By PH. D. Raymond J McCall

One of the issues integrated are induction and deduction, easy apprehension, phrases, judgment, propositions, and hypothetical and specific syllogisms.

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R (u) = {u}, then p R (u, u) = 1, which means that the system stays in u with probability 1 (and with zero probability of leaving it). In turn, to each atomic action A ∈ A on (W, R) and to each pair (u, w) ∈ W × W a number c A (u, w) ∈ [0, 1] is assigned so that c A (u, w) = 0 if and only if w ∈ f A (u). The number c A (u, w) · p R (u, w) is interpreted as the probability that performing action A in the state u will move the system to the state w. 19) which is assumed to exist, is the probability that performing the action A in the state u will lead the system to a state which belongs to Y .

Proof Assuming the Axiom of Choice, every binary relation is the set-theoretic union of partial unary functions. Hence, for every action A ∈ A, there exists a family B(A) := {Bi : i ∈ I (A)} of partial functions on W such that A = {Bi : i ∈ I (A)}(= B(A)). Let B := {B(A) : A ∈ A}. The action system N := (W, R, B) is deterministic and equivalent to M. Note The Axiom of Choice, abbreviated as AC, states that: For every nonempty family X of nonempty sets there exists a function f defined on X such that f (A) ∈ A for all A ∈ X.

What does action performability really mean? What is the process that brings the system to a state that is the effect of a given action? Two options arise here. One looks for the possibility of control over the course of events brought about by the agent, no matter who he is. Such a notion of performability is close, though not tantamount, to the notion of controlling the system. 1 is devoid of probabilistic or stochastic connotations. Performing an action thus resembles a chess player’s deliberate move across the chessboard—it is a conscious choice of one of many possibilities of the direct continuation of the game.

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