Anomaly set theoretic equivalent to material implication

  • Thread starter nomadreid
  • Start date
  • #1
nomadreid
Gold Member
1,456
144
OK, this is embarrassing, but I never looked carefully at this elementary point. We say that if
p implies q
P is the set of all things for which p is true
Q is the set of all things for which q is true
then Q ⊆ P.
Also that the set of all things for which p&q is true equals P∩Q
But p & q implies p, so (from the above) P ⊆ P ∩ Q, which is in general false.
What is wrong?
Thanks
 

Answers and Replies

  • #2
22,089
3,286
OK, this is embarrassing, but I never looked carefully at this elementary point. We say that if
p implies q
P is the set of all things for which p is true
Q is the set of all things for which q is true
then Q ⊆ P.
Are you sure that shouldn't be ##P\subseteq Q##?
 
  • Like
Likes nomadreid
  • #3
nomadreid
Gold Member
1,456
144
that was also my reaction, micromass, but I read a number of sources which stated it as I have put it, including HallsofIvy on this forum: https://www.physicsforums.com/threads/set-theory-representation-of-material-implication.206811/ :
/If P is the set of all things for which statement p is true and Q the set of all things for which statement q is true, then "If p then q" can be represented as "Q is a subset of P"./
This is apparently the reason that the symbol for material implication is often the (backward) subset sign: "p implies q" is often written as p⊃q, as you can see in http://en.wikipedia.org/wiki/Material_conditional and other sources. (In fact, this usage is what got me started on this question.)
So .....????
 
  • #4
22,089
3,286
Let p = divisible by 4, let q = divisible by 2. Then ##P = 4\mathbb{Z}## and ##Q = 2\mathbb{2}##. We have ##p\rightarrow q## and we have ##P\subseteq Q##. So I don't see how it could be differently.

The answer of Halls in that thread is wrong on another level to. It is right that the set theoretic version of ##p \wedge q## is ##A\cap B## and so on. But the set theoretic version of a operation should be a set. So it is false that the set theoretic analog of ##p\rightarrow q## is ##A\subseteq B## (or ##B\subseteq A##), since that is not a set. The set theoretic analog is rather equal to ##A^c \cup B## (in classical logic, the theory becomes much more complicated and more beautiful in other logics).

The reason why we sometimes use ##p\subset q## for ##q\rightarrow p## is an unfortunate historical coincidence, see http://math.stackexchange.com/quest...the-symbol-supset-when-it-means-implication-a
 
  • Like
Likes nomadreid
  • #5
nomadreid
Gold Member
1,456
144
Thanks very much, micromass. Whew, that's a relief; my initial doubt was like suddenly thinking that maybe 1+1 isn't really 10. This historical accident sounds a bit like the unfortunate convention in physics that switched positive and negative in current. Also thanks for pointing the very good and easily overlooked point about sets versus proper classes.
 

Related Threads on Anomaly set theoretic equivalent to material implication

Replies
2
Views
6K
Replies
4
Views
2K
Replies
1
Views
5K
  • Last Post
Replies
1
Views
2K
  • Last Post
Replies
2
Views
3K
Replies
1
Views
907
  • Last Post
Replies
7
Views
4K
Replies
7
Views
2K
Replies
3
Views
3K
Replies
14
Views
458
Top