Is P a Necessary and Sufficient Cause of Q?

[nickto21]

Hey All,
I've just started reading a discrete math book, and in the beginning the book covers logic.
One concept I'm finding hard to understand is certain conditional propositions.

When the example uses a word problem, I mostly get it.
The statement is "If The Mathematics Department gets an additional $20,000, then it will hire one new teacher.
p is: "The Mathematics Department gets an additional $20,000.
q is :"The Mathematics Department hires one new faculty member.

If p and q are true, I get why it's true.
If p is true and q is false, I get it.
If both are false, I get it.

But, when p is false and q is true, why is the proposition true?

THEN, I don't seem to have trouble with the word problems, but when numbers are used in place of sentences, I cease to get the concept.
For example, if p is 1>2 and q is 4<8, then the proposition is supposed to be true. I am just not seeing the connection. letter p doesn't seem to relate to q.
Even if both statements are true, let's say p:2>0 and q:3<8, just because one is true, it doesn;t say anything about the other. I know both are true, but the statement " If 2>0, then 3<8" doesn't seem to mean anything to me. I can memorize the truth table, but I'd also like to understand why?

I know I'm just thinking about this badly. I'm not even sure if I'm getting my question across. I'll check back later. Maybe someone will provide some clarity. Maybe I'll understand my confusion better, and ask better questions later.

Thanks for your time,
Steve

 

[SW VandeCarr]

Hey All,
I've just started reading a discrete math book, and in the beginning the book covers logic.

If p and q are true, I get why it's true.
If p is true and q is false, I get it.
If both are false, I get it.

But, when p is false and q is true, why is the proposition true?

In the material implication relation P->Q, P is the antecedent and Q is the consequent. If the consequent is true, P->Q is true. It doesn't matter what P is.

THEN, I don't seem to have trouble with the word problems, but when numbers are used in place of sentences, I cease to get the concept.
For example, if p is 1>2 and q is 4<8, then the proposition is supposed to be true. I am just not seeing the connection. letter p doesn't seem to relate to q

P says 1 is greater than 2. That's false. Q says 4 is less than 8. That's true, so P->Q is true although we can't say why Q is a consequent of P (and it doesn't matter). Just because it doesn't make sense doesn't mean it's not logical. Who told you formal logic is supposed to make sense?

Thanks for your time,
Steve

You're welcome. Stay sane.

 

[CRGreathouse]

I can memorize the truth table, but I'd also like to understand why?

My preferred interpretation when explaining implication is that is is the promise "If A, then B". If A does not happen, then the promise is kept.

"If you clean your room, you can have ice cream."

If you don't clean your room, the promise is kept (whether you get ice cream or not). The only way for the promise to be broken is if you clean your room but don't get ice cream.

 

[SW VandeCarr]

My preferred interpretation when explaining implication is that is is the promise "If A, then B". If A does not happen, then the promise is kept.

"If you clean your room, you can have ice cream."

If you don't clean your room, the promise is kept (whether you get ice cream or not). The only way for the promise to be broken is if you clean your room but don't get ice cream.

Why would you give someone ice cream if he/she didn't clean their room? Here's a seemingly more socially acceptable situation:

Implication: If you do work for the government, you get paid.

Antecedent: I did work for the government.

Consequent: I got paid.

The consequent is true even if the antecedent is false. Read the contract.

Therefore P -> Q is materially (if not strictly) true even if I did no work for the government.

 

[EnumaElish]

I think of it as a set of promises:

a. we get new money ==> we'll hire you.
b. you agree to work for free ==> we'll hire you
c. we lose a staff member ==> we'll hire you
d. parents like you as a teacher ==> we'll hire you
e. students think you're good ==> we'll hire you
f. ...
g. ...
h. ...

Some of the above may be known, others unknown. If I am the candidate then I may know (a), can guess (b), may not know the rest. If I get hired even though they don't have new money and I don't agree to work for free, then it must be because some other (unknown) premise must have become true.

Alternatively, that just the definition of "==>" as a binary relation.

 

[EnumaElish]

If 2>0, then 3<8

0 < 2 ... add 2 to both, to get:
2 < 4 ... multiply both by 1.5, to get:
3 < 6.

Since 0 < 2 (the "if"), I can add this to both sides, and end up with 3 < 8.

if p is 1>2 and q is 4<8

Note that "if 1 < 2 then 4 < 8" is TRUE (multiply both sides by 4). Therefore:

Known or seen: 1>2 ==> 4<8
"unknown" or unseen: 1<2 ==> 4<8

4<8 can be true even though 1 is not > 2 because some other (unseen) relationship happens to be the case (1 < 2), which (also) implies 4<8.

 

[SW VandeCarr]

0 < 2 ... add 2 to both, to get:
2 < 4 ... multiply both by 1.5, to get:
3 < 6.

Since 0 < 2 (the "if"), I can add this to both sides, and end up with 3 < 8.Note that "if 1 < 2 then 4 < 8" is TRUE (multiply both sides by 4). Therefore:

Known or seen: 1>2 ==> 4<8
"unknown" or unseen: 1<2 ==> 4<8

4<8 can be true even though 1 is not > 2 because some other (unseen) relationship happens to be the case (1 < 2), which (also) implies 4<8.

I guess logicians find material implication useful, but strict implication (from modal logic) is more intuitive. The only strict relation admitted is P->Q iff P^Q and P^~Q is impossible.

 

[vici10]

Another way to think about P -> Q is \neg P \vee Q. They have the same truth table. So if P is false then \neg P is true, hence proposition is true. If P is true then the proposition depends on value of Q.

 

[Pere Callahan]

Ex falso quodlibet

 

[SW VandeCarr]

Ex falso quodlibet

Yes.~P^Q doesn't exactly fit this description, but P can be any false proposition at all for the material implication to be true provided Q is true..

Strict implication (see my last post) seems more rigorous although more restricted.

If P^~Q is necessarily false, then P might be considered a necessary and sufficient cause of Q. However, there might be other sufficient causes of Q. If we modified the definition of P to be the necessary, sufficient and sole cause of P we might have a strict causal logic. So it seems we could have two axioms for a strict modal causal logic:

1. P^~Q is necessarily false

2. ~P^~Q is necessarily true.

If both of these hold, then P is the necessary, sufficient and sole cause of Q.

~P^Q would be necessarily false under this type of logic.