Peirce lays down the law!

I happened across Peirce's Law the other day; it's a logical form that does some neat stuff. I'd seen it before in the textbooks, but it's only now become apparent to me that it's full of import. Peirce's Law is basically a complex conditional form, like you'd find in algebra.

A simple conditional goes like this:

P → Q

P and Q are variables, where you could substitute - let's put "logicians" for P and "cool dudes" for Q. The arrows (→) show that these are if-then statements, i.e. showing implication; so
may be read variously as

If somebody is a logician, they're a cool dude (or dudette).

All logicians are cool dudes (or dudettes).

Peirce's law reads something like this:

((P → Q) → P) → P

If the case of logicians being cool dudes (or dudettes) implies logicians, then there are logicians.

What's interesting about this is the fact that it fixes an antecedent (P); by this I mean it establishes something (logicians, for instance) as being inevitably there. In ordinary conditionals, logicians are only cool if there are logicians; they have to exist in order to be cool. Which makes for a very contingent thing - we have to hope for some logicians out there. Peirce's Law says that the relation of logicians and coolness determines that there are logicians out there. So if we understand what it means to be a logician, we can confidently say that they are pretty darn cool.

"Big deal," I can hear you say. "Whatta ya gonna do widdit? Here's a suggestion: drink three pots of tea, then go write your formula in the snow. "

I would, but there's no snow. So let me unpack this a little more instead.

Anybody who reads Peirce in Philosophy 101 probably gets his article "How to Make Our Ideas Clear." This is where he formulates his famous pragmatic maxim: "Consider what effects, that might conceivably have practical bearings, we conceive the object of our conception to have. Then, our conception of these effects is the whole of our conception of the object."

To talk about effects presupposes a cause. So you're presupposing a causal relation. Now if a causal relation implies a cause, it follows that the cause is already in play. So think effects, and you think cause.

This is important because we never observe causes. Effects are what we observe; they're reactions, and what we see is other than us. Our eyeballs react to the light, which comes from a source. Causes are never observed, they are inferred. So if it's a new thing, you guess at the cause, and see whether it really is so. That's what hypothesis is all about.

Concepts also have a causal relation. Think "2 + 2," and you'll probably think "4." Antecedent, consequent. Think Belgium, and you may think waffles, chocolates and beer. Stimulus, response - cause, effect. I won't go into the habit-thing here, that's far afield (though closely related - !) to the matter at hand.

"Whaddaya gettin' at?"

This: knowledge begins with hypotheses, so we need the strongest way to frame hypotheses. Peirce wanted to ground knowledge, so he gave us pragmatism. Now from every appearance, it looks like Peirce got this maxim from his laboratory experience. (He was a scientist and a logician - as well as a lot of other cool stuff.) The trouble is, any scientist knows there's a margin for error in observation, meaning even the most accepted laws are susceptible to revision or rejection. But try this on for size:

((P → Q) → P) → (Q → P)	((P → Q) → P) → (Q → P)	((P → Q) → P) → (Q → P)
If a conception of an object has effects,	thereby implying that conception of an object,	then the effects imply the conception of the object.

The pragmatic maxim fits Peirce's law almost to a T - which is what you'll get if you make a truth table for this eminently valid form. This is how Peirce could equate the effects of a conception of an object with the object itself. The pragmatic maxim didn't just come out of lab work, nor even psychology; it's pure logic, baby.

It's a handy tool to have around, this Peirce's Law. Girls, proceed at once to reel in the guys and rock their world:

Guy: What's a nice girl like you doing in a place like this?
You: It's Ladies' Night, and I think you're gonna get me drunk, cheap.
Guy: Somebody stop me, I'm in love with the coolest dudette ever.

And boys, you can really impress the chicks, just like the pros:

You: You know how you keep tellin' me heavy metal sucks 'cuz it makes you headbang? Well, that's what you think!
Chick: Dude, you're so cool.

Now go out there and Lay down the Law! Hoo-whee! This gun's so hot, it's sweatin' bullets!

(Image nefariously doctored using originals unwittingly donated by www.iep.utm.edu/p/PeirceAr.htm and www.tvcrazy.net/tvclassics/americantv/bj.htm)