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drinking in the bar
Posted 03 August 2013 - 08:24 PM
Posted 03 August 2013 - 08:46 PM
P (someone is drinking) is true. Q (everyone is drinking) is false. So "if P then Q" is false.
Posted 03 August 2013 - 10:01 PM
Someone is drinking = at least one person is drinking.
Someone is not drinking = at least one person is not drinking.
These statements are not mutually exclusive. If you are drinking and I am not drinking, then both are true.
Posted 03 August 2013 - 10:44 PM
Suppose that at least one person is not drinking. For any particular nondrinking person, it still cannot be wrong to say that if that particular person is drinking, then everyone in the pub is drinking because that person is, in fact, not drinking. In this case the condition is false, so the statement is vacuously true due to the nature of material implication in formal logic, which states that "If P, then Q" is always true if P (the condition or antecedent) is false.  Either way, there is someone in the pub such that, if he is drinking, everyone in the pub is drinking. A slightly more formal way of expressing the above is to say that if everybody drinks then anyone can be the witness for the validity of the theorem. And if someone doesn't drink, then that particular non-drinking individual can be the witness to the theorem's validity.  The proof above is essentially model-theoretic (can be formalized as such).
Edited by BMAD, 03 August 2013 - 10:44 PM.
Posted 04 August 2013 - 08:13 AM
That's not the same statement. "There is someone in the pub, such that if he is drinking, everyone is drinking" is not the same as "if someone in the pub is drinking, then everyone is drinking". The first statement checks each person individually for the "if-then" statement, and asserts that at least one of those is true. The second statement is just another way of saying "no one is drinking or everyone is drinking", which is often false.
To illustrate the difference, suppose persons A and B are in the bar. A is drinking, B is not drinking. So someone in the bar is drinking. But everyone is not drinking.
The statement in the OP, "if someone is drinking, then everyone is drinking", is false (P is true, Q is false).
The statement on wikipedia, "there is someone in the pub, such that if he is drinking, then everyone is drinking", is in fact true. Let us check these two statements:
- If A is drinking, then everyone is drinking. A is drinking, but everyone is not drinking. So this one is false.
- If B is drinking, then everyone is drinking. Since B is not drinking, this statement is true by default.
So there is indeed such a person in the pub that the conditional statement is true, and that person is B. Hence there is someone in the pub, such that if he is drinking, then everyone is drinking.
In the OP, the man said something to the effect of "I noticed the first man was drinking, and if he is drinking then everyone is drinking". This is false unless everyone is in fact drinking. But, if he had seen someone who was not drinking, he could have safely pointed to the non-drinker and said "if that man is drinking, then everyone is drinking". That statement is true because the "if" part is false.
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