Oog.

[logomancer]

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.

Comments

How it works

[matthiasrat.livejournal.com]

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.

Re: How it works

[robertliguori.livejournal.com]

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?

Re: How it works

[cxi162.livejournal.com]

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();

Re: How it works

[matthiasrat.livejournal.com]

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.