# An Arithmetization of Logical Oppositions

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## Abstract

An arithmetic theory of oppositions is devised by comparing expressions, Boolean bitstrings, and integers. This leads to a set of correspondences between three domains of investigation, namely: logic, geometry, and arithmetic. The structural properties of each area are investigated in turn, before justifying the procedure as a whole. To finish, I show how this helps to improve the logical calculus of oppositions, through the consideration of corresponding operations between integers.

### Keywords

*k*-base system Bitstring Chasles’ relation Opposite Opposition Question-answer semantics Vectors

### Mathematics Subject Classification

Primary 03B35 Secondary 03B05 03B65### References

- 1.J.-Y. Béziau, New light on the square of oppositions and its nameless corner. Log. Investig.
**10**, 218–233 (2003)Google Scholar - 2.R. Blanché, Sur l’opposition des concepts. Theoria
**19**, 89–130 (1953)CrossRefGoogle Scholar - 3.R. Blanché, Sur la structuration du tableau des connectifs interpropositionnels binaires. The J. Symb. Log.
**22**, 178 (1957)CrossRefGoogle Scholar - 4.T. Cze\( \.{z}\)owski, On certain peculiarities of singular propositions. Mind
**64**, 392–395 (1955)Google Scholar - 5.A. Moretti, The Geometry of Logical Opposition, PhD Thesis, University of Neuchatel, 2009Google Scholar
- 6.
- 7.F. Schang, Oppositions and opposites, in
*Around and Beyond the Square of Opposition*, ed. by J.Y. Béziau, D. Jacquette (Birkhäuser/Springer Basel, 2012), pp. 147–173Google Scholar - 8.F. Schang, Logic in opposition. Stud. Hum.
**2**(3), 31–45 (2013)Google Scholar - 9.F. Schang, in
*No, No, and No*. Submitted draftGoogle Scholar - 10.H. Smessaert, On the 3D-visualisation of logical relations. Log. Univers.
**3**, 303–332 (2009)Google Scholar - 11.H. Smessaert, L. Demey, Logical geometries and information in the square of oppositions. J. Log. Lang. Inf.
**23**(4), 527–565 (2014)Google Scholar

## Copyright information

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