"... no one has, so far as we know,
proposed
modelling propositions as
vectors
while retaining classical logic."
-- Jonathan Westphal and Jim Hardy
in the February 2004 paper excerpted below.
"This note discusses a 'geometry of logic'....
the 16 elements of the affine
4-space A over the two-element Galois field GF(2)."
-- Steven H. Cullinane, The Geometry of Logic, March 2007.
(Recall that an affine space is, essentially,
a vector space (plus translations).)
The following is from "Logic as a Vector System," by Jonathan Westphal
and Jim Hardy, Journal of Logic and Computation
2005 15(5): 751-765. Received by the Journal on 16
February 2004.
Of course the idea that logic might be visualized or represented
geometrically is not at all new.
Venn diagrams are commonly used in beginning logic instruction [8] and
can themselves be seen as
improvements on Euler diagrams. Lewis Carroll presents diagrams that he
sees as further improvements
[5]. There have also been attempts to understand the logic of diagrams
generally. The work
of Jon Barwise and John Etchemendy [3, 2] is important here. Finally,
many studies have attempted
to spell out the logical workings of already existent diagrammatic
methods, for example, Sun-Joo
Shin’s work on the logic of several diagrammatic systems [12, 13, 14].
More closely related to our work are systems such as arrow logic2
and
digraphs [7]. However,
these systems do not quantify the magnitude or direction of the arrows
they use. As a result, neither
can be seen as a vector system.
We do not wish, however, to suggest that our work occurs in a vacuum,
nor that we are the first
to envision a relation between vectors and logic. Widdows and Peters
[16] have argued that logical
operations within certain kinds of vector spaces provide a perspicuous
and computationally efficient
way of modelling word meaning within the context of document retrieval
operations. Vectors, in their system, are models of word meaning.
Mizraji [10] has developed a
vector-based logic in which
truth-values are vectors and logical operators are matrix operations on
the space. More closely related
to the present work is von Neumann’s quantum logic [4]. Von Neumann
models propositions
as sets of vectors in Hilbert Space. However, in contrast to the
present work, von Neumann’s logic
is non-classical. More recently, Aiello and van Benthem [1] have
suggested that mathematical morphology
provides a model for linear logic. On such a model, formulas are
interpreted as arbitrary
subsets of vector space. However, the logic is again not classical.
Given the history, it would be surprising if anyone were to deny that
there was a relation between
vectors and logic. Nonetheless no one has, so far as we know, proposed
modelling propositions as
vectors while retaining classical logic. We hope that by the end of
this paper the reader will agree
that doing so is both interesting and fruitful.
2 The best introduction to the subject
is [15]
References
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