Problem Orientable Evaluations as L-Subsets
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– On L–subsets S, S ’ of a set X we introduce L–inclusion as follows:
S ⊆ L S
⇐⇒ ∀x ∈ X : S(x) ≤ S
(x).
– Intersections of two such L-subsets can be defined, using t-norms
τ : L × L → L, mappings with symmetry, monotony, associativity and side
condition τ (x,1 L ) = x. They yield τ –intersections I on L X with M and N as
arbitrary membership functions (Pollandt 1997).
I(x) = (M ∩ τ N) (x) = τ (M(x), N(x)) .
One of the most important t–norms is:
– The standard norm s, defined as
s (x, y) = x ∧ y.
Other t-norms are the drastic norm, the algebraic product and the bounded
difference, see (Kerber 2006, 2017)
We use a notion of truth, based on τ and its residuum:
– τ * : L × L → L is a residuum of τ , iff
τ (x, y) ≤ ν ⇐⇒ x ≤ τ
∗ (y, ν) .
In this case τ is called a residual t–norm.
As an example of a residuum for L = [0, 1] we select the standard norm, s(α,β)
s
∗ (α, β) =
1, if α ≤ β
β, otherwise
This means that we have choices, and that we can use a problem orientation:
– Choose a suitable lattice L as set of values; pick a suitable residual t-norm
τ obtaining a set theory. Its residuum τ * gives the corresponding logic, i.e. a
quantification of the subset-set-relation. Apply that to E ∈ L O×A , the evaluation
considered, and get a basis of the implications (see below)!
1.3 Exploration
For the exploration of the evaluation E we can use that object o has attribute a if and
only if E(o,a) > 0. We put
A
(o) = τ
∗ (A ⇒ E) =
a∈A
τ
∗ (A(a), E (o, a)) ,