## Friday, September 29, 2006

### Negatives

In the movie Kelly's Heroes, there's a completely anachronistic character called Odd-ball, played brilliantly by Donald Sutherland (Mr. Bennet in the Keira Knightley Pride and Prejudice). He's famous for delivering, during World War II, such sixties hippy phrases as, "What's with the negatives waves?"

While such a thing is certainly hilarious, I wonder if there is not something philosophically deeper going on in today's culture. To illustrate what I'm talking about, I'm going to dive into some technical material which I hope I explain clearly. I would STRONGLY encourage you to try to follow my train of thought.

In symbolic logic, there is a concept called "adequacy of a set of connectives." The idea goes like this: in certain kinds of logic, you have what are called Boolean variables; they're just like an on/off switch or a bit in computers. The Boolean variable can hold the value "true" or it can hold "false". So there are two options for one Boolean variable. Suppose you have two Boolean variables that you can distinguish one from the other. How many different ways can those variables be set? Well, you could have (true, true) or (true, false), or (false, true) or (false, false). So that makes four possibilities. If you look at three variables, you'll see that you can have eight possibilities. So in general, if you have n Boolean variables, there are 2 to the n possible ways of setting those variables.

Now introduce the concept of a function of Boolean variables. What is that? Well, we'll build up from scratch again. Suppose you have one Boolean variable, call it x. There are, as we already discussed, two possible ways to set that variable. Now a function f of the Boolean variable x, which we will write as f(x), is another variable whose value depends on the value of x. How many functions of one Boolean variable are there? Well, you could have the function that is always true no matter what x is (the true function), or also the function that is always false (the false function), or you could have just x right back at you (the identity function), or you could have the function that changes true to false and false to true (the negation function). So that's four possible functions of one Boolean variable. It turns out that if you have n Boolean variables, there are 2 to the 2 to the n possible functions on those variables.

Let us examine quietly one possible function, call it q, on two Boolean variables, call them x, y. So we'll say that q is true only when both x and y are true. So it looks like this:

q(true, true) = true
q(true, false) = false
q(false, true) = false
q(false, false) = false.

We have a notation for this function: ^. So we write q(x,y) = x ^ y. This is called the "and" connective, because it connects x and y in a functional way.

There are many such connectives, including "or", "exclusive or", "nand", "nor", "Sheffer stroke", and others. One question that logicians have concerned themselves with in the past is the following: given an arbitrary function on n Boolean variables, can I write that function using only one or possibly two known connectives? This is the question of adequate connectives.

It turns out that there are only two adequate connectives (of the possible sixteen functions on two variables). Both of them have the word "not" in their description.

My patient readers may be wondering why I dragged them through so much symbolic logic. The reason is this: the conclusion many have come to is that in order to be complete, or adequate to express things, you must be able to say "no" in some manner or other. No language is going to be adequate to express everything worth expressing if it cannot do negation.

This grates against the sensibilities of the post-mods, and the politically correct, who want to employ a "both-and" kind of logic. "You can embrace both this and that," regardless of the fact that they're contradictions. They can't do this consistently. As Ravi Zacharias said once (I paraphrase), "Either you use the both-and logic, or you don't, isn't that right?" And the person he was speaking to, a professor of Eastern religions, said, "The either-or logic does seem to emerge, doesn't it?" Whereupon Zacharias replied, "Yes, and I've got some shocking news for you. Even in India, we look both ways before crossing the street. EITHER the bus, OR me, not both of us." You can listen to the whole broadcast by going here, and clicking on the link for 2006-01-30. This acceptance of contradiction is nonsense, and leads to intellectual suicide. You must be able to distinguish between things. If you read the Bible at all, this should become very clear; just read Leviticus. In Leviticus, one central theme is the holiness of God. God was to be approached only by what was "clean," and NOT by anything that was "unclean." Clean and unclean was defined in excruciating detail by divine fiat. God gave priests a great many rules about how to decide if a thing or a person was clean or unclean.

R. C. Sproul once said, "If it's the prerogative of the woman to change her mind, it's the prerogative of the theologian to make distinctions." Distinctions are absolutely necessary in order to talk intelligently about just about anything.

So my little analogy from symbolic logic is that you have to be able to negate things in order to fully realize the expressive capabilities of your system. But this carries over into language, philosophy, and theology. I would claim that this concept comes from the Bible, and not from logic. I only started with the logic point for rhetorical effect: to argue from the lesser to the greater.

If someone comes to you, claiming that you are "intolerant of their views," first ask them what they mean by that. What they most likely want you to do is to accept all these views as equally true. It's just relativism thinly disguised, or not disguised at all. Christianity, though, makes absolute claims. "Jesus said to him, 'I am the way, and the truth, and the Life. No one comes to the Father except through me.'" - John 14:6 (ESV). That is an exclusive statement; if it is true, no other claims can also be true. Your detractor might say, "Well, that's true for you, but not for me." This is nonsense! They are claiming there is no absolute truth. In which case you just ask them whether the statement, "There is no absolute truth" is true for everyone, or not. The fact is, they've just uttered an absolute truth.

Philosophically, we are "trapped" into believing in absolutes. We cannot consistently do otherwise. But let us remember that the truth will set us free. But falsehood, the negation of truth, will do no such thing.

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At 10/02/2006 03:13:00 PM ,  Susan said...

My patient readers may be wondering why I dragged them through so much symbolic logic.

*chuckle* Ah, so true :). I must say, this was an interesting post. I even followed it, I think.

This sort of reminds me of something I cover in Geometry: definitions! One of the best ways to begin discussing or defining something is to think of what a thing is not, not just what a thing is. It's all well and good to define something directly, but when I begin asking my students is it _______?, or is it _______?, then they start seeing a fuller picture of the geometric object and are able to more eloquently define a term.

Another tie to geometry and negativity: proof by contradiction! It's far easier (in my estimation) to prove something by contradiction than directly. It forces the mathematician to consider why all the other conclusions fail, thus leaving only the valid conclusion. Beautiful.

At 10/02/2006 03:29:00 PM ,  Adrian C. Keister said...

You are correct in thinking that proofs by contradiction are often easier. However, they are not as useful as constructive proofs, because constructive proofs actually give you the answer. It's also easier to make a mistake in proofs by contradiction (did you really rule out all the possibilities?). For these reasons, many mathematicians rank various subjects according to age by how many constructive proofs there are in the field. I remember taking Graph Theory here at Tech, and remembering that there were a great many proofs by contradiction; it is also a young field. I also remember that the few constructive proofs were very beautiful indeed. On the other hand, all the analysis courses I ever took had loads of constructive proofs.

Contradiction proofs are certainly there, and they are useful at times. But if there is a constructive proof, I think overall, they are to be preferred.

In Christ.

At 10/02/2006 09:16:00 PM ,  Susan said...

It's funny to hear you refer to VT as "Tech," because in Georgia, "Tech" is Georgia Tech.

Well, I'm certainly not claiming that proof-by-contradiction is the most elegant method, just that it fits well with my brain waves. I relish finding inconsistencies, so those types of proofs are fun. I like straight-forward proofs as well, mind you, and of course you are right that it is always a possibility to forget to check all alternatives. I've done that a few times :).

At 10/04/2006 11:34:00 AM ,  Adrian C. Keister said...

Aha. So you're a heresy hunter, eh? ;-)]

Hey, like my new look? I can't say that your efforts to get me to have a nice look weren't relevant, but I will say that I eventually decided the black look was a little too hard to read.

In Christ.

At 10/04/2006 12:59:00 PM ,  Susan said...

I just logged onto Google Reader, clicked to check my starred posts, and I was so disoriented when I clicked on this post! What's this? Cumberland Island is no longer a dungeon? My eyes do not have to adjust in order to read your posts. Ah, sweet bliss ;-). I quite like the new look. Nothing fancy or nice by way of design, but I'll take it. . . and Mother Dear's eyes will be grateful. Your blog actually looks inviting now. Before it looked like a dungeon that was trying to keep people away ;-).

Yes, I guess you could call me a heresy hunter of sorts. Actually I used to be one to an extreme. Now I pray for moderation in that area. I think Presbyterianism lends itself to heresy hunting, don't you know? Our greatest strength is also our greatest weakness.

At 10/04/2006 03:40:00 PM ,  une_fille_d'Ève said...

Hooray! No more black!

At 10/04/2006 04:38:00 PM ,  Anonymous said...

I am all astonishment! I love the new look! My old eyes thank you.

-S's and H's Mother Dear

At 10/04/2006 05:20:00 PM ,  Adrian C. Keister said...

*smiles* Well, the G. Dears are all very welcome, I'm sure. I was wondering something: is there any way of using a template that is not provided by Blogger? Or of tapping into a few more choices than are given?

Heresy hunting is, alas, necessary. But let it be done only by the most mature Christians, the most humble ones, the ones most grounded in Scripture. That rules me out, though I have no qualms whatsoever about having my own opinions on things like the Auburn Avenue heres... I mean, viewpoint. ([sic] the whole last sentence.) Ditto for N. T. Wright, Norman Shepherd, and the New Perspective on Paul.

In Christ.

At 10/05/2006 07:44:00 AM ,  Susan said...

See, you should have done this long ago. . . just to thrill us. I forgot to mention before, but I love the sidebar with your book reads :). When I registered my blog, that was the first thing I wanted to include on the sidebar.

I am completely ignorant when it comes to non-canned websites, so I can't answer your question about templates. I knew absolutely no HTML before I started blogging, and I had to get Ashley to help me set up my blog.

I agree, by the way, that heresy hunting is necessary - certainly for the movements listed above, for example. What I was bemoaning was heresy hunting for things that aren't heresies or an ungodly zeal for heresy hunting. When we approach heretics, it should be with brokenheartedness, which I haven't done in the past.

I like, for example, Francis Schaeffer's remarks when the PCA was formed by splitting from the theological liberals (heretics, if there ever were any).

When it is no longer possible to practice discipline in the church courts, then you must practice discipline in reverse and leave. But your leaving must be with tears, not with flags flying and bands playing.

At 10/05/2006 10:58:00 AM ,  zan said...

Oh, this look is SO much better. Before I felt like I was going cross-eyed when I read your posts.

I think this color sceme serves you well. Very no nonsense, masculine and practical, like your blog posts.

Getting phoofie colors wouldn't suite your blog, in my opinion.

At 10/06/2006 11:42:00 AM ,  Adrian C. Keister said...

You like my book reads? That's a page from - ha - your book. Question: I like having the title of my blog, Cumberland Island, to be a link for itself. I've managed to do this via altering the template HTML. The only drawback is that the tab in Firefox now says all that gobbledy-gook code I had to type in instead of just good ol' Cumberland Island. Is this a terrible thing? I mean, loads of web sites do that so that it's easy to get back to home. Maybe the link to My Blog on the right is sufficient. What think you?

Agreed about heresy hunting. It's never a cause for celebration, like some think. If it is, then there's surely pride involved: "Look at me. Aren't I great for discovering this heresy?" Good quote there.

Why, thank you for the compliment on my masculine and practical posts. So they're not too theoretical, are they? ;-)]

In Christ.

At 10/07/2006 06:36:00 AM ,  Susan said...

I personally think it's convenient to have the tab read the title of the post, so if I minimize a few I can see what I minimized. But then, I'm just an ignorant IE user who doesn't understand the advantage to your blog having a link for itself :).

At 10/07/2006 10:14:00 AM ,  Mr. Baggins said...

Adrian, you can type in "templates for blogger" and find some other templates that others have used. I like your new look much better than the black. Blogger itself has instructions for how to change the template (which code you have to change).

At 10/09/2006 10:12:00 AM ,  Adrian C. Keister said...

As you can see, I've decided to just get rid of the self-link and keep the template as is. I'm fine with it... Zan convinced me it was good. :-)]

In Christ.

At 10/09/2006 12:27:00 PM ,  Susan said...

I see the alteration. Goodie for Zan :). Is it alright to TIOC?

At 10/09/2006 07:50:00 PM ,  Adrian C. Keister said...

Sure, let's TIOC.

In Christ.