Modelling Informational Entropy

Abstract

By 'informational entropy', we understand an inherent boundary to knowability, due e.g. to perceptual, theoretical, evidential or linguistic limits. In this paper, we discuss a logical framework in which this boundary is incorporated into the semantic and deductive machinery, and outline how this framework can be used to model various situations in which informational entropy arises.Keywords: Lattice-based modal logic · Epistemic logic · Concept lattice · Graphbased semantics · Polarity-based semantics ⋆ The research of the third author is supported by the NWO Vidi grant 016.138.314, the NWO Aspasia grant 015.008.054, and a Delft Technology Fellowship awarded in 2013. PreliminariesNotation. We let ∆ U denote the identity relation on a set U, and we will drop the subscript when it causes no ambiguity. The superscript (·) c denotes the relative complement of the subset of a given set. Hence, for any binary relation R ⊆ S × T , we let R c ⊆ S × T be defined by (s, t) ∈ R c iff (s, t) R. For any such R and any S ′ ⊆ S and T ′ ⊆ T , we let R[S ′ ] := {t ∈ T | (s, t) ∈ R for some s ∈ S ′ } and R −1 [T ′ ] := {s ∈ S | (s, t) ∈ R for some t ∈ T ′ }, and write R[s] and R −1 [t] for R[{s}] and R −1 [{t}], respectively. Any such R gives rise to the semantic modal operators R , [R] : P(T ) → P(S ) s.t. R W := R −1 [W] and [R]W := (R −1 [W c ]) c for any W ⊆ T . For any T ⊆ U × V, and any U ′ ⊆ U and V ′ ⊆ V, letKnown properties of this construction (cf. [14,) are collected below.. 4. T (1) [U ′ ] = T (1) [T (0) [T (1) [U ′ ]]] and T (0) [V ′ ] = T (0) [T (1) [T (0) [V ′ ]]]. 5. T (0) [ V] = V ′ ∈V T (0) [V ′ ] and T (1) [ U] = U ′ ∈U T (1) [U ′ ].

Modelling competing theories

“…Informational entropy can be due to many factors (e.g. technological, theoretical, linguistic, perceptual, cognitive), and in [4], examples are discussed in which the nature of these limits is perceptual and linguistic. In the present section, we discuss how the theoretical frameworks adopted by empirical scientists can be a source of informational entropy.…”

“…Moreover, in the context of the graph-based structures above, an empirical theory is characterized by (and here identified with) a certain subset X of variables which are relevant to the given theory; also, in what follows, for all databases z j ∈ Z, we let X j denote the set of variables structuring the data contained in z j . 4 Hence, the A-relation E encodes to what extent database z 2 is similar to z 1 (e.g. by letting E(z 1 , z 2 ) record the percentage of variables of z 1 that also occur in z 2 ), while the relations R X encode to what extent one database is similar to another, relative to X (e.g.…”

Modelling socio-political competition

“…This can be equivalently expressed as follows: 5 Notice the inversion: formulas of type SD (social demands) are evaluated (tested) on the P-side of the model, i.e. on political parties, and conversely, political promises are evaluated on social groups.…”

“…In particular, via formal context semantics, in [7], the basic non-distributive modal logic and some of its axiomatic extensions are interpreted as epistemic logics of categories and concepts, and in [8], the corresponding 'common knowledge'-type construction is used to give an epistemic-logical formalization of the notion of prototype of a category; in [6,19], formal context semantics for non-distributive modal logic is proposed as an encompassing framework for the integration of rough set theory [23] and formal concept analysis [16], and in this context, the basic non-distributive modal logic is interpreted as the logic of rough concepts; via graph-based semantics, in [5], the same logic is interpreted as the logic of informational entropy, i.e. an inherent boundary to knowability due e.g.…”

Rough concepts

“…In the first paragraph of this section we remarked that the formal environment of rough concepts provides a new sets of interpretations for the lattice-based modal logic of Section 2.5. Several other interpretations for the same logic are proposed in [12,11,21], based on a different but related semantics (referred to as graph-based semantics) which is grounded on Ploščica's duality and representation theory for bounded lattices [65], (see also [25,24]). Inspired by the rough set theory approach, the relation E in the graphs (Z, E) on which the graph-based models are based is interpreted as an indiscernibility relation.…”