The meaning of the logical rules is to be found in the rules themselves.
— J. Y. Girard in “On the Meaning of Logical Rules I: syntax vs. semantics”
There are two types of dualities in linear logic. Linear negation (⊥) carries the conjunctions to the disjunctions and back again, as equivalences (≡), like De Morgan’s laws in classical logic.
Additionally, the exponentials ? and ! link the additives and the multiplicatives, as linear biconditionals (o—o).
But, now again to weave the tale begun,
All nature, then, as self-sustained, consists
Of twain of things: of bodies and of void
In which they’re set, and where they’re moved around.
For common instinct of our race declares
That body of itself exists: unless
This primal faith, deep-founded, fail us not,
Naught will there be whereunto to appeal
On things occult when seeking aught to prove
By reasonings of mind. Again, without
That place and room, which we do call the inane,
Nowhere could bodies then be set, nor go
Hither or thither at all- as shown before.
— From On the Nature of Things by Lucretius
A book on the rediscovery of the ancient Epicurean poem “On the Nature of Things” by Lucretius has recently been published. “The Swerve: how the world became modern” by Stephen Greenblatt looks quite interesting. The fourfold above was inspired by the previous fourfold Spacetime. A very nice NPR review can be found below.