Allow me to present a preview to my upcoming research article Modal group theory, a portion of which relates to a recent publication titled The modal logic of abelian groups, authored by Sören Berger, Alexander Christensen Block, and Benedikt Löwe. Namely, the authors posed the following question.
What are the propositional modal validities of group theory?
The reader who is closely acquainted with my work will realize that this inquiry was a subject of my 2019 research, during my time as a Recognised Student under the supervision of Professor Joel David Hamkins at the Faculty of Philosophy, University of Oxford.
And so, given my inclination to ponder mathematics with enthusiasm but sometimes delay in documenting it systematically (though I guess I am in a good company considering that the results of Berger, Block, and Löwe date back to 2015), and the multitude of inquiries I have received regarding the propositional modal validities of groups, I have decided to offer a preview of this segment of my work ahead of releasing the entire preprint.
Let’s commence with the foundation:
We define that a group satisfies the assertion ◇ϕ if it can be embedded in a larger group where ϕ holds (and □ϕ if ϕ is true in all such groups).
A group validates a propositional modal assertion ϕ(p₀, p₁, …, pₙ) if, for all substitution instances pᵢ ↦ ψᵢ in the language of groups, the resulting assertion ϕ(ψ₀, ψ₁, …, ψₙ) holds true in that group.
The propositional modal theory of interest today is known as S4.2, and it is defined as the smallest propositional modal theory containing the axioms for S4.2, which are:
- □p → p
- □p → □□p
- ◇□p → □◇p,
and that is closed under modus ponens, substitution, and necessitation (□).
Lemma. Every group validates the propositional modal theory S4.2.
Proof. Based on the work of Hamkins and Löwe (with a minor correction in Hamkins and Wołoszyn), a commuting directed system of L-structures validates the propositional modal theory S4.2. It’s evident that the cone above any group G, that is the cocone of G, forms such a commuting directed system. Indeed, a free product with amalgamation of a pair of overgroups over any group within the cone remains within the cone. Furthermore, the definition of the product ensures that the diagram formed by these groups and their embeddings commute.
Dealing with the lower bounds was straightforward. The real challenge lay in determining the upper bounds on propositional modal validities, which was the essence of Berger, Block, and Löwe’s question. In particular, a crucial result by Hamkins and Löwe underpins this problem.
Lemma. Suppose M belongs to a family of L-structures in a common first-order language L. If M admits arbitrarily large families of independent buttons independent of arbitrarily long dials, then the propositional modal validities of M constitute precisely the propositional modal theory S4.2.
I should clarify what buttons and dials are and what it means to be independent. Buttons and dials are control statements, a technology used by a multiverse traveler to control the modal nature of a world they are about to visit. In our context, of groups, an unpushed button is an assertion not yet true in a given group but becomes true in some overgroup and all subsequent groups that extend it. Once such a button becomes necessarily true, it is considered pushed. On the other hand, a dial is a list of assertions dᵢ where exactly one assertion is true in any given group within the cone above the base group, but for any other assertion on the list, there is always an overgroup where it becomes true. Control statements are independent if their underlying assertions do not interfere with one another. For instance, buttons are independent of each other and independent of a dial if one can push a selected subset of buttons and set the dial value to any assertion dᵢ without interfering with other buttons.
Theorem. Suppose G is a group. The propositional modal validities of G precisely constitute the propositional modal theory S4.2.
Proof. The lower bound on the propositional modal validities follows from the lemma. To complete the proof, we must establish the existence of an arbitrarily large family of independent buttons independent of an arbitrarily long dial. The choice of buttons is straightforward: we take assertions bₙ expressing the existence of an element of order n, where n is the nth prime number. For the dial, let dᵢ state that the group’s center has size m, where m is much larger than any of the primes used for the buttons. We can introduce as many distinct values of m as needed, modifying the last dᵢ to be the negation of the disjunction of the former ones. It’s clear that each bₙ constitutes a button since once an element of a given order exists, it continues to exist in all overgroups. Moreover, there is no interference with the dial if we realize bₙ by embedding the base group G in the group (G ⊕ Cₚ). Similarly, dᵢs form a dial that does not interfere with the buttons, as witnessed by the embeddings G ↦ Cᵢ ⊕ (G*Z). This concludes the proof.
Since the argument applies regardless of the choice of G, we deduce the following corollary.
Corollary. The propositional modal validities of groups precisely constitute the propositional modal theory S4.2.
Let me end the post by noting that there exist individual groups that validate more than S4.2, a topic I delve into further in my upcoming paper.