# Math Help - This logic is nonsense

1. The “if-then” form is the most important is mathematical proofs!
Here is a summery: True statements must imply true statements.
False statements can imply any statements .
Thus the only non-allowable form is a true statement implies a false statement.

2. Originally Posted by Doktor_Faustus
Hi guys,

Just to continue this discussion about if/then:

I've managed to confuse myself again, but I was thinking about the statement (1) " if it rains, then there are clouds."

If the premise is true and the conclusion is false, then the implication is false - If it's raining, there must be clouds.
If the premise is false and the conclusion is true, then the implication is true - it may be cloudy, but that does not mean that it is raining.
If the premise is false and the conclusion is false, then the implication is true - No rain, no clouds.

Makes sense.

Then I thought about it another way.

(2) "If there are clouds, then it is raining. "

what I'm confused about is this: Lets say the premise in (2) is false, but the conclusion is true. According to the truth table, the implication is true. However, as far as I know, there can't be rain without clouds.

I know I'm missing something. Anyone mind clarifying this for me ? Does it just mean that the assumption - or premise - is incorrect ? That is, if it's raining, then there must be clouds, and anyone who says there are no clouds is lying ?

Cheers.
What you have done here is take the converse of $P \implies Q$ , that is, $Q \implies P$.

The two do not necessarily have the same truth values. "Implies" is not a commutative operation.

"If it is raining, then there are clouds" does not have the same truth value as "If there are clouds, then it is raining".

We can re-interpret $P \implies Q$ "If P then Q" as meaning "P being true and Q being false can not happen." Any other combination of their truth values can. That is, "It can not be the case that P is true and Q is false."

That is, "it can not be the case that it is (at the same time) raining and that there are no clouds."

This is not the same as "it can not be the case that there are clouds and that it is not raining."

Clearly in this case (with this particular assignment of statements to P and Q):
$P \implies Q$ holds;
$Q \implies P$ does not hold.

3. Originally Posted by Plato
The “if-then” form is the most important is mathematical proofs!
Here is a summery: True statements must imply true statements.
False statements can imply any statements .
Thus the only non-allowable form is a true statement implies a false statement.
We can add to this:
"All statements (true or false) imply true statements."

"If your name is Matt Westwood, then you have been offered a special sales opportunity!" screams the rubbish on the envelope that I find on my doormat. No, the fact is that *everybody* receiving this envelope is the recipient of this scam, and whether or not my name is Matt Westwood has nothing to do with the truth value of whether "I have been offered a special sales opportunity" or not.

4. Originally Posted by Matt Westwood
What you have done here is take the converse of $P \implies Q$ , that is, $Q \implies P$.

The two do not necessarily have the same truth values. "Implies" is not a commutative operation.

"If it is raining, then there are clouds" does not have the same truth value as "If there are clouds, then it is raining".

We can re-interpret $P \implies Q$ "If P then Q" as meaning "P being true and Q being false can not happen." Any other combination of their truth values can. That is, "It can not be the case that P is true and Q is false."

That is, "it can not be the case that it is (at the same time) raining and that there are no clouds."

This is not the same as "it can not be the case that there are clouds and that it is not raining."

Clearly in this case (with this particular assignment of statements to P and Q):
$P \implies Q$ holds;
$Q \implies P$ does not hold.

Cheers Matt. Something for me to digest over the next few hours.

5. Originally Posted by Matt Westwood
We can add to this:
"If your name is Matt Westwood, then you have been offered a special sales opportunity!" screams the rubbish on the envelope that I find on my doormat. No, the fact is that *everybody* receiving this envelope is the recipient of this scam, and whether or not my name is Matt Westwood has nothing to do with the truth value of whether "I have been offered a special sales opportunity" or not.
I like the example. It's quite funny. It's similar to

If 3 is even, then 57 is prime.

3 is not even, but it does not change the fact that 57 is prime.

6. Originally Posted by novice
I like the example. It's quite funny. It's similar to

If 3 is even, then 57 is prime.

3 is not even, but it does not change the fact that 57 is prime.

$57=19\cdot 3$ ....surprise!

Tonio

7. Originally Posted by tonio
$57=19\cdot 3$ ....surprise!

Tonio
I was sloppy.
Let's change it to
If 3 is even, then 7919 is prime.

8. I just want to point out that the $P\implies Q$ confusion is actually a fair criticism in logic. Some logicians believe that $P$ and $Q$ must somehow be related.

See Relevance logic - Wikipedia, the free encyclopedia

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