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Vector Logic

Background:

"... 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  and can themselves be seen as improvements on Euler diagrams. Lewis Carroll presents diagrams that he sees as further improvements . 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 . 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  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  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 . 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  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 

References

 M. Aiello and J. van Benthem. A modal walk through space. Journal of Applied Non-Classical Logics, 12, 319–363,
2002.
 J. Barwise and J. Etchemendy. Visual information and valid reasoning. In Visualization in Mathematics, W. Zimmermann and S. Cunningham, eds, pp. 9–24. Number 19 in MAA Notes, Mathematical Association of America, 1991.
 J. Barwise and J. Etchemendy. Heterogenous logic. In Diagrammatic Reasoning: Cognitive and Computational Perspectives. MIT Press, 1995.
 G. Birkhoff and J. von Neumann. The logic of quantum mechanics. Annals of Mathematics, 37, 823–843, 1936.
 L. Carroll. Symbolic Logic and the Game of Logic. Dover, 1958.

 M. Gardner and F. Harary. The propositional calculus with directed graphs. Eureka, 34–40, 1988.
 P. J. Hurley. A Concise Introduction to Logic. Wadsworth, 2002.

 E. Mizraji. Vector logics: the matrix-vector representation of logical calculus. Fuzzy Sets and Systems, 50, 79–185, 1992.

 S.-J. Shin. Peirce and the logical status of diagrams. History and Philosophy of Logic, 15, 1994.
 S.-J. Shin. A situation-theoretic account of valid reasoning with venn diagrams. In Logical Reasoning with Diagrams, J. Barwise and G. Allwein, eds. Oxford, 1996.
 S.-J. Shin and E. Hammer. Euler’s visual logic. History and Philosophy of Logic, 19, 1998.
 Y. Venema. A crash course in arrow logic. In Arrow Logic and Multi-Modal Logic, L. P´ol´os, M. Marx and Masuch, eds, pp. 3–34. CSLI, Stanford, 1996.
 D. Widdows and S. Peters. Word vectors and quantum logic: Experiments with negation and disjunctions. In Proceedings of Mathematics of Language, Vol. 8. R.T. Oehrle and J. Rogers, eds, pp. 141–154. 2003.