Understanding P -> Q: An Example

[Omid]

In the statement P -> Q , Q is necessary for P.
I just don't get it ,why don't we say P is necessary for Q, can you illustrate it in an example ?

 

[quantumdude]

Example: If you live in Miami, then you live in Florida.

This is equivalent to: A necessary condition for living in Miami is living in Florida.

It is not eqivalent to: A necessary condition for living in Florida is living in Miami.

(One may live elsewhere in Florida).

 

[HallsofIvy]

"If you get an A on every test, then you will get an A in the course".

Getting an A on every test is NOT "necessary" to getting an A in the course (if you get all A's except for one B, I might give you an A in the course) but, assuming that the statement is true, it is impossible for you to have gotten an A on every test without getting an A in the course. Getting an a is the course IS a necessary condition for getting an A on every test.

 

[Omid]

Your examples clear things up.
Can you give me some more examples for the truth values of P->Q ?
I know when we have P , Q there are 4 possible truth value for p-> Q :
P Q P->Q
1. T T T
2. T F F
3. F T T
4. F F T
Fore the first and second I have some examples but for the third and forth I haven't found any.
Thanks

 

[HallsofIvy]

I'll use my example again: If you get an A on every test, you will get an A in the course:

1) You get an A on every test and you get an A in the course: yes, the statement is true.
2) You get an A on every test but do NOT get an A in the course: No, the statement is not true, it's false.
Now the hard ones:
3) You do NOT get an A on every test but do get an A in the course.
Well, yes, that's possible- maybe you got an A on every test but one and a high B on that one. I would certainly give you an A in course. The original statement says NOTHING about what happens if you do NOT get an A in the course.
4) You do NOT get an A on every test and do not get an A in the course.
Well, that's perfectly reasonable isn't it?

In three and four, we really have no evidence as to whether the person making the original statement was telling the truth or not. Since we did NOT get an A in the course, we don't know what would happen if we did!

I like to think of it as "innocent until proven guilty": if the hypotheses are false, the statement itself is true no matter what the conclusion is.

 

[Omid]

Thank you very much

 

[metacristi]

In the statement P -> Q , Q is necessary for P.
I just don't get it ,why don't we say P is necessary for Q, can you illustrate it in an example ?

I want to stress (hope it is not a too difficult subject) that there are two totally different ways of interpreting the implication.One stems from the Russelian (material) definition of implication,needed for the formalization of logic,where the definition of implication merely says that (P -> Q) means that is false that P is true and Q is false,the other says that it is impossible for P to be true and Q to be false.The first definition does not take into account the relations between the terms in the two statements,the consequences being contingent,the second means that between P and Q is a necessary link.The material (russelian) definition of implication has a greater scope,everytime when the strict implication holds it holds also,the reverse is not valid.

For example there is no necessary connection between P='2+2=4' and Q='Washington is the capital of USA',still the inference P -> Q is valid,that is always a TRUE implies a TRUE irrespective of the relations between the terms of the propositions P and Q in the russelian (material) definition.On the other hand there is a necessary one between P='Bucharest and Washington are cities,Washington being the capital of the USA' and Q='Iasi is not the capital of USA' (this means that under the strict definition of implication the inference that '2+2=4' implies 'Washington is the capital of USA' is not valid).