# Logic of Propositions in Topos Quantum Theory

## Abstract

In Chap. 10 of the first series of lecture notes on topos quantum theory [26] we showed that quantum propositions were represented by clopen sub-objects of the spectral presheaf \( \underline {\Sigma }\) [26, Ch.10, Sec.1]. The collection of all such clopen sub-object, which we denoted by \(Sub_{cl}( \underline {\Sigma })\), was shown to form a Heyting algebra [26, Th.10.2], hence the logic of quantum theory derived from the topos approach is an intuitionistic logic. In this chapter we will explain some recent results obtained in [15] in which it is shown that \(Sub_{cl}( \underline {\Sigma })\) is not only a complete Heyting algebra , but also a complete co-Heyting algebra , therefore quantum logic is represented by a complete bi-Heyting algebra where two types of implications and negations are present.

## References

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