Fuzzy Logic

A few days ago I was trying to explain what logic is, what metalogic is, and what fuzzy logic is. After I was done I decided I'd try to make a quick post and explain those terms....and I also decided that while I was at it I'd explain some of the more useful logic terms and their definitions. A lot of these logic words have creeped into our language (but we usually use them wrong)...so, here's a quick lesson in logic.

Valid: a valid argument is one where you say that given the premises the conclusion must follow. This is an argument like

All dogs are mammals
All mammals have fur.
Therefore, all dogs have fur.

This is valid. Validity has nothing to do with truth. Validity only means that the premises support the conclusion. I could say:
All women are kind people
All kind people cook for me
All women cook for me.

I can tell you from my fast food diet that this is not true. But, it is a valid argument (given the premises).

Sound: When you have a valid argument with true premises it is sound. So, the first argument is Sound (in that the premises are true and it's valid). The second argument is valid but it is not sound (because all women don't cook for me).

Deductive: all arguments that are valid are deductive arguments. Deductive arguments have necessary conclusions (conclusions that must follow from the premises).

Inductive: inductive arguments are arguments that present something as likely. For example, if I said the weatherman said it will likely rain today, I think I smell rain in the air, therefore it will likely rain...this is an inductive argument. In an inductive argument you present premises that support a conclusion, but even if your premises are true, the best you can get is a conclusion that is likely. Almost every argument you make is an inductive argument. In an inductive argument there is no necessity. Inductive arguments come in two types:

Strong (or cogent) and weak.

Strong inductive arguments are likely and the premises support the conclusion well.

Weak arguments have premises that don't really support the conclusion.

Fallacy: Many people use the word fallacy. They say "oh yeah, that's fallacious." What is important to remember is that just because an argument is fallacious doesn't mean it's wrong. A fallacious argument is this:

Fallacious: an argument that appears stronger than it actually is.

So, if I said "You want universal healthcare. Did you know that Hitler wanted universal healthcare too? Yeah, that's why universal healthcare is a bad idea." I'd be using the Hitler fallacy (it really is a fallacy, it gets used so much they've named it).

This is fallacious because I'm using Hitler (and all of his negative connotations) to strengthen my argument. I'm not saying anything about the issues...I'm only scaring people into agreeing with me because they don't want to be compared to Hitler. This makes my argument fallacious.

However, as you can see, just because an argument appears stronger than it actually is...it could still be right. So, that's what a fallacy is.

I think that's probably enough that you can throw out some terms and impress your friends (as if anyone is impressed by using these words correctly).

Now for metalogic. Metalogic is logic used to prove logic. In metalogic we make proofs that show the limits of what we can do with logic. In metalogic you talk about set theory, infinite sets, power sets that are larger than infinite sets (yeah, something larger than infinity) you talk about decision procedures in logic (whether we can make positive and negative claims about properties). I could try to explain more but it wouldn't make very much sense. All metalogic is is a way of proving what logic can and can't do.

Fuzzy logic. My dad always says this. It's actually a real type of logic. Fuzzy logic isn't just weird thinking, it's talking about things that are a mixture of true and false.

Here's a paradox: I'm telling you a lie right now.

This is the liar's paradox. It's very famous. It goes like this. If I'm telling you I'm lying, then I'm telling you the truth, but it's only the truth if I'm lying, but if I"m lying I"m telling you the truth, and if I'm telling you the truth I"m lying, and if that's a lie then I'm lying and it's truthful........forever and ever and ever and ever. This is a paradox. Can't fix it. People have tried, and some people have come up with some very elaborate systems (often using metalogic) to discuss the way we predicate things like lying and making truth claims. Some of the greatest minds in the word have stumbled over that.

Anyway, if you wanted to say whether that statment is true or false you couldn't. In logic this is called "a glut." It's a statement that is both true and false.

In fuzzy logic you deal with things that aren't true or false...they're somewhere in between. The formal definition of fuzzy logic is this:

0 equals false. 1 equals true. Fuzzy logic deals with claims with truth values between 0 and 1.

(in logic it works like a computer with 0 and 1 being false and true respectively).

So, what would a fuzzy logic problem be? Well, take for an example a child. I could see a 5 year old boy and say "that boy is a child." I just predicated to that boy that he's a child. This is a vague predicate. For when is the boy not a child? Think about it this way.

At 5:00 and 1 second I say "that boy is a child."
At 5:00 and 2 seconds I say "that boy is a child."

And if I did that a hundred times, adding a second each time, it would still be true that that boy is a child. So, you'd like to say that "it is true that if a child is a child, they will be a child at the next second." But, we have to admit that enough seconds add up to 30 years, and a 30+ year old person is not a child. So somewhere along the way that child is no longer a child. But what second is that? Can you point to some second and say "oh, that's the second he became a man"? Nope, it's a vague predicate (not referring to legal definition of adulthood). Most of the predicates we use everyday are vague. These are the predicates that fuzzy logic explores.

Well, that's about it for my writing. There is a lot more to write, but I've gotta get ready for tonight (and by get ready I mean put shoes on and walk out the door (great being a guy)).

Have a great weekend!