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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