Peirce's Pragmatic Law - The Point of It All

I’m afraid I got ahead of myself. For anyone who’s been patient enough to follow me through this working-out, you may be wondering what the benefit of all this is. Let me put the point right up front: The pragmatic maxim gives you a real good tool for sharp thinking. You could say it’s the Ginsu knife of logical tools. What I’ve been doing is spelling out exactly how it works. (The maxim, not the knife.)

It might help to situate this finding in the context of Peirce’s view of reasoning, since he was a big fan of logic – to say the least. As a scientist and philosopher, good thinking is essential; it comes in handy in lots of situations, and everyone knows how sexy good thinking is. So every little bit that helps thinking, helps your life. Peirce knew this. In 1876 he wrote,

All thought rolls upon one thing following from another. That which follows is inferred deductively from that which it follows. This from which something else follows is inferred inductively or by hypothesis, from the consequent. Thus the relation of antecedent to consequent is the most important of all relations to us…[1]

So what is the object of the pragmatic maxim? To clarify concepts and weed out hypotheses that don’t help thinking. The PM is specifically for governing the logic of abduction, or hypothesis – guessing. Hypotheses are basically educated guesses. As readers know, some guesses are sensible, but others may just be off-base; scientists know this too. The reason for the difficulty in guessing is that there seems to be no real rule for it.

This calls for a quick-and-dirty guide to reasoning. There are basically three forms of reasoning:

1. deduction,
2. induction, and
3. abduction.

Deduction is what most people think of, when they think of logic – stuff like

Rule	All detectives are brilliant reasoners.	Premiss
Case	Sherlock Holmes is a detective.	Premiss
Result	Therefore he is a brilliant reasoner.	Conclusio

Rule –-> Case –-> Result. What’s cool about deduction is that the conclusions are certain – no doubt about it. Admit the premisses, and you have to admit the conclusion too. But you can see that knowledge isn't really extended at all here; we've just picked out something in our rule.

Induction is what you do when you generalize. The more evidence you’ve got, the more support for your conclusions:

Result	Sherlock Holmes is a brilliant reasoner.	Premiss
Case	Sherlock Holmes is a detective, like Miss Marple, Hercule Poirot, Maigret,…	Premiss
Rule	Therefore all detectives are probably brilliant reasoners.	Conclusion

Result –-> Case –-> Rule. Sometimes we can make assertions about every member, but not always: “John Bull is a terrible cook. He’s an Englishman, and so is Tony Blair, John Cleese, and… – but Jamie Oliver is pretty good. So not all Englishmen are terrible cooks (just most of them).” So induction is handy for testing the extent of a notion. But, like deduction, it doesn't really extend knowledge either. We don't get new ideas from it, we just test the ones we've already got.

Abduction is how we extend knowledge. As we saw above, it's basically guessing. Peirce was extremely interested in the thinking going on behind guesses, because that's how you extend knowledge. What you see here is a more fleshed-out version of what we do when we take a guess:

Rule	All detectives are brilliant reasoners.	Premiss
Result	Sherlock Holmes is a brilliant reasoner.	Premiss
Case	Therefore Sherlock Holmes is a detective.	Conclusion

Rule –-> Result –-> Case. Another example might help make things clearer:

Abbott: Who’s on first.

Costello: I dunno.

Abbott: No, dummy, Who’s on first.

Costello (thinking): “Who” might be the guy’s name (because everyone has a name, and this guy on first base has a name too).

There are some pretty good guidelines for inductive reasoning, but not many - or any – for guessing. We’re pretty much left to our own devices. Learn from experience. Heuristics are nice, but not foolproof: sometimes they work, sometimes they don’t.

Enter the pragmatic maxim. The PM was designed specifically as a rule for making abductions.[2] What it does is make us aware of what we’re thinking and what we can do with it, so that we can tell whether a certain guess is worth making or not. What’s extremely cool about the PM is its form: it delivers truth in spades.

Granted, it's a sort of "This is true as far as it goes" kind of truth, but that's something. But when making guesses, every bit helps. What's more, that support lends to the surety of the concept - and a few well-done concepts are much better than a bunch of half-assed ones. Think the old story of the fox and the hedgehog: the fox knows lots of things, the hedgehog only one big thing. (In case you were wondering, I think Peirce could be called a hedgehog. And I think he'd take that as a compliment.)

So the PM gives us the conclusive power of a deductive reasoning, while allowing the freedom needed for abductive reasoning. And you can use it in a whole lot of different walks of life. Clarifying ideas – it’s not just for philosophy anymore.

What I’ve done is simply show the logical form underlying the PM, which is what gives it its power. I’ve also drawn some conclusions that seem to follow from the maxim itself. Hopefully my thinking has improved after chewing on this stuff for a while, and if it is helpful to you, then all the better.

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[1] W3,203 (1876).

[2] EP2:234–5 (1903).

Further Extensions - Out on the Leafy Edge

Before the intermission, I argued that Peirce's Law/pragmatism leads to a realism. I'd like to cap that argument with the mother of all reals: the categories. I hope this won’t take as long as the last post did.

Readers of Peirce are familiar with the three categories – Firstness, Secondness, and Thirdness. This here is not an exposition of those, but rather a demonstration that the signs we use to refer to them imply that they are real. Peirce argues that they are. It will involve a little of his theory of signs, so bear with me.

Signs involve an object, the sign, and an interpretant. The object can’t be the sign itself; it has to be something else, something other than the sign. Now for interpreting the sign, other signs are referred to. We have to use words to talk about something, and we do the same with the categories. But that has to do with the interpretant. The object is simply the thing pointed at: there's just brute reaction here, no intellectifying.

You: What are you talking about?

Tarzan (pointing): Unh!

The categories are genuine signs. That means they definitely refer to something apart from the signs. By the general structure of the signs, it follows that the categories must be real objects and not nonsense, as he determined the Liar Paradox to be. (For those who care, the Liar is nonsense because it refers exclusively to itself. But as pointed out, a sign cannot refer to itself, so it must be nonsense.[1]) The fact that it is the object of the sign means that it is other.

We further suppose that the object caused the sign, and not the other way around. Thus the sign points toward its cause, meaning it tacitly assumes a causal relation. And if that’s the case, and such a relation implies a cause, it follows that the categories must be real.

Now we can try it another way: by tracing back the logical chain. Start wherever you please: pick a thought, any thought. Call it Q. That idea must have been triggered by something else – an idea, no doubt – so let it be a Q to the P that set it off. Assuming that there is a relation, Peirce’s Law determines that it must have been stimulated by the idea P. You check out the effects of that concept in order to know that concept, meaning you’re killing two birds with one stone. That idea in turn was triggered by another one, leading to the same setup as before. This goes on, ad infinitum.

If we start generalizing, as we sure can with ideas, ultimately we’ll arrive at the simplest ones possible. Peirce will argue that these are the irreducible Big Three: Firstness, Secondness, and Thirdness. This is evidently arrived at by way of the pragmatic maxim/Peirce’s Law.

If the categories are real, everything else has to be too. So this portion of the conjecture should be the clincher. So ballsy conjecture #5 is complete.

Of course I'd like to think all this is correct. But knowing myself, I can't be 100% sure. So I'm airing it out to see what needs fixing. If it doesn't need fixing, maybe it'll be useful to someone.

Be fair. Use, and cite this! Plagiarize me, and I'll have yer damn scalp quicker'n you can say "Ward Churchill!"

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[1] This he concludes as early as 1865 (see W1,174), and he reaffirms it in 1903 (see EP2:166–9).

An Intermission

Still working on the next addition to the Peirce series. Meantime, enjoy the show. I probably shouldn't find this amusing, and yet...I hope you do too.

Extensions on Peirce's Pragmatic Law

In a previous post I went out on a limb to argue that Peirce's pragmatic maxim is formally identical to Peirce's Law, a form found in many logic textbooks today. In a later post, I went a bit further out by suggesting an avenue for (re-)constructing a logical proof of pragmatism; and still further by stating that pragmatism is basically correct if my shpiel happens to be right.

Today I'm going to inch closer to the leafy edge by making a further conjecture, tediously illustrated.

Ballsy conjecture #4: The development of Peirce's philosophy can be seen to assume the pattern of Peirce's Law. In other words, it thoroughly informs his thinking.

If pragmatism is the rule for right reasoning, as he argues in the 1903 Harvard lectures,[1] and if he duly acts on that, we should be able to discern it in the form of his reasoning.

This may be only of interest to scholars, but it seems to me that it would make sense of the trajectory of his thinking. That's got to count for something. Here's a case I've found so far. In 1902 Peirce criticizes Aristotle's venerable argument for first principles.[2] The Stagirite states that every premiss depends on scientific arguments rest on more general principles; those may well in turn rest on still more general principles. If anything is to be true at all, there must be a stop to the regress - i.e. most general principles. Peirce disagrees. He likens the Achilles paradox except we're going backward in time rather than forward (This would be because thinking takes time. Nope, there are no two ways about it - go ahead, try.)

Reasoning would be like what Lewis Carroll pictures in "What the Tortoise Said to Achilles", first published in Mind in 1895.[3] Suppose, the tortoise says, you argue that premisses A and B lead to conclusion Z: what if he accepts the premisses but refuses the conclusion? Well, Achilles can introduce premiss D: If A and B, then Z must follow.[4] What if the tortoise refuses that? Simple – add premiss E: If A, B, and D, then Z. And so on.

Effectively this is attempting to state that if a rule of inference is accepted, along with certain premisses, the conclusion necessarily follows. It would be equivalent to saying that if the rule is accepted, the consequent follows out of necessity. Formalized, it lo0ks like this:

{(C	→	Z)	→	Z}	→	Z	…
T	T	T	T	T	T	T	…
T	f	f	T	f	f	f	…
f	T	T	T	T	T	T	…
f	T	f	f	f	T	f	…
	⇑		⇑		⇑		…

You can see the pattern that is set into motion here: as long as we are trying to derive the conclusion from the premisses, we will never actually reach it by means of the reasoning alone. The arrows ⇑ at the bottom of the table mark the truth values accruing to the conditionals as each successive ramification is added; one can also see that the alternation will continue infinitely. In posing this paradox, Carroll has questioned the notion of implication.

Peirce would reply that the question is misdirected at the outset. He appeals to the act of thinking things out: in the process of thinking we are so busy focusing on the object under consideration that we cannot attend to the process. Once we have reached a point where we can pause and reflect on the path leading up to this conclusion, we search for propositions that do lead up to this conclusion. But the argument represents the last stage of his thought, not what led up to it: we say X is true if Y and Z are true – it follows, granting the premisses. The key element here is thought: thought is thinking about something, whatever that may be. If we try to clarify this, we find that thought is starting on a percept. But percepts cannot be articulated in propositions, for they are not products of thought; therefore the first logical particles are perceptual judgments or facts. But these are radically different from the percepts themselves, meaning that thought is developing on its own processes. What this means for the tortoise’s challenge is that he is right – but only if his picture of reasoning is right. And this Peirce would deny. We do not employ logical rules in order to reach conclusions, we use them to articulate the conclusions we have reached already. So the form (P → Q) → Q gives an improper picture of how logical rules are applied.

The error lies in supposing that the conclusion Q can follow from the relation P → Q alone. A more fruitful inference to make is this: If a relation of inferences to the conclusion implies that we took the steps leading up to that conclusion, then those steps were taken. In one fell swoop, both the tortoise’s paradox is dissolved and the validity of the premisses affirmed. And in doing that, there is no need to justify implication; it is merely posited hypothetically. That it happens to work every time enables us to say that although there is no reason for granting implication, there is no reason against it. Meanwhile all evidence supports its validity, and it has been astoundingly successful. Therefore, until further notice, we have every reason to accept implication as a valid form of inference

We have seen this sort of reasoning – in Peirce’s Law. Much of Peirce’s argumentation is of the form ((P → Q) → P) → P; indeed, it is arguably the framework of all of his researches. Assuming he takes his own pragmatic maxim to heart – a maxim which bears this form as well – his analysis of conceptions follows this train of reasoning. The form has been proven valid, and it explains the maxim; for an explanation is “the adoption of a simpler supposition to account for a complex state of things”, and the logical form is more general. This suggests, then, that the elusive proof of pragmatism is to be found in the proof of Peirce’s Law. The modifications of the maxim are but refinements of the great project which he maps out in the Popular Science Monthly series.

This is a belabored explanation of a single point. But I'd like to think it's typical of the sort of reasoning Peirce employs throughout his career; at least the conjecture I'm making suggests it. It's actually not so controversial as the next assertion I'm holding out for examination:

Ballsy conjecture #5: Peirce's pragmatism leads to realism, and does so by means of its form.

It's pretty well known that Peirce was a nominalist early on in his career. He eventually adopts a realistic stance, but it takes some time. (To be honest, I'm not so sure it's that simple - my own inclination is that he always was a realist, though his intellectual professions said otherwise. But that's another post.)

Taking the conjecture a step further, it seems to me that it leads to realism of its own accord. Suppose that the things we perceive - percepts, as Charley called them - are effects. This means that where's a cause, there's an effect. If that's the case, then it follows that they're caused by something real. This is exactly how we conceive them: images are caused by something.

Note that I am not saying that the percepts themselves necessarily imply a cause, only that we think them so. So I'm not misapplying the pragmatic maxim here. I'm saying that we conceive percepts as part of a causal relation; this means we suppose that real things cause those percepts. And what holds them together is this relation of cause and effect. It is our link to an external reality, and a refutation of nominalism.

In another post, I wish to show that this realism extends to the broadest conceptions possible: the categories.

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[1] CP 5.196 (1903).

[2] See CP 2.27 (1902), of which this is a boring reiteration.

[3] Mind, N. S. vol. 4 (1895), p. 278

[4] W4,547 (1883–4).

Peirce's Pragmatic Law - a Conjecture

In a previous post I noticed how useful and versatile Peirce's Law can be: not only can you understand the pragmatic maxim, you can score with babes. Does it sharpen knives? Well, no, but it might help you figure out how.

In this post I'd like to demonstrate something rather far-ranging about this finding.

Peirce's Law, you will recall, is "((P -->Q) --> P) -->P." If the relation of implication implies the antecedent of that relation, then the antecedent is the case. Peirce coins a certain symbol of his own, which we'll have to represent here as -<; nowadays we render it as an arrow -->. So don't panic, just think "All right, -<." So Peirce himself describes it like this: That it is true appears as follows. It can only be false by the final consequent x being false while its antecedent (x -< y) -< x is true. If this is true, either its consequent, x, is true, when the whole formula would be true, or its antecedent x -< y is false. But in the last case the antecedent of x -< y, that is x, must be true.

This can be succinctly represented in a truth table like so:

x	y	{(x -< y)	-<	x}	-<	x
T	T	T	T	T	T	T
T	f	f	T	T	T	T
f	T	T	f	f	T	f
f	f	T	T	f	T	f

What this means is, given any case of two entities x and y, the statement "((x -< y)" must be true.

Now recall that the pragmatic maxim is like this: “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.”[1] This is easily shown to have the same form as Peirce's Law. Here is the pragmatic maxim, framed in the most assumption-free manner:

((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 those effects imply the conception of the object.

This has the necessary validity as Peirce's Law, without loss or distortion of the maxim. And it does so without presupposing anything. In other words, pragmatism has a firm, presuppositionless foundation. The fewer the presuppositions, the stronger the system.

What does this mean? If the pragmatic maxim is formed in terms of Peirce's Law, then it must be true in any case. In other words, the proof of pragmatism is to be found in its logical form, which is a variation of Peirce's Law. The truth table is a quick-and-dirty logical proof of pragmatism, which turns out to be a self-supporting structure - and the one rule for making hypotheses.

Peirce himself promised a proof of pragmatism, and especially built his case in 1903, but a proof-proof never quite emerged. He intended to use his Existential Graphs for that, but for some reason... Several scholars think it's because he never settled on hammering out a satisfactory account of continuity.[2] For my own part, if Peirce didn't actually go about rendering the logical form of pragmatism in EG, it could well be the way to go.

If EG's validity as a logical notation is demonstrated, Peirce's Law could be mapped out and unpacked by means of EG. This would afford a visual, iconic proof of the proposition.

OK, time now for some ballsy conjectures. I've made two already (WARNING: Pun ahead): the first conjecture started the ball rolling: pragmatism is best formulated as above. The second follows from that: Peirce is and must be correct to the extent that concepts are developed along the lines of the pragmatic interpretation of Peirce's Law.

Ballsy conjecture #3: Peirce seems unaware of the logical form of his maxim. If he was aware of the connection I'm asserting, he didn't let on. So far I haven't read anything where Peirce says, "Folks, the pragmatic maxim has this form, which is necessarily true." This tells me that

(a) I haven't found anything of this sort, but only because I need to keep reading;
(b) nobody has found anything of this sort, meaning we all need to keep reading; or
(c) he didn't write anything of this sort.

Now (a) is definitely true, I've got a lot of work to do; (b) is possibly true, though I won't bet on it. What about (c)? That can't be said until everything he ever wrote has been read, and I dare say nobody has accomplished that. But until further notice, it seems to be the case. He did argue for a long time that categorical propositions were of the same logical form as conditionals (which is common practice in logic today), and perhaps after publishing his belief he took it for granted. That's just a speculation, though.

In 1873 when he was developing the insights that would later emerge as the pragmatic maxim, he repeatedly points out that effects are what earmark something; a cognition has no existence if it generates no further cognitions.

In 1878, Peirce announces the maxim without, however, going into its form. Instead he frames it in terms of belief – which is consistent with a logical mode, but does not prove it.

I don't see evidence that Peirce was thinking of his Law when explaining pragmatism in the 1903 Lowell lectures. And if he was, why wouldn't he mention something about it?

That's my notion, such as it is. Some extensions of it I'll post soon…

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[1] Collected Papers 5.402 (1878).

[2] Essential Peirce 2:xxix

But serious, folks

The last post is, of course, silly - the last half, anyway. I do want to bring out some more serious points regarding it later. But that's another post - stay tuned...

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)