Finite
Geometry Notes |

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

-- 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,

© The Author, 2005. Published by Oxford University Press. All rights reserved.

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 logic

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.

References

[1] M. Aiello and J. van Benthem. A modal walk through space.

2002.

[2] J. Barwise and J. Etchemendy. Visual information and valid reasoning. In

[3] J. Barwise and J. Etchemendy. Heterogenous logic. In

[4] G. Birkhoff and J. von Neumann. The logic of quantum mechanics.

[5] L. Carroll.

[7] M. Gardner and F. Harary. The propositional calculus with directed graphs.

[8] P. J. Hurley.

[10] E. Mizraji. Vector logics: the matrix-vector representation of logical calculus.

[12] S.-J. Shin. Peirce and the logical status of diagrams.

[13] S.-J. Shin. A situation-theoretic account of valid reasoning with venn diagrams. In

[14] S.-J. Shin and E. Hammer. Euler’s visual logic.

[15] Y. Venema. A crash course in arrow logic. In

[16] D. Widdows and S. Peters. Word vectors and quantum logic: Experiments with negation and disjunctions. In