In the paper the basic concepts of extended probability theory are introduced. The basic idea: the concept of an event as a subset of Ωis replaced with the concept of an event as a partition. The partition is any set of disjoint non-empty subsets of Ω(i.e. partition=subset+its decomposition). Interpretation: elements inside certain part are indistinguishable, while elements from different parts are distinguishable. There are incompatible events, e.g {{e1},{e2}} and {e1,e2}. This is logical incompatibility analogical to the impossibility to have and simultaneously not to have the which-way information in the given experiment. The context is the maximal set of mutually compatible events. Each experiment has associated its context. In each context the extended probability is reduced to classical probability. Then the quadratic representation of events, partitions and probability measures is developed. At the end the central concept of quadratic probability spaces (which extend Kolmogorov probability spaces) is defined and studied. In the next paper it will be shown that quantum mechanics can be represented as the theory of Markov processes in the extended probability theory (Einstein's vision of QM).
Partition logics -- non-Boolean event structures obtained by pasting Boolean algebras -- provide a natural language for situations in which a system has a definite latent state but can be accessed and resolved only through mutually incompatible coarse-grained modes of observation. We show that this structure arises in a range of social-science settings by constructing six explicit examples from personnel assessment, survey framing, clinical diagnosis, espionage coordination, legal pluralism, and organizational auditing. For each case we identify the latent state space, the observational contexts as partitions, and the shared atoms that intertwine contexts, yielding instances of the
L12 bowtie, triangle, pentagon, and automaton partition logics. These examples make precise a notion of social complementarity: different modes of inquiry can be incompatible even though the underlying system remains fully value-definite. Complementarity in this sense does not entail contextuality or ontic indeterminacy. We further compare the classical probabilities generated by convex mixtures of dispersion-free states with the quantum-like Born probabilities available when the same exclusivity graph admits a faithful orthogonal representation. The framework thus separates logical structure from probabilistic realization and suggests empirically testable benchmarks for quantum-cognition models.