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Re: Affine


	From: Vladimir Alexiev <vladimir@cs.ualberta.ca>
	Date: 	Thu, 26 Jun 1997 10:21:08 -0600
	I think the original question is still not answered: why is
	affine logic called affine? I would be very disappointed if the
	answer is "because you obtain it by loosening up linear logic a
	bit, just like you obtain affine geometry by loosening up
	linear geometry a bit". I'd imagine that the *character* of the
	two "loosening-ups" should be similar.

Affine logic is linear logic with weakening.  Weakening may be
understood as a form of translation.  The algebraic rationale for
rejecting weakening is that the signature contains a constant which is
preserved by homorphisms.  Weakening is refuted in this way by many
closed categories: pointed sets, abelian groups, finite dimensional
vector spaces, etc.  The binary operation of the latter two also
refutes contraction, which however is ok for pointed sets.  Affine
algebra/geometry is the other way round, having a binary operation that
refutes contraction but no constant refuting weakening (affine
geometry's missing origin).

>From the perspective of Chu spaces over K, weakening asserts the
continuity of the unique function

	A -> 1

which is equivalent to the requirement that A\perp contain all K
constants.  For K >= 2 this implies that A contains no constants.

Vaughan Pratt
References: