Conditional Statments and Truth Value

[toboldlygo]

So, I know that P ⊃ Q is a true statement even if P is false as long as Q is true. However, I don't understand why that is, or how that is logically sound. Is it because I'm stuck in thinking of these types of statements as "If P, then Q," and they are not supposed to be thought of that way? How else can I approach this to have it make logical sense to me? Thanks. Also, I'm sorry if this is supposed to go to the HW section (I thought this fit here); please let me know and I'll move it.

 

[jfizzix]

Given the statement P⊂Q (i.e., the elements of the set P are contained in the set Q),
it is not illogical to have an event where P is false and Q is true.

It could be the case that there are multiple elements in the set Q that are not also in the set P.
If that were the case, than an event could be in Q and not in P.

 

[Hornbein]

So, I know that P ⊃ Q is a true statement even if P is false as long as Q is true. However, I don't understand why that is, or how that is logically sound. Is it because I'm stuck in thinking of these types of statements as "If P, then Q," and they are not supposed to be thought of that way? How else can I approach this to have it make logical sense to me? Thanks. Also, I'm sorry if this is supposed to go to the HW section (I thought this fit here); please let me know and I'll move it.

If the moon is blue then the Earth is round.
If the moon is not blue then the Earth is round.
THEREFORE
The Earth is round.
If the first statement were false, then the deduction wouldn't follow.

 

[toboldlygo]

Given the statement P⊂Q (i.e., the elements of the set P are contained in the set Q),
it is not illogical to have an event where P is false and Q is true.

I really like thinking of it this way! Would it be accurate, then, for me to think of it like this: if P = {a, b, c} and Q = {P, d}, then P can be false even if Q is true?

 

[WWGD]

See also the paradox of the material conditional. There are modal operators where $p \rightarrow q$ only if q can be derived logically from p.

 

[jfizzix]

I really like thinking of it this way! Would it be accurate, then, for me to think of it like this: if P = {a, b, c} and Q = {P, d}, then P can be false even if Q is true?

Yes.