This an example of how careful one needs to be in selecting the correct logic, and the limitations of classical logics with their ‘law of the excluded middle’.
The problem statement
The New Scientist (30 March 2013) has the following question, under the heading ‘Stupid is as stupid does’:
Jack is looking at Anne but Anne is looking at George. Jack is married but George is not. Is a married person looking at an unmarried person?
Possible answers are: “yes”, “no” or “cannot be determined”.
According to the New Scientist ‘the vast majority’ (presumably including financiers, whose thinking is being criticised) think the answer is “cannot be determined”, but:
careful deduction shows that the answer is “yes”.
IF we apply classic logic and – as seems reasonable – suppose that ‘unmarried’ is synonymous with ‘not married’, then Anne is either married or unmarried and in either case the answer is “yes”. But should we make these assumptions? Is the question about an idealised situation, or reality?
But presumably of the following alternatives, ‘cannot be determined’ means the last:
- I cannot determine.
- There is no method that gives an answer “yes” or “no”‘.
- I would not gamble on the questioner, or some other judge, thinking that “yes” or “no” is the ‘correct’ answer.
- Neither “yes” or “no” or no cannot be proven without making some unproveable assumptions.
Possible conditions under which Anne become widowed.
If we are truly being careful, we should look out for hidden assumptions, such as that the law of excluded middle applies to marriage. Now, Newtonian Physics is consistent with classical logic, but it is not obvious that ‘to be married’ is a physical property, deducible from a state or attribute of Anne. Rather, is it not a relationship?
Suppose that Anne marries Henry, and then Henry dies. When does Anne cease to be married?
- Straight away
- When she learns / knows that Henry has died?
- When she knows that some legal process has confirmed that Henry is ‘legally dead’?
- Something else?
On death of husband
Does ‘the time at which Anne is widowed (and hence ceases to be married)’ have any definite meaning? In Newtonian Physics yes, assuming that life is a binary attribute. But even before Einstein, the validity or otherwise of Newtonian Physics was not (despite Kant) a logical or mathematical truth. No theory of Physics can be ‘logically determined’ (except in a sociological sense), and any deduction from such a theory is conditional on that theory. Thus the best one can do is to say “yes, provided that marriage have always have definite start and end times”, which rather begs the question. “Cannot be determined” is surely the correct answer if one cannot give caveats.
On ”learning’ of husband’s death?
If the determining factor is Anne’s perception of her husband’s death, then Anne’s marriage is uncertain when her perception of her husbands death is uncertain. Moreover, is Anne is Married to a Henry Smith Age 44 of Malvern she might not realise that there were two such, and believe herself to be widowed when her husband was actually alive. So this seems an unsatisfactory notion of marriage.
Knowledge is ‘justified true belief’, which requires that Anne has some evidence that Henry is dead, that he is dead, and that she believes him to be dead. It follows form the above that this is not determined.
A legal process can absorb uncertainty by declaring Anne to be ‘legally married’ unless and until there were conclusive evidence that Henry had died, or some other rule. (E.g. when a husband goes missing for 7 years.) Anne also has to know that there has been such a legal process, which may be achieved by a meeting or an exchange of letters. Thus the ideal would be for the legal process to create certainty. But even lawyers can get things wrong, or be less than straightforward.
For example, suppose that Anne has been married to Henry in a British registry office and a Sharia court. Henry then divorces her Sharia-style, but not according to British law. Jack is a British lawyer. George is a Pakistani. Jack (married) is looking at a married person. Anne is looking at an unmarried person. Is Anne married or not? Which jurisdiction is relevant? What if she has flown to Pakistan to marry George? A lawyer would know, but for me the answer is uncertain.
We might allow “yes” as an answer if it was always understood to be subject to various caveats, such as these:
- There are some very rare exceptions.
- There is a possibility that I am always wrong, but I don’t think so.
- This conclusion could be conditional upon
- everything that is considered to be ‘common knowledge’ or authoritative on the subject.
- everything that I have previously said on the subject.
- everything that you have previously said on the subject.
- everything that I believe.
- everything that I believe that you believe.
These may be ‘pragmatic’, but if we allow such answers to what seem to be logical issues, it begs the question of how we communicate an unconditional “yes”.
The New Scientist blames financier’s ‘bad thinking’ in part for the financial crash, where others blame mathematicians:
Employees leaving logic at the office door helped cause the financial crisis.
But it seems to me (as a mathematician) that even if financiers had only made ‘careful deductions’ in the same way as the New Scientist, then they would still have been in error. What financiers and their advisers were saying and doing seems to me to have been pragmatic. An underlying problem was that (as Keynes pointed out) classical logic does not apply to economics: it includes complex relationships that are more like marriage than those ideal types assumed by classical logic. Being married is a relation, not a proposition about Anne. Anne has no state or attributes from which her marital status can be determined, any more than terms such as crash, recession, money supply, inflation, inequality, value or ‘the will of the people’ have any correspondence in real economies. These are simply (mostly) convenient tales we tell ourselves. Unless you know different?
One could argue that the question ‘must have meant’ something other than my interpretation. But as far as I can see, one could still adapt the example.
Some people object that the term ‘is married’ may not be well-defined, or that the time of death may not be determinable. But these not my concerns. Even supposing that one has definitions of marriage and determinations of the time of death that are as complete and precise as possible, one has a problem.
According to some accounts of relativity theory (such as Russell’s) one definitely cannot determine the time at which Anne is widowed if Henry dies while undergoing great acceleration and velocity relative to Anne. Perhaps this aspect of the theory has not been conclusively proven, or perhaps the theory has been superseded by one in which the time can be determined. My remarks still stand: the conclusion depends on Physics, not logic. We cannot know that there really is an ‘objective’ time at which Anne is widowed.
(Most people thought I was being unduly picky. But isn’t that what logic is all about. I agree that ‘intuitively, a married person ‘must’ be looking at an unmarried one, but logic is supposed to correct intuition, or what is the point?)