Algebra of Observation

Birkof£-von Neumann's contention') that observational propositions in Quantum Mechanics do not obey the Boolean logic, but the modular logic was suggested by the structure of the Hilbert space, and only indirectly related to the nature of observation in physics. Husimi2 ) was the first to undertake the task of establishing directly experiment:J.l foundation of quantum logic. The present paper has the same aim as Husimi's early paper,') but adopts a different approach. A feature of the present paper is its methodology of extracting an algebraic structure from a pro­ babilistic structure. A concrete observational interpretation is given for the first time to the conjunction of two incompatible propositions. A side benefit of this work is clarification of the origin of the Booleanity of the ordinary logic, which has to be sought outside the implicational lattice.