Decisions, Decisions

Decisions, Decisions

Neil J. Gunther

Created Jan 13, 2009
Updated Jul 19, 2013

Contents

1  The Proposition
2  Logic Notation
3  Truth Evaluation
    3.1  Antecedent
    3.2  Consequent
    3.3  Implication
    3.4  Truthhood
4  Interpretation
5  Implication Redux
    5.1  Ternary Operation
    5.2  Venn Diagrams
6  Conclusions
    6.1  The Bottom Line
    6.2  Other Implications

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 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.

2  Logic Notation

A proposition is composed of statements or variables. In this case, Proposition 1 is composed of two sub-propositions which can be re-expressed as:
Proposition 2
  1. If the evidence is good and the consequences are bad, then the decision is good

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

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 or variables in Table 2
Table 2: Logic statements
  C   good consequences 
 ¬C   bad consequences  
  D   good decision  
 ¬D   bad decision  
  E   good evidence  
 ¬E   bad evidence  
we can represent Proposition 2 symbolically as:

H1
=  if (E ∧ ¬C) then D
(1)
H2
=  if (¬E ∧ C) then ¬D
(2)
Proposition 3 [Formal Logic] Since the if-then combination corresponds to the implication symbol in Table 1, expressions (1) and (2) can be written entirely symbolically as:

H1
=  (E ∧ ¬C) ⊃  D
(3)
H2
=  (¬E ∧ C) ⊃  ¬D
(4)
The original aphorism in Proposition 1 (denoted hereafter by H) is then the conjunction of these two sub-propositions:

H = H1 ∧ H2
(5)
Remark 1 [Halve the Problem] The conjunction in (5) means that H is true only when both sub-propositions H1 and H2 are true. From the symmetry of Proposition 2, it's a safe bet that we can examine the truth functionality of H1 and H2 separately in order to determine the truth functionality of H. The same or almost identical truth conditions should hold for either H1 or H2. In other words, the problem is really only half as big as it first appears.
Remark 2 [Complements] Note that H2 is not the complement of H1. The complement is the same as negation and if H2 were simply the negation of H1, then H = H1 ∧ ¬H1 in (5) would produce a contradiction. What we can say is: H2 is the complement of each of the statements in H1.

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
(6)
for H1 and examine its truth table (TT). Similarly, we can also write

B = ¬E ∧ C
(7)
for statement H2.
Table 3: Truth table for the antecedent A
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
We see from Table 3 that A is only True in one case (row 2), viz., when E and ¬C are both True.
Remark 3 Having the antecedent A be True was quite likely the key assumption underlying the construction of the original aphorism in Proposition 1, irrespective of whoever actually came up with it.

3.2  Consequent

H1 in (3) is a material implication, which can be written as A ⊃ D in formal logic where D is the consequent. (See Section 5 for more about implications.) 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 antecedent 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  H1 = A ⊃ D 
1 True True True
2 True False False
3 False True True
4 False False True
In English, row 1 of Table 5 corresponds to (1) re-expressed as:

H1 = If A then D
(8)
which is the first part of Proposition 2. In English:
Proposition 4 [Row 1] If A is good, then the decision is good
H1 is True when both the antecedent A and the consequent D are True.
But H1 is also True in rows 3 and 4, where the antecedent A is False and independent of the truth of D. How can that be?

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 5 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 antecedent. A false antecedent ¬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 6 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 (8) as:

H1 = if A then D else X
(9)
where X stands for any (undefined) proposition that can be either true of false. If the antecedent 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 antecedent 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 when A is False (i.e., ¬A is True), H1 gets the logical value True.

3.4  Truthhood

What can we do with this analysis?
  1. We can examine what values of the variables make H true.

  2. We can see if H can be reduced to a simpler expression.

  3. We can see what H looks like in disjuctive normal form (DNF), conjuctive normal form (CNF), etc.

  4. We can consider other representations such as, Venn diagrams and boolean logic circuits.

Let's start with the first of these. What values of the variables C, D, E make 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 ¬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
Since H is not True in every row of Table 6 it is a contingent proposition. If were True in every row, H would be a tautology. If it were False in every row, H would be a contradiction. Since 2 of the rows are False, H has a probability density 6/8 or 75% chance of being True. Compare that with Table 5 for H1, which has a probability density 7/8 or 87.5% chance of being True on its own. Similarly for H2. In that sense, H has a stronger truth value due to the conjuction of H1 and H2.
Because we are interested in truthhood here, and H is False in rows 3 and 6, we can ignore those cases. This follows from the fact that for H to be True, both H1 and H2 must be True because they are and-ed together. Therefore, if either sub-proposition were False, then the overall proposition H would be False. That is the case in rows 3 and 6.

4  Interpretation

What does Table 6 say in English? Taking into account the truth values of E, C and D respectively, H1 is True in the following rows:
  1. The evidence is good, the consequences are good but the decision is bad.
    In this row, the variables have the values E = True, C = True and D = False and make H = True. This looks contradictory. How can evidence and consequences that are both good be consistent with a bad decision?
    Referring to Table 5, a bad decision (i.e., D = False) can make H1 = False. (see row 2 of Table 5) Similarly for H2. And that suggests H should be False. But, as we can see, H1 and H2 are both True in row 1.

  2. The evidence is bad, the consequences are good but the decision is bad.
    This statement is consistent with sub-proposition H2.

  3. Ignore.

  4. The evidence is bad, the consequences are bad and the decision is bad.
    This statement seems self evident. Since there are no surprises, it's probably not what Herodotus was intending to convey.
    Nonetheless, it's a member of the possible truth states of H.

  5. The evidence is good, the consequences are good and the decision is good.
    This statement also seems self evident.

  6. Ignore.

  7. The evidence is good, the consequences are bad but the decision is good.
    This statement is consistent with sub-proposition H1.

  8. The evidence is bad, the consequences are bad yet the decision is good.
    This proposition is counterintuitive. How can evidence and consequences that are both bad be consistent with a good decision?

Let's remove the items that have satisfactory truth conditions, and consider only those rows that still look inconsistent, viz.,
 1. The evidence is good, the consequences are good but the decision is bad. 
 8. The evidence is bad, the consequences are bad yet the decision is good. 
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.

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?

5.1  Ternary Operation

The way I like to view the values in Table 4 is to think of p ⊃ q as being represented by a ternary operation:

If p Then q Else Anything
(10)
Ironically, although this is not the usual way implication is represented in formal logic notation, it is a fundamental construct in all modern computer programming. I used Mathematica to generate the truth tables seen here. Since Mathematica is a functional programming language, the standard If function

v = If [p, q, Anything];
(11)
can be applied directly. The boolean result is returned to the variable v. The same is true with the R language for statistical programming, which defines the ifelse function:

v = ifelse(p, q, Anything)
(12)
in its base code. Both functions (11) and (12) return a boolean value. Other computer languages that define a standard if-else control construct (e.g., C, Perl), require a separate expression2:

v = p ? q : Anything;
(13)
to evaluate the ternary operation as a statement.

5.2  Venn Diagrams

From eqn. (11) 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 (13).

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 (10).
If p is True, then the truth value of the implication is determined solely 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
(14)
And you can see that eqn.(14) 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.
Here also are the Venn diagrams for H1, H2 and H:
Table 7: Venn diagrams for Proposition 1

H1 =  (E ∧ ¬C) ⊃  D H2 =  (¬E ∧ C) ⊃  ¬D H = H1 ∧ H2
Proposition 1 corresponds to the intersection or overlap of the blue areas in the diagrams for H1 and H2.

6  Conclusions

Returning now to Proposition 3, we write the truth values as determined by the antecedents A (6) and B (7) in the sub-statements:

H1
= A ⊃ D
(15)
H2
= B ⊃ ¬D
(16)
for the "anomalies" in rows 1, 4 and 8 of Table 6. The results are collected in Table 8.
Table 8: TT for anomalous cases 1, 4 and 8
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
The antecedent A in (15) corresponds to the conjunction E and ¬C and we note that it is always False in Table 8. So, the value of D is ignored in favor of Anything in the Else branch of (10). Thus, H1 is True.
Similarly, the antecedent B in (16) is always False, so the value of ¬D is ignored in favor of Anything and H2 is also True. We can summarize these points in Table 9.
Table 9: Final TT for H
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 
Antecedent A is True:
When the antecedent A is True, the truth value of H1 is determined by the truth value of D. This case only occurs in row 7, where D also happens to be True3. Row 7 is a unique case and could also be considered the essential Herodotus proposition H1.
Antecedent A is False:
When the antecedent A is False, the truth value of H1 is determined by Anything and is therefore always True. That is the case for every other row in Table 9.
A similar analysis using the antecedent B and the consequent ¬D produces the truth values of H2 in Table 9.
Remark 4 [Row 5] It is worth noting that row 5 is only an apparent truism. That E and C are True only appears to be consistent with D being True. In fact, the antecedent A is False, so the truth of H1 comes from Anything and not from D.

6.1  The Bottom Line

Finally, because of the way the truth functionality of implication works, we can say the following about Proposition 1:
  1. Proposition 1 is true, even in certain senses which appear contradictory, viz., rows 1 and 8 of Table 6.

  2. This follows from the unintuitive result of an implication being True even when the antecedent is False. (cf. Table 4)

  3. Although H2 appears redundant in the presence of H1 it really is not. Neither H1 nor H2 can be eliminated. The difference in the symmetry of H1 and H2 can be seen in the corresponding Ven diagrams of Table 7.

  4. The contingency of H being true is lower than either of H1 or H2 taken alone. This is reflected in the lower probability density for H.

  5. There are no significant Boolean reductions or simplifications for H. It has the same TT as:
    1. DNF: (C ∧ E) ∨ (¬C ∧ D) ∨ (¬D  ∧ ¬E)

    2. CNF: (¬C ∨ ¬D ∨ E) ∧ (C ∨ D ∨  ¬E)



6.2  Other Implications

  1. Let's not hastily 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
    (17)
    But (p ∧¬p) is a contradiction and therefore always False. Using equation (10) we can interpret (17) 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 (10), since p is False we can ignore q altogether. The truth of proposition q is irrelevant 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 (10), 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)
    (18)
    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.
2Consult the Wikipedia pages Ternary operation and ?: for more examples.
3A is also True in row 3, but that row was already eliminated because H1 and H are False in that case.


File translated from TEX by TTH, version 3.38.
On 19 Jul 2013, 10:16.