Linear logic is a substructural logic invented (or discovered, if you’re a Platonist) by logician Jean-Yves Girard. Many other logics can be embedded into it, including classical and intuitionistic logic, so in a sense it is a “logic behind logic”. Linear logic can be partially derived from the rejection of the structural rules of weakening and contraction, the first of which adds arbitrary propositions and the second reduces duplicated propositions to single occurrences. Due to these changes in the logical rules, logic is transformed from being transcendental (truth transcends its use) to pragmatic or materialistic (truth is restricted by use). Therefore linear logic can be given a “resource interpretation” that makes it a logic not of truth but of *things*: producing and consuming, giving and taking, pushing and pulling, like the desiring machines of Deleuze and Guattari (see Hjelmslev’s Net).

The fragment of linear logic I show here is called MALL, for Multiplicative-Additive Linear Logic. The two exponentials that interconvert additive and multiplicative operations are not shown, which also allow for the weakening and contraction rules to be reintroduced.

Note that the two additive operations allow for propositions to be created and destroyed and the two multiplicative operations contain exactly the same propositions. One could say the additive operations allow for change, and the multiplicative operations allow for bearing. In the resource interpretation, note that additive disjunction (⊕) is creative and additive conjunction (&) is destructive. Both additive conjunction (&) and multiplicative disjunction (⅋) are reversible, whereas additive disjunction (⊕) and multiplicative conjunction (⊗) are irreversible.

Linear logic was a major inspiration for naming this blog “Equivalent Exchange” (see Introduction), since it is a logic of production and consumption. Linear implication, written as A –o B (and equivalent to A^{⊥} ⅋ B), can be thought of as exchanging A for B.

Linear logic has also been adopted as the logic for “radical anti-realism”. How can it have both a physicalistic interpretation, and yet describe an anti-realism more radical than ordinary anti-realism? I will need further study to understand these claims.

References:

http://en.wikipedia.org/wiki/Linear_logic

http://plato.stanford.edu/entries/logic-linear/

[*5.68-*5.70]

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Linear logic fascinates the hell out of me, particularly as it bears on two of the problems you hit on here: the ease with which it seems to accommodate both anti-realist and materialistic interpretations (which, I think, don’t really contradict one another, once the underlying concepts are refined a bit), and the question as to what it means to call linear logic “a logic behind logic” as Girard does in his “Linear Logic” monograph (where it’s also called “la secrète noirceur du lait”—that’s stuck with me for a while now).

The “behind”, whatever it means, means something more complicated than embeddability. Girard’s intention seems to be that linear logic is, in some sense (perhaps transcendental), “constitutively prior” to intuitionistic logic, which itself carves more deeply into the logos than does classical logic. (Ludics and the Geometry of Interaction, in turn, are meant to uncover still “deeper” strata…) I need to give this much more thought, still, but some work towards conceptually clarifying this notion of “depth” can be gleaned from Girard’s most recent text, “Syntaxe transcendentale, manifeste” (See http://iml.univ-mrs.fr/~girard/syntran.pdf )

I like the idea of thinking through at least the perfective fragment of linear logic as a logic of “equivalent exchange”. This tempts me to think of the imperfective in terms of surplus value… but I’m not yet sure how to articulate this idea. In a capitalist, or absolute exchange economy–where everything can, in principle, be exchanged for anything–the rule of equivalent exchange by and large holds true, but with one necessary exception: there must exist a commodity which is worth more than its cost, a commodity whose use-value generates more exchange value than the exchange value paid for it. This is what it means to commodify labour-power. If the ordinary commodity presents itself in the form of a perfective formula, it’s tempting to think of labour-power as the “imperfective commodity”, the exponentiated commodity. Of course, this goes too far! It makes the worker into an infinite power, a sort of proletarian demiurge. Labour-power is neither a perfective commodity, nor an absolutely imperfective one, but something which exceeds (is brutally made to exceed) the principle of equivalent exchange, something which the exchange economy pushes towards imperfection. The exponentiation of the proletariat is an impossible ideal for the capitalist, but an ideal which is nevertheless regulative.

Taking these ramblings a bit further, now: Girard’s four-stratum hierarchy, whose axis gives us the dimension of depth or anteriority evoked in his remark that “linear logic is a logic behind logic”, bottoms out in a stratum where the logic of equivalent exchange—linear logic—is not yet constituted: an anomic or paralogical arena where “anything goes” (think of ludics), and where the forms of exchange-logic emerge in virtue of certain dynamical symmetries. What this tempts me to do is to search this “level -4” for a way of formally articulating something like the process of Real Abstraction (as understood by Sohn-Rethel, e.g.), the emergence of logic’s “invisible hand”, in a more or less anarchical arena of generally unequal exchange relations… But maybe I’m getting carried away.

Anyway, enough of my rambling. I’m curious to hear more of your thoughts on linear logic, etc.

Thanks for your intriguing comments! Linear logic has also been an interest of mine, but unfortunately my knowledge is very cursory. There is a huge wealth of logical and mathematical material available, but philosophical and broader applications are just starting to emerge.

What I seem to be working towards is that the “behindedness” of LL indicates that information is the ground of materiality and non-materiality, the objective and the subjective, syntax and semantics. This would also allow it to serve to join realistic and anti-realist interpretations.

I do not know enough about Ludics to comment on them at this time. However, Chu Spaces and Game Semantics also look quite promising.

I’ll have to wait for an English translation of Girard’s latest. It looks quite interesting, though. I tried to read it via Google Translate, but didn’t make much progress.