Uncertain Logics and Variables

Abstract

Our considerations are based on multi-valued logic. To introduce terminology and notation employed in our presentation of uncertain logic and uncertain variables, let us remind that multi-valued (exactly speaking — infinite-valued) propositional logic deals with propositions (α ₁, α ₂, ...) whose logic values w(α) ∈ [0, 1] and

$$ \begin{gathered} w(\neg \alpha ) = 1 - w(\alpha ), \hfill \\ w(\alpha _1 \vee \alpha _2 ) = \max \{ w(\alpha _1 ),w(\alpha _2 )\} , \hfill \\ w(\alpha _1 \wedge \alpha _2 ) = \min \{ w(\alpha _1 ),w(\alpha _2 )\} . \hfill \\ \end{gathered} $$

(1.1)

Multi-valued predicate logic deals with predicates P(x) defined on a set X, i.e. properties concerning x, which for the fixed value of x form propositions in multi-valued propositional logic, i.e.

$$ w[P(x)] \triangleq \mu _p (x) \in [0,1] for each x \in X. $$

(1.2)

For the fixed x, μ _p(x) denotes degree of truth, i.e. the value μ _p(x) shows to what degree P is satisfied. If for each X ∈ X μ _p(x) ∈ {0, 1} then P(x) will be called here a crisp or a well-defined property, and P(x) which is not well-defined will be called a soft property. The crisp property defines a set

$$ D_x = \{ x \in X:w[P(x)] = 1\} \triangleq \{ x \in X:P(x)\} . $$

(1.3)

Consider now a universal set Ω, ω ∈ Ω, a set X which is assumed to be a metric space, a function g: Ω → X, and a crisp property P(x) in the set X. The property P and the function g generate the crisp property ψ (ω, P) in Ω: “For the value $\bar x = g(\omega ) \triangleq \bar x(\omega )$ assigned to ω the property P is satisfied”, i.e.

$$ \Psi (\omega ,P) = P[\bar x(\omega )]. $$