Probability theory and fuzzy logic have been presented as quite distinct theoretical foundations for reasoning and decision making in situations of uncertainty. This paper establishes a common basis for both forms of logic of uncertainty in which a basic uncertainty logic is defined in terms of a valuation on a lattice of propositions. The (non-truth-functional) connectives for conjunction, dis...

♣ Rings . A ring is a non-empty set R with two binary operations ( + , · ) , called addition and multiplication, respectively satisfying : Axiom 1. Closure ( + ) : ∀x, y ∈ R , x + y ∈ R . Axiom 2. Commutative ( + ) : For every x, y ∈ R , x + y = y + x . Axiom 3. Associative ( + ) : ∀x, y, z ∈ R , x + (y + z) = (x + y) + z . Axiom 4. Neutral ( + ) : ∃ θ ∈ R , such that ∀x ∈ R, x + θ = θ + x = x ...

The following notion of forcing was introduced by Grigorieff [2]: Let I ⊂ ω be an ideal, then P is the set of all functions p : ω → 2 such that dom(p) ∈ I. The usual Cohen forcing corresponds to the case when I is the ideal of finite subsets of ω. In [2] Grigorieff proves that if I is the dual of a p-point ultrafilter, then ω1 is preserved in the generic extension. Later, when Shelah introduced...

A new axiom is proposed, the Ground Axiom, asserting that the universe is not a nontrivial set forcing extension of any inner model. The Ground Axiom is first-order expressible, and any model of zfc has a class forcing extension which satisfies it. The Ground Axiom is independent of many well-known set-theoretic assertions including the Generalized Continuum Hypothesis, the assertion v=hod that...

Development of new algorithms in the area machine learning, especially clustering, comparative studies such as well testing according to software engineering principles requires availability labeled data sets. While standard benchmarks are made available, a broader range sets is necessary order avoid problem overfitting. In this context, theoretical works on axiomatization clustering algorithms...

In this paper, the Urysohn and completely Hausdorff axioms in general topology are generalized to L-topological spaces so as to be compatible with pointwise metrics. Some properties and characterizations are also derived