A concept of stochastic transitivity based on conditional probabilities

A new notion of probability logical consequence is introduced. Given a probability algebra of propositions hL, P i, proposition ψ of L is called probability logical consequence of φ1 ∈ L and φ2 ∈ L (written φ1,φ2 ° ψ) iff P (ψ |φ1∧φ2 ) > max {P (ψ |φ1 ) , P (ψ |φ2 )}. A new definition of stochastic transitivity is put forward in terms of probability logical consequence. An empirical binary relation  is called stochastically transitive if a  b, b  c ° a  c. It is proved that a composition of a linear (or strictly linear) order and an independent binary random noise is stochastically transitive.