Negation of a statement If P then Q

[fk378]

"If P then Q"
Why is this logically equivalent to "Either P is false, or Q must be true"?? Can anyone explain this in plain English?

If P is "It is raining" and Q is "you will get wet", then how can Q be true if P is false ("It is not raining, and you will get wet")?

 

[Dick]

That's not the negation. It's logically equivalent. If P is false, then saying "If P then Q" has no useful information. To say "If it is raining then you will get wet" doesn't tell you anything about whether you will get wet if it's not raining.

 

[fk378]

Oh sorry, I meant to ask why it is logically equivalent. Disregard the negation part, I will edit it now.

 

[HallsofIvy]

"If P then Q"
Why is this logically equivalent to "Either P is false, or Q must be true"?? Can anyone explain this in plain English?

If P is "It is raining" and Q is "you will get wet", then how can Q be true if P is false ("It is not raining, and you will get wet")?

OR! "either it is not raining OR you get wet".

 

[fk378]

Yes but isn't this the inclusive OR? In which case one of the situations is that BOTH P and Q are true...

 

[Dick]

Yes but isn't this the inclusive OR? In which case one of the situations is that BOTH P and Q are true...

Yes. In that case both the implication and the or statement are both true. Why is that a problem? Start making truth tables, ok?

 

[fk378]

Well "Either P is false, OR Q must be true"..."If it is not raining, you will get wet"

That is supposed to be true?

 

[Dick]

No! The implication is "P implies Q". NOT "not P implies Q". "If it IS raining, you will get wet." And WHAT is supposed to be true?? Any of these statements can by either true or false. What you are trying to show is that two of them are logically equivalent. Those two are not.

 

[HallsofIvy]

Inclusive or means A is true OR B is true OR both are true. As long as anyone of those is true, the statement is true.

Assuming the statement "if it is raining then you will get wet" is true, it certainly also true that "either it is not raining or you will get wet" is true. It happens that here they can't BOTH be true but they don't have to: only one of them has to be true.