Modelling socio-political competition

Abstract

This paper continues the investigation of the logic of competing theories (be they scientific, social, political etc.) initiated in [4]. We introduce a many-valued, multi-type modal language which we endow with relational semantics based on enriched reflexive graphs, inspired by Ploščica's representation of general lattices. We axiomatize the resulting many-valued, non-distributive modal logic of these structures and prove a completeness theorem. We illustrate the application of this logic through a case study in which we model competition among interacting political promises and social demands within an arena of political parties social groups.

Introduction

This paper is a continuation of the investigation into competing theories started in [4]. Its technical contributions are rooted in the generalized Sahlqvist canonicity and correspondence for normal lattice-based logics [10], [9], i.e. nonclassical propositional logics for which the distributive laws between ∧ and ∨ do not need to hold. Via algebraic and duality-theoretic techniques, these logics, and non-distributive normal modal logics in particular, have been endowed with complete relational semantics based on formal contexts [18] and reflexive graphs [3], [5]. These semantic structures have a well developed theory, both algebraic and proof-theoretic [20], [13], [15] and model-theoretic [11], and have facilitated new insights on possible interpretations and use of lattice-based modal logics.

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], [21], formal context semantics for non-distributive modal logic is proposed as an encompassing framework for the integration of rough set theory [25] and formal concept analysis [18], 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. to perceptual, theoretical, evidential or linguistic limits, and in [4], many-valued graph-based semantics is introduced for non-distributive normal modal logic, and its potential is explored as a formal framework for modelling competing theories in the empirical sciences.

Both in the crisp and in the many-valued setting, in the graphs $(Z,E)$ on which the relational structures are based, the relation E is interpreted as an indiscernibility relation, which makes the present approach similar to that of approximation spaces in rough set theory [25]. However, the key difference is that, rather than generating modal operators which associate any subset of Z with its definable E-approximations, E generates a complete lattice in which the distributivity laws do not need to hold. This lattice is defined as the concept lattice of the formal context $(Z,Z,E^c)$ arising from the graph $(Z,E)$. In the approach proposed in [5], [4] and followed in the present paper, concepts are not understood as definable approximations of predicates, but rather they represent ‘all there is to know’, i.e. the theoretical horizon to knowability, given the inherent boundary encoded into E. Interestingly, E is required to be reflexive but in general neither transitive nor symmetric, which is in line with what observed in the literature in psychology (cf. [27], [24]) and business science [17].

In this paper, we start exploring a semantic setting for non-distributive modal logics that is not only many-valued, as the setting of [4] is, but unlike [4] is also multi-type. The main motivation and starting point of the present contribution is to introduce a formal environment in which to analyse the similarities between the competition among political theories (both in their institutional incarnations as political parties, and in their social incarnations as social blocks or groups) and the competition between scientific theories as treated in [4].

In [4], scientific theories are identified with the sets of their relevant variables (e.g. mass, speed, position are relevant variables for gravitation theory); hypotheses formulated in the background of a given theory X establish connections between variables in X and are captured as formulas which can be tested (i.e. evaluated) on different databases (i.e. states of the domain Z of a graph-based model), with a greater or lesser degree of confidence in the outcome of the test (captured in the truth-value in the many-valued semantics). Since databases themselves are built according to a given theory (“observations are theory-laden”), the degree of confidence in the outcome of tests is formulated in terms of how compatible the background theory of the given hypothesis is with the theory according to which the given database has been built. Theories compete in the arena of databases by their key hypotheses being tested on different databases. Then the criteria establishing whether theory X outcompetes theory Y need to assign different weights to the performances of hypotheses on databases that have high compatibility with the theories to which each hypothesis pertains, and to the performances of the same hypotheses on databases with low compatibility. In the present paper, we propose the following analogies:

Here the “issues” could be things like distribution of wealth, access to education and progressive taxation in the context of e.g. a socialist theory. The main difference between the competition of scientific theories outlined above and that of socio-political theories is that competition among the latter plays out not on a single arena but on at least two arenas simultaneously: that is, political parties (incarnating socio-political theories) compete with each other by testing how well their promises (phrased in terms of issues) score on different social groups, while at the same time, social groups (also incarnating socio-political theories) compete with each other by testing how well their demands score on political parties. The double-sidedness of this situation calls for a multi-type formal framework, both in respect to the language and the models. However, there is another interesting similarity between the socio-political case and the scientific case: as discussed above, the fact that databases are theory-laden results in different degrees of confidence in the outcomes of tests of different hypotheses, depending on the degree of compatibility between their underlying theories; likewise, the fact that each social group has an underlying theory (captured by the set of issues which are relevant to that social group) results in different degrees of confidence when the promises of different political parties are tested on different social groups, which again depends on the degree of compatibility between their underlying theories. Conversely and symmetrically, the fact that each political party has an underlying theory results in different degrees of confidence when the demands of different social groups are tested on different political parties, which again depends on the degree of compatibility between their underlying theories.

The paper is organized as follows: Section 2 collects some preliminaries on multi-type non-distributive modal logic and the many-valued enriched formal contexts and many-valued graphs upon which its semantics will be built. Section 3 introduces the notion of a many-valued heterogeneous frame – structures we will use to model sets of political parties and social groups and the relations among them. In Section 4 we define the formal semantics of our logic in terms of many-valued heterogeneous models and formulate a completeness theorem, the proof of which is given in Appendix A. Section 5 presents a case study in which we use a many-valued heterogeneous model to capture and reason about a scenario loosely inspired by the British socio-political scene. We conclude in Section 7 by discussing some potential applications of our work and outlining some further research directions.