We introduce the subject of modal model theory, where one studies a mathematical structure within a class of similar structures under an extension concept that gives rise to mathematically natural notions of possibility and necessity. A statement φ is possible in a structure (written $\mathit{\phi }$) if φ is true in some extension of that structure, and φ is necessary (written $\mathit{\phi }$) if it is true in all extensions of the structure. A principal case for us will be the class $Mod(T)$ of all models of a given theory T—all graphs, all groups, all fields, or what have you—considered under the substructure relation. In this article, we aim to develop the resulting modal model theory. The class of all graphs is a particularly insightful case illustrating the remarkable power of the modal vocabulary, for the modal language of graph theory can express connectedness, k-colorability, finiteness, countability, size continuum, size ${\mathrm{\aleph }}_1$, ${\mathrm{\aleph }}_2$, ${\mathrm{\aleph }}_{\mathit{\omega }}$, ${\beth }_{\mathit{\omega }}$, first ℶ-fixed point, first ℶ-hyper-fixed point, and much more. A graph obeys the maximality principle $\mathit{\phi }(a)\to \mathit{\phi }(a)$ with parameters if and only if it satisfies the theory of the countable random graph, and it satisfies the maximality principle for sentences if and only if it is universal for finite graphs.

In 2008, Graff Fara presented relative-sameness semantics, a semantics for a first-order modal and temporal language with the explicit aim of being able to render true certain contingent/temporary identity claims (relative to certain contexts). Graff Fara achieves this aim by abandoning a straightforward analysis of de re modal/temporal claims in terms of identity. Instead, such a claim is analyzed in terms of her relative-sameness relations (which need not be the identity relation), with the relevant relative-sameness relations in play determined by context and individual terms occurring in the claim. However, there is a significant problem for relative-sameness semantics concerning (i) iterated modality/temporality, and (ii) existence that renders it an inappropriate account of our modal and temporal talk. In this paper, I present that problem and a solution to it.

In epistemic logic, the beliefs of an agent are modeled in a way very similar to knowledge, except that they are fallible. Thus, the pattern of an agent’s true beliefs is an interesting subject to study. In this paper, we conduct a systematic study on a novel modal logic with the bundled operator $⊡\mathit{\varphi }:=\square \mathit{\varphi }\wedge \mathit{\varphi }$ as the only primitive modality, where ⊡ captures the notion of true belief. With the help of a novel notion of ⊡-bisimulation, we characterize the expressivity of this new language on relational models, and offer various completeness results. We also discuss some interesting connections between our work and previous works on reflexive-insensitive logics and the so-called boxdot conjecture. Finally, we study two generalizations of the ⊡-operator: one is the iterated true belief operator $\square n\mathit{\varphi }:=\mathit{\varphi }\wedge \square \mathit{\varphi }\wedge \cdots {\wedge \square }^n\mathit{\varphi }$, the other is the neighborhood semantics for the ⊡-operator.

We present tableau proof systems for the annotated version of propositional justification logics, that is, justification logics which are formulated using annotated application operators. We show that the tableau systems are sound and complete with respect to Mkrtychev models, and some tableau systems are analytic and provide a decision procedure for the annotated justification logics. We further show Craig’s interpolation property and Beth’s definability theorem for some annotated justification logics.