4 ms·

I have a notation question: if I am reading correctly, the article uses "A ⊃ B" to mean "A implies B" -- but subset/superset notation defined at https://en.m.wi

I have a notation question: if I am reading correctly, the article uses "A ⊃ B" to mean "A implies B" -- but subset/superset notation defined at https://en.m.wikipedia.org/wiki/Subset https://en.m.wikipedia.org/wiki/Subset seems to be just the opposite (i.e., "A implies B" should mean B is a superset of A, as in "B ⊃ A". Are these two conflicting notations, or am I just confused?

gwern 7y ago
It's not subset, but https://en.wikipedia.org/wiki/Material_conditional https://en.wikipedia.org/wiki/Material_conditional
- repsilat 7y ago
  Sure, though the grandparent is kinda correct in one sense -- if one were to use a set operator to mean logical implication, you could use "is a subset of". Imagine the set universe has just one element, "truth". Every statement is a set, and that set contains "truth" or it doesn't. Now, if A implies B, then one of three things is true: - A contains "truth" and so does B, - B contains "truth" but A does not, or - Neither A not B contains "truth". So implication would mean that B is a superset of A (or A⊆B.) There are senses in which the reverse operator could make sense, though, like for logical corrolaries. Any statement provable from B is also provable from A, but not necessarily vice-versa. If each statement corresponds to the things it implies, then if B can be proven from A then A⊇B. I guess this difference is natural. Think of these logical statements as constraining the universe of possible things. More constraints means fewer possible universes. (More to the point, a subset of constraints leads to a superset of universes.) The direction of Set relationships between logical statements will depend on which you put in your sets. EDIT: I just re-read this and fixed a couple of errors, leading to maybe the opposite conclusion!?
- sn9 7y ago
  Could you please include a note explaining that in the text? I, too, thought it was referring to a subset relationship. Ctrl-F 'Material' also doesn't turn up anything in the page.
ubavic_nikola 7y ago
The symbol ⊃ is an old notation for material conditional, which is now usually symbolized using ⇒ or →. And yes, use of ⊃ in symbolic logic and set theory is conflicting, as you noted.

gwern 7y ago

It's not subset, but https://en.wikipedia.org/wiki/Material_conditional https://en.wikipedia.org/wiki/Material_conditional

repsilat 7y ago
Sure, though the grandparent is kinda correct in one sense -- if one were to use a set operator to mean logical implication, you could use "is a subset of". Imagine the set universe has just one element, "truth". Every statement is a set, and that set contains "truth" or it doesn't. Now, if A implies B, then one of three things is true: - A contains "truth" and so does B, - B contains "truth" but A does not, or - Neither A not B contains "truth". So implication would mean that B is a superset of A (or A⊆B.) There are senses in which the reverse operator could make sense, though, like for logical corrolaries. Any statement provable from B is also provable from A, but not necessarily vice-versa. If each statement corresponds to the things it implies, then if B can be proven from A then A⊇B. I guess this difference is natural. Think of these logical statements as constraining the universe of possible things. More constraints means fewer possible universes. (More to the point, a subset of constraints leads to a superset of universes.) The direction of Set relationships between logical statements will depend on which you put in your sets. EDIT: I just re-read this and fixed a couple of errors, leading to maybe the opposite conclusion!?
sn9 7y ago
Could you please include a note explaining that in the text? I, too, thought it was referring to a subset relationship. Ctrl-F 'Material' also doesn't turn up anything in the page.

ubavic_nikola 7y ago

The symbol ⊃ is an old notation for material conditional, which is now usually symbolized using ⇒ or →. And yes, use of ⊃ in symbolic logic and set theory is conflicting, as you noted.