Modalities in Sociological Arguments

Abstract

This article introduces modal logics for a sociological audience. We first provide an overview of the formal properties of this family of models and outline key differences with classical first-order logic. We then build a model to represent processes of perception and belief core to social theories. To do this, we define our multimodal language and then add substantive constraints that specify the inferential behavior of modalities for perception, default, and belief. We illustrate the deployment of this language to the theory of legitimation proposed by Hannan, Pólos, and Carroll (2007). This article aims to call attention to the potential benefits of modal logics for theory building in sociology.

Keywords:

• logic
• methodology
• organizations

ACKNOWLEDGMENTS

We thank the Stanford Graduate School of Business and Durham Business School for financial support.

Notes

¹An interesting exception is Montgomery's (2005) reconstruction of role theory in a (standard) nonmonotonic logic.

In the original discussion of this idea, Frege (1893/1964) introduced the distinction between sinn (sense or meaning) and bedeutung (reference) using the example of the “morning star” and the “evening star,” which astronomers discovered are the same object—Venus. He argued that the sentences “The ‘morning star’ is the ‘morning star’” and “The ‘morning star’ is the ‘evening star’” have the same meaning, because the two names point to the same object. However, the first sentence seems to be true merely by virtue of the law that every object is identical to itself; but the second sentence states an astronomical discovery.

A combination of the syntactic and semantic approach tries to prove the completeness theorem, to show that a given syntactic rendering of the relation and a semantic definition actually depict exactly the same relation.

A functional expression takes certain linguistic expressions and combines them into a new expression. For example, the conjunction takes two formulas and produces a new formula with two argument slots: one for a noun phrase (a name) and one for a sentence (a proposition).

Those who assume that two scenarios that are mutually part of each other are identical would find the first consideration redundant. We prefer to follow a noncommittal approach on this issue.

Opinions differ on the ontological status of the possible worlds. At one extreme, Lewis (1993) argues that the possible worlds are a set of actual worlds, much as some contemporary physicists argue for the existence of parallel universes. Others regard the set of worlds as nothing more than an index over a set of possible alternative assignments of truth values. We adopt the latter view, as we will make clear.

To define deontic logics one also needs to impose seriality.

For the sake of the argument, assume that in the world w □ϕ is true and □□ϕ is false. Since □□ϕ is false in w, there must be a w′ such that ℛ(w, w′) and □ϕ is false in w′. However if □ϕ is false in w′, then there exists a w″ such that ℛ(w′, w″) and ϕ is false in w″. Now, due to transitivity we have ℛ(w′, w″), that is, ⋄ ¬ ϕ is true in w, which contradicts the fact that □ϕ is true in w.

The terms postulates and axiom are often used as synonyms. But we use these terms in parallel to indicate the following difference: Postulates describe, in terms of the accessibility relations, what a belief frames look like. However some of their properties related to the interaction between modal and temporal considerations are too complex for us to offer easily understandable postulates that support to these properties. So we propose two axioms that describe the interaction between the modal and temporal constructions.

Strictly speaking this is not true: In such a scenario all worlds are accessible from themselves so the accessibility relations are reflexive, while Postulate 3.2 requires the opposite. At least one of the arrows that point back to a world has to be removed, and there is a multiplicity of alternatives to choose from. For the sake of simplicity we do not discuss these details.

Exactly eight arrows begin in each world. Any one of them might be in the accessibility relation or not, which gives a total of 2⁸ possibilities. To satisfy the seriality condition at least one of these arrows have to be chosen, which gives us 2⁸ − 1 possibilities per possible worlds, and there are 8 possible worlds.

The powerset of a set is the collection of all of its subsets. We refer the powerset here because an agent can apply multiple labels to the same object.

Suppose we have a set x = {x ₁, x ₂,…, x _i } and a set i containing the first i natural numbers: i = {0, 1, 2, …, i}. We can express the indexed set as x _i  = {x _i ∣i ∈ i}.