A little while ago Richard Zach, whose LogBlog offers an excellent view into the world of serious logic, pointed out something interesting: a new blog aggregator, “Blogging on PseudoScientific DoucheBags“, features as its logo the sequent calculus for Girard’s linear logic. Professor Zach wonders why they would choose a perfectly legitimate and coherent logical formalism as a symbol of quackery; I have to say that my experience has definitely been that the Internet “skeptic” community has a limited appreciation of logical research (I used to get told on Usenet that Alfred Tarski was a fraud, and even the cleverer sort of autodidact often has an insufficient understanding of how slick people like Tarski and Roman Jakobson were). However, I believe in making people’s dreams happen, so here’s a speculative application of linear logic that’s been knocking around in my head for a while: I’ll skip the diagrams as a gesture of friendliness, but I expect those looking for pseudoscience will find it.
Linear logic is sometimes explained (as in the helpful Wikipedia entry) as a logic of resources. In ordinary logic, premises are indefinitely available for use in inference: but there are situations where we only want to draw conclusions from “fresh” information. By modifying the “structural” inference rules of Weakening (where an irrelevant premise can always be added to an inference) and Contraction (where a redundant premise can always be removed from an inference), and splitting apart the connective rules into “context-sharing” and “context-free” versions, linear logic makes it possible to set precise bounds on the applicability of a piece of information. This is very useful for reasoning about the behavior of programming languages with state, but there’s a “real-world” phenomenon it resembles as well: industrial production. In manufacturing goods from raw materials, we use the materials up — and this places constraints on how economic systems arising around production can operate.
Now, a man famously tried to show how all this worked: Karl Marx, in the three volumes of Capital. Marx’s “circuits of capital”, like the sequence of production for profit M-C-C’-M’, symbolically represented the processes of industrial capitalism; the consequences he spun out from the basic processes of production and exchange under capitalism (putatively) showed the fundamental limits to this economic system. What would Marx think of linear logic? I think he would be impressed by the way linear implication takes resources that are available simultaneously (determined by applications of the “context-sharing” rules) and resources that are only available discretely (determined by applications of the “context-free” rules) and represents situations where information, or whatever, is consumed in the production of a logical result.
I think one could make a strong case that in the context-sharing rules we have a formalization of Marxian “use-value”, those features of the world (including the natural world) that make a real contribution to life, and in the case of context-free rules we have a formalization of exchange-value, the rationalistic processes by which economic agents (capitalists and laborers) shape the direction of economic activity. I don’t know enough about linear logic yet to say whether there is also a “tendency of the rate of proofnets to fall”, but its “economy of the sign” is a real one — and should perhaps be especially interesting for people tracking the dynamics of the capitalist system in the order of thought, as well as that of reality.