Here is a tweet from Tim Gowers It is a poll. The question is
I’ve rolled a die and not looked at it yet. The statement, “If the number I rolled equals 2+2 then it equals 5,” is …
The two answers:
a) Probably true
b) Definitely false
I've rolled a die and not looked at it yet. The statement, "If the number I rolled equals 2+2 then it equals 5," is …
— Timothy Gowers (@wtgowers) October 18, 2020
Here is a tweet from Tim Gowers It is a poll. The question is
I’ve rolled a die and not looked at it yet. The statement, “If the number I rolled equals 2+2 then it equals 5,” is …
The two answers:
a) Probably true
b) Definitely false
Nice…i just started to teach today today discrete math to electrical engineers (undergraduate 1st year)
And spent a non trivial amount of time on the truth value of an “if A then B” statements….
Is the answer here related to the universal constant 5/6???
😉
I dont see a test of intuition here, just an exercise in propositional logic: if the antecedent is false, then the conditional is true.
….and long experience teaching this material indicates that the formal truth table behavior of conditionals is pretty unintuitive.
Here is a variant: False or True (or probably true): “If it will snow this coming winter in Jerusalem all the cats in the city will start speaking fluent Spanish”
Here is a variant that tests also combinatorics skills
How many functions f in [n]^[n] there exists for which the statement
“if f is not a bijection then f(1)=1” is TRUE?
Regarding the first question: Since you’ve already rolled the die, the statement is either true or false. There’s no probability involved; it doesn’t matter that you haven’t looked at the die yet.
Dear Ira, that’s a nice point. But you can make the same statement also before you role the die. And it seems reasonable that the answer would not change.
I’m going to roll a die. The statement, “If the number I will roll equals 2+2 then it equals 5,” is …
But I think you are right that the statement is not a statement about probability. For example if you have a contract where its written: In case of emergency our agency will send an ambulance. And then there was an emergency but an ambulance was not sent. The agency cannot claim that their statement is probably true because emergencies are rare.
On the other hand at the time of the drawing up of the contract, one could correctly say that (for trivial reasons) it is unlikely that the agency will break it.
Of course, if it has many such contracts and no intention of sending an ambulance, then it will almost certainly break some of them, and in practice that is what matters.
The answer here depends on whether you interpret the “if … then” as material implication or as conditional implication (and, to a lesser extent, on whether you view the probability as expressing your belief, in which case the fact that the outcome has been determined is irrelevant). In the former case, it is probably true; in the latter, it is definitely false.
First, the case of material implication. In that case, as Albert observed, it is vacuously true if the antecedent (“the number I rolled is 2+2”) is false, which it is with probability 5/6 (from my subjective viewpoint, since I don’t know how the die landed). It is false only if the die landed 4. On the other, to me a more natural interpretation of the “if .. then” is that it is a statement of conditional implication, where we treat counterfactuals quite differently. Suppose you’re in court after an automobile accident. You were drunk, but your brakes were faulty. Consider a statement like “If the brakes weren’t faulty, then I wouldn’t have had the accident” (so I am suing GM). Is that true? If we interpret the “if .. then” as material implication, then it’s vacuously true. The brakes were faulty. But that isn’t how we typically interpret the statement. The standard way of dealing with statement that involve counterfactuals (due to two philosophers, David Lewis and Bob Stalnaker — I’m giving you the Lewis variant) is to consider the closest worlds where the antecedent is true (according to some notion of closeness) and see whether the conclusion is true there. In this case, we imagine a world “just like the actual world” except that the brakes are not faulty. Would I have still had the accident in that world? That depends on how well I drive when I’m drunk. If I drive really well when I’m drunk, then in the closest world where my brakes are OK, I don’t have the accident, so the statement “if my brakes weren’t faulty I wouldn’t have had the accident” is true. If in the closest world I would have had the accident anyway, due to being drunk, then the statement is false. If you have a probability on these worlds then you can also talk about the probability of the conditional implication being true. Now going back to the original question, in all the closest worlds where the number rolled is 2+2, it’s not 5, so the statement would be false if we viewed it as a conditional implication, no matter how the dice landed. English doesn’t do a good job of signaling how to interpret the implication (although a subjunctive is often a signal of a conditional implication: “if the number I rolled were equal to 2+2” rather than “if the number I rolled is equal to 2+2”).
Bottom line: the statement is ambiguous, and both conclusions are reasonable.
I wonder if Tim Gowers would care to disambiguate his intended meaning.
I don’t think counterfactual conditionals are relevant here. If I looked at the die and saw that I had rolled a 3, then the statement “If the number had equalled 2+2 then it would have equalled 5” would still be clearly false, whereas it is hard to interpret the conditional in the statement “If that number equals 2+2 then it equals 5” as anything but a material conditional.
But maybe one could interpret what I wrote as meaning “In every possible world in which I have just rolled a number equal to 2+2, I have just rolled a number equal to 5,” which is definitely false. But by giving “probably true” as an option I was trying to encourage the material-conditional interpretation.
None of it was meant to be taken seriously — the main reason for the poll was to make a joke about the 2+2=5 pseudo-controversy that’s been going on on Twitter.