4 ms·

Are those statements really the same? "For all x, if x is a marble and x is in the bag, then x is white" I would question if nothingness is a marble. Or would

Are those statements really the same? "For all x, if x is a marble and x is in the bag, then x is white" I would question if nothingness is a marble. Or would you say that the following statement is also true? I would say that x is a marble and therefore can be neither even or odd because it is not a number? "For all x, if x is a marble and x is in the bag, then x is even" That is just a nonsense question same as asking if all the non-existing marbles are of a particular color.

fasterik 3y ago
According to the rules of logic, we can always replace (P -> Q) with (~P v Q). In other words, if P is false then (P -> Q) is always true. This is known as a vacuous truth. https://en.wikipedia.org/wiki/Vacuous_truth https://en.wikipedia.org/wiki/Vacuous_truth Agreed, sometimes this leads to true statements that sound like nonsense. One example from the article is "if Tokyo is in France, then the Eiffel Tower is in Bolivia". This is a true statement because Tokyo is not in France. >"For all x, if x is a marble and x is in the bag, then x is even" We can rewrite this as "For all x, x is not a marble or x is not in the bag or x is even". We can see that this is true if "x is not in the bag" is true for all x. It's not really nonsense because we can quantify over all objects, so some of them will be marbles, some of them will be even, etc. Going back to the more general question we can look at the statement "For all x, F(x)" where F is any predicate. Call this statement P. The negation ~P is equivalent to "There exists some x such that ~F(x)". But if we are quantifying over the empty set, ~P is false, so P must be true in that case.
- VonGallifrey 3y ago
  I think my problem here is that everyone just assumes an extra condition. The question "if Tokyo is in France, then the Eiffel Tower is in Bolivia" is not the same as "is the Eiffel Tower in Bolivia" and the statement of all(m=="white" for m in marbles) is not the same as "For all x, if x is a marble and x is in the bag, then x is white". The statement is more "For all x in the bag, is x white?" and the answer to that is either "No, nothing is not white" or "Question is not applicable".
  - fasterik 3y ago
    >"For all x, if x is a marble and x is in the bag, then x is white". The statement is more "For all x in the bag, is x white?" They're the same, just quantifying over different sets. I can quantify over objects in the universe, in which case I need the clause about being in the bag. Or I can quantify over objects in the bag, in which case I'm quantifying over the empty set. all(m=="white" for m in marbles) By De Morgan's laws, this is equivalent to not any(m != "white" for m in marbles) If you want all(m=="white" for m in marbles) to evaluate to False for the empty set, then any(m != "white" for m in marbles) has to evaluate to True for the empty set, or you have to give up on De Morgan's laws. This is how standard first-order logic works. Again, you're welcome to try to make your own system of logic that works differently, but good luck doing that while preserving all of our theorems in set theory and mathematics.

fasterik 3y ago

According to the rules of logic, we can always replace (P -> Q) with (~P v Q). In other words, if P is false then (P -> Q) is always true. This is known as a vacuous truth. https://en.wikipedia.org/wiki/Vacuous_truth https://en.wikipedia.org/wiki/Vacuous_truth Agreed, sometimes this leads to true statements that sound like nonsense. One example from the article is "if Tokyo is in France, then the Eiffel Tower is in Bolivia". This is a true statement because Tokyo is not in France. >"For all x, if x is a marble and x is in the bag, then x is even" We can rewrite this as "For all x, x is not a marble or x is not in the bag or x is even". We can see that this is true if "x is not in the bag" is true for all x. It's not really nonsense because we can quantify over all objects, so some of them will be marbles, some of them will be even, etc. Going back to the more general question we can look at the statement "For all x, F(x)" where F is any predicate. Call this statement P. The negation ~P is equivalent to "There exists some x such that ~F(x)". But if we are quantifying over the empty set, ~P is false, so P must be true in that case.

VonGallifrey 3y ago
I think my problem here is that everyone just assumes an extra condition. The question "if Tokyo is in France, then the Eiffel Tower is in Bolivia" is not the same as "is the Eiffel Tower in Bolivia" and the statement of all(m=="white" for m in marbles) is not the same as "For all x, if x is a marble and x is in the bag, then x is white". The statement is more "For all x in the bag, is x white?" and the answer to that is either "No, nothing is not white" or "Question is not applicable".
- fasterik 3y ago
  >"For all x, if x is a marble and x is in the bag, then x is white". The statement is more "For all x in the bag, is x white?" They're the same, just quantifying over different sets. I can quantify over objects in the universe, in which case I need the clause about being in the bag. Or I can quantify over objects in the bag, in which case I'm quantifying over the empty set. all(m=="white" for m in marbles) By De Morgan's laws, this is equivalent to not any(m != "white" for m in marbles) If you want all(m=="white" for m in marbles) to evaluate to False for the empty set, then any(m != "white" for m in marbles) has to evaluate to True for the empty set, or you have to give up on De Morgan's laws. This is how standard first-order logic works. Again, you're welcome to try to make your own system of logic that works differently, but good luck doing that while preserving all of our theorems in set theory and mathematics.