Archive for March, 2011

J.-Y. Girard’s Linear Logic

March 9, 2011

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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