logomancer | Oog.

Just came back from Discrete Math. Brown was in rare form today, as he explained to us the mechanics of conditional statements. My biggest beef with the class is this: It claims to be a class to teach reasoning skills, but the concepts he presents sometimes are not logical. For example, we learned about the conditional statement, "p implies q". p is what we called a hypothesis and q is a conclusion. However, Brown tells us that if p is false and q is true, then the conditional statement is true. Which puts me off because if one's hypothesis is false, how can one derive a conclusion that is true? And even if that's the case, how does that prove that the conditional statement is true? It's confusing.

The rest of the class was somewhat easier to follow. Nothing compared to Chemistry, where Prof. Amateis played with liquid nitrogen today, freezing a banana and breaking it (although it didn't shatter -- it wasn't cold enough). She also splashed some LN₂ on the floor, which freaked out the first row ("Oh, and by the way, class, be sure to wear closed-toe shoes in your labs.") Very cool.

Anyway, I need folders, dinner, and Spiel.

Flat | Top-Level Comments Only

I probably have this wrong, but imagine this:

p = "Ed is a horse" and q = "Ed has four legs". (p implies q) works out to: "Ed is a horse" implies "Ed has four legs". So far, so good.

Now, what if Ed is a dog? p is false, yet q is true. However, the conditional statement is still true, because it is not claiming he is a horse. It's just claiming that if he were, he'd have four legs.

Another example: p = "All science-fiction fans go to Virginia Tech" and q = "Virginia Tech is crowded". p isn't true, and q is (by the accounts I've heard), yet the statement only claims that Tech would be crowded is all science-fiction fans were there, and is therefore true.

Yes, but according to Wacky Brown Logic(tm), the statement (if Ed was a dog) Ed is a horse implies Ed has 17 legs would be true. 0->0 is true, oddly enough. Only 1->0 is false.

Also, what about statements like I am male->I am a CS major? Both are true, but the statement itself is false, as my CS-majordom does not derive from my manhood.

Well, with further thought, I was beginning to wonder if I missed something, because "p implies q" can't be true for all values of p and q. If (p implies q) is true for selected values of p and q, p need not necessarily be true for q to be. Conversely, even if p and q are true, it does not always follow that p implies q - there may be no causal connection. "I have 10 fingers" and "I like pizza" are both true, but neither implies the other.

Maybe the professor accidentally skipped over a paragraph or two of lecture notes today.

It's really quite simple. I was amazed at how many people thought that it wasn't simple and were confused for days after Shockley explained it.
Here's how simple it is:

If p then q.

That's it. That's the whole thing. If p is true, then q has to be true. If p is false, then all bets are off and q could go either way.

Don't try to rationalize it by using words. Words have no place in logic. It's just a function.

You were right, it isn't always true. Here's a truth table:

See, it's true whenever p isn't, and it's true whenever both p and q are true. It's false, however, when p is true but q isn't, because that means that clearly p doesn't imply q (if p can be true without q).

[b]
If p is false, then all bets are off and q could go either way.
[/b]

That's the thing. If this were trying to accurately model the universe, than p->q would only have a truth value for 1,0, because it requires extra random evaluations to discover if p->q is true in the verbal sense, instead of q depending on, perchance, r.

If this were trying to accurately model the universe

Allow me to paraphrase Shockley, from when I took the course:
"The most dangerous assumption you can make is that logic has any bearing on actual events. A proof is only as useful or as true as the propositions that go into it."

This isn't trying to accurately model the universe, not in the way that you're thinking. Forget the "verbal sense", the verbal sense is ambiguous and imprecise, which is why this notation was made in the first place.

P.S. I know I basically just repeated the same things you said, but I needed to to get them lined up in my own head :)

The statement (if p then q) is always true if p ends up being false, regardless of what q is. Why?

Notice that the statementes begins IF p...

So, everything that follows in the statement depends on p being true. If q is false and p true, then the statement is quite obviously false, because when p is true, q has to be true.

But, what happens when p is false? Then q does not matter. So the statement (if p then q) itself is still logically true, as the if condition was never actually triggered (because p is false).

Why is the statement true though? Because it is NOT false. When p is false, then the truth of q is irrelevant and can do nothing to contradict the conditional. Thus, the statement is NOT false. Thus, it must be true.

I hope that was clear.

I understand what you're saying, but this truth-by-default is tripping me up. The sentence "This sentence is false" is not false, but it's not true, either. It has no truth value. I therefore don't understand how we can assume that p->q for p=0 evaluates to 1, and not .5 or whatever the official mathematical syntax for no truth value is.
This is one of those "It's defined that way" situations, isn't it?

You are attempting to add a third state to binary logic. You can't have a value of "not true and not false" just the same as you can't have a value which is both true and false.

if (true && false) abort_universe();

Boolean mathematics (true/flase) is a binary system in some sense, as it only has two states (true or false). Thus, if something is not false, it has to be true. An implication can only be proven false if you start from a true statement and get a false one. Thus by default, every other form of the statement becomes true.

I think the example about the Ed and the dog below explains it very well. Check that out.

Because sometimes q is only true *because* p is false.

BTW, at this point, the necklace you're wearing is supposed to start blinking and smoke needs to come out of your ears...

here's another thing. assume p -> q. like those guys up there said a million times, if p is true, then the entire statement p -> q is true. q is not necessarily true, and is not necessarily false. it's just that p -> q as a statement is still valid, because it has not been proven to be false.

for example: assuming the statement "'Ed is a dog" implies "Ed has four legs'" is true:
if Ed is, indeed, a dog, then he has four legs
if Ed is not a dog, then he may or may not have four legs, but since you haven't proven the implication false, it is still true, since you have not proven that there exists an Ed that is a dog but does not have four legs.

my main point is, you're saying "if one's hypothesis is false, how can one derive a conclusion that is true?", and you're way off base. your hypothesis, that Ed is a dog, isn't saying that your conclusion, that Ed has four legs, is correct if Ed isn't a dog. it's just saying that, as a statement, "Ed is a dog implies Ed has four legs" is still valid, and makes no comment whatsoever on whether or not Ed the non-dog, indeed, has four legs.

Oog.

Oh, shut up, Data

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How it works

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