Why is it called linear logic?

Question

Why is it called "Linear" Logic? What's linear about it?

Maybe useful: Jean-Yves Girard & Paul Taylor, PROOFS AND TYPES, Cambridge UP (1989), Ch.12.3 Linearity, page 98. — Mauro ALLEGRANZA, Jun 28 '17 at 09:26
See also Jean-Yves Girard, Linear logic (1987), page 16, for the def of linear function (associated to coherent spaces) and the definitions (page 19) of linear negation and linear implication. — Mauro ALLEGRANZA, Jun 28 '17 at 10:03

Peter Heinig · Accepted Answer · 2017-06-28T12:34:57.400

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Girard himself, in his native language (cf. Girard, Cours de Logique I, Hermann, 2006, Section 1.B.2), writes:

"La logique linéaire est issue d'une prise en compte systématique de l'interprétation catégorique. En particulier, les espaces cohérents [...], proches des espaces vectoriels [...] font apparaître des structures logiques familières en algèbre linéaires [...]"

Translation: "Linear logic developed from systematically taking into consideration the categorical interpretation. In particular, coherent spaces [...], similar to vector spaces [...] give rise to logical structures which are familiar from linear algebra."

So, in one sentence, it is called linear logic because it involves semantics which resemble structures from linear algebra.

edited Jun 28 '17 at 12:34

answered Jun 28 '17 at 09:13

Peter Heinig

1,288

May someone with a French keyboard care to add in all the diacritical signs? – Peter Heinig Jun 28 '17 at 09:16
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I see. So for instance because times distributes over plus? I'm curious, what's the analogue of "with" and "par" in linear algebra? – psquid Jun 30 '17 at 07:07
Well, an analogy does not necessarily imply that there is the correct answer to your question. You can see the similarity you named as one. – Peter Heinig Jun 30 '17 at 14:56
@psquid: “with” is the Cartesian product. The “times” is tensor product of vector spaces. – phadej May 10 '20 at 16:06

Why is it called linear logic?

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