value of True. Therefore, even A is False (i.e., ¬A is True), H1 gets the logical value True.

3.4  Generalizing

We ask, under what truth values of C, D, E, is H true? Since there are 3 propositions, each of which can be True or False, we need a TT with 23=8 rows.
Table 6: Truth table for H
E C D H1 H2 H
1 True True False True True True
2 False True False True True True
3 True False False False True False
4 False False False True True True
5 True True True True True True
6 False True True True False False
7 True False True True True True
8 False False True True True True
Since H is False in rows 3 and 6 of Table 6, we can eliminate them.
Table 7: Reduced truth table for H
E C D H1 H2 H
1 True True False True True True
2 False True False True True True
3
4 False False False True True True
5 True True True True True True
6
7 True False True True True True
8 False False True True True True
From this its clear that for H to be True, both H1 and H2 must be True because they are and-ed together and therefore, if either sub-statement were False, then H would be False.

4  Interpretation

What does Table 7 say in English?
  1. H is true when the evidence was the best available, the consequences were the happiest but the decision was unwise.
    This looks contradictory.

  2. H is true when the evidence was the worst, the consequences were the happiest but the decision was unwise.
    Corresponds to H2.

  3. Eliminated.

  4. H is true when the evidence was the worst, the consequences were the unhappiest but the decision is unwise.
    This is counterintuitive because the expectation is that H should be False.

  5. H is true when the evidence was the best available, the consequences were the happiest and the decision is wise.
    This seems obvious and, although it is not wrong, it is not what Herodotus was trying to say.

  6. Eliminated.

  7. H is true when the evidence was the best, the consequences were the unhappiest and the decision is wise.
    Corresponds to H1.

  8. H is true when the evidence was the worst, the consequences were the unhappiest but the decision is wise.
    This looks contradictory.

Let's remove everything that appears satisfactory and leave only those rows that still look strange.
 1.   H is true when the evidence was best, the consequences were the happiest but the decision was unwise. 
 4.   H is true when the evidence was worst, the consequences were the unhappiest but the decision was wise. 
 8.   H is true when the evidence was worst, the consequences were the unhappiest but the decision was wise. 
Notice that in rows 1 and 4 of Table  7, D is False. Formally speaking, 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 have to remind ourselves about the truth functionality of the implication connective in the last row of Table 2.

5  Implication Redux

For material implication, p ⊃ q is True in all cases except when the consequent q is False (i.e., row 2 in Table 4). That seems a bit strange 1. Moreover, how can p ⊃ q be True when both the antecedent (p) and the consequent (q) are False?
The way I like to view the values in Table 4 is to think of p ⊃ q as being represented by a ternary operator:

If p Then q Else Anything
(8)
In C, Perl, and some other computer programming languages, such a ternary operator exists and is written as

v = p ? q : Anything;
(9)
where the variable v is the value of the conditional statement itself. In Mathematica (used here to generate the TTs), the If function

v = If [p, q, Anything];
(10)
can be used as a ternary-valued operator.
From (10) it should be clear that when p is True, the expression v gets the truth value of q. Otherwise, we ignore q and evaluate the Else branch which contains Anything. The question remains, what truth value do we assign to Anything? The intent here is to allow that Anything is possible (even if it seems logically absurd in English), so we always assign it a value True, or `1' in the case of equation (9).

Figure 1: Venn diagram for p ⊃ q is the dark blue region
If the concept of "Anything" still seems shady, consider the Venn diagram for p ⊃ q in Figure 1. There, the members of p are shown in red and q in yellow. What about ¬p? That is shown in green (the complementary color of red) and, of course, includes everything that is not a member of p viz., everything else in the universal set (the rectangle); including some the members of q. Let's see how this matches up with equation (8).
If p is True, then the truth value of the implication is determined soley by q, the yellow set in Figure 1. Notice that there is only a small overlap with the members of p. That means q could be True for reasons other than p being True. See row 3 in Table 4. When p is False, that is equivalent to ¬p or the green set. The green set is everything that is not a member of p or everything else in the universe. Equivalently, we can say anything else is true.
The complete truth functionality of p ⊃ q must also include the case where q is True (yellow). In a Venn diagram, this corresponds to the union of the set for the complement of p with the set for q, and this combined set is shown as the blue region (the other complementary color) in Figure 1. Furthermore, using set-theoretic notation, we can write the union of ¬p and q as ¬p ∪ q. The ∪ symbol resembles the ∨ symbol in Table 2 in that they both point downward; at least, that's how I like to remember it. This suggests making the replacement: ¬p ∪ q → ¬p ∨ q, which leads to the logical equivalence:
p ⊃  q ≡ ¬p ∨ q
(11)
And you can see that eqn.(11) is correct by watching the color changes in Figure  1. The TTs are also equivalent.
This view of p ⊃ q is completely consistent with the way computer programs work. They use Boolean logic and that's all formal logic reflects. That's why I prefer this explanation rather than those that use example propositions expressed in English (or any spoken language, including the Greek of Herodotus). More often than not, such semantic examples are confusing because they tend to reinforce the paradoxical appearance of p ⊃ q, rather than explain it. Nonetheless, if you don't like my explanation, try this.

6  Conclusion

Returning now to Herodotus, we can break out the subcomponents of H1 and H2 for the strange looking cases.
Table 8: Expanded TT for H1 and H2
E C (E∧¬C) D H1 (¬E ∧C) ¬D H2
1 True True False False True False True True
2
3
4 False False False False True False True True
5
6
7
8 False False False True True False False True
We see that the antecedent (E∧¬C) is always False, so the value of D is ignored in favor of Anything in the Else branch, and thus H1 is True. Similarly, the antecedent (¬E ∧C) is always False, so the value of ¬D is ignored in favor of Anything, and thus H2 is True. We can summarize these points in Table 9.
Table 9: Final TT for H
E C D H1 H2 H Reason
1 True True False True True True Anything and ¬D
2 False True False True True True Herodotus
3
4 False False False True True True  Anything and Anything 
5 True True True True True True Obvious truism
6
7 True False True True True True Herodotus
8 False False True True True True D and Anything
So, the proposition H is more broadly valid than Herodotus intended because of the way implication truth functionality works.

7  Postscripts

  1. Let's not assume that everything is now completely tidy with regard to Herodotus or implication. Looking at Table 4 once again, I can construct a special case by replacing p with (p ∧¬p). The implication now reads

    (p ∧¬p) ⊃ q
    		(12)
    But (p ∧¬p) is a contradiction and therefore always False. Using equation (8) we can interpret (12) as
    Anything follows from a contradiction.
    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.

  2. Let's test our understanding of all this. Consider the statement:
    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.
    Table 10: Propositions
     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 
    Here, we have chosen the symbol "A" to stand for "anything." Since it is not possible for Bush to hold office for more than two terms (legally), we can be quite sure that proposition p is False. According to equation (8), since p is False we can ignore q altogether. The truth of proposition q is irrelavent to the truth function of the statement. By the same token, however, that means A = "there is life on Mars" must be True! But we really don't know that. It seems that, from a logical standpoint, A could be either True or False. According to equation (8), however, the Else branch should be True; always! What's going on here?
    We can remove this problem by drawing on the same device we used in Postscript 1. There, we noted that a contradiction is always False. Conversely, a tautology is always True. Therefore, if we replace A by "there is life on Mars or there is no life on Mars" then the Else branch must always be True. More generally, we can write:

    If p Then q Else (A ∨ ¬A)
    		(13)
    and now, the interpretation of Section 5 and the propositions in Table 10 are completely consistent. This is the correct logical encoding of Anything is possible.

References

[1]
D. S. Shiva, Data Analysis: A Bayesian Approach, Clarendon Press (1997)
See e.g., p. 2.
[2]
D. J. Bennett, Logic Made Easy, Norton (2004)
[3]
A. Hodges, Alan Turing: The Enigma, Walker & Company (2000)

Footnotes:

1Deborah Bennett [2] points out that entire theories and countless papers have been written about how people (i.e., non-logicians) reason using the word if.

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On 7 Oct 2010, 17:39.