The Bristol model is an inner model of $L[c]$, where c is a Cohen real, which is not constructible from a set. The idea was developed in 2011 in a workshop taking place in Bristol, but was only written in detail by the author in [8]. This article is a guide for those who want to get a broader view of the construction. We try to provide more intuition that might serve as a jumping board for those interested in this construction and in odd models of $\mathsf {ZF}$. We also correct a few minor issues in the original paper, as well as prove new results. For example, the Boolean Prime Ideal theorem fails in the Bristol model, as some sets cannot be linearly ordered, and the ground model is always definable in its Bristol extensions. In addition to this we include a discussion on Kinna–Wagner Principles, which we think may play an important role in understanding the generic multiverse in $\mathsf {ZF}$.

It is a well-known empirical phenomenon that natural axiomatic theories are pre-well-ordered by consistency strength. Without a precise mathematical definition of “natural,” it is unclear how to study this phenomenon mathematically. We will discuss the significance of this problem and survey some strategies that have recently been developed for addressing it. These strategies emphasize the role of reflection principles and ordinal analysis and draw on analogies with research in recursion theory.

We study dependence and independence concepts found in quantum physics, especially those related to hidden variables and non-locality, through the lens of team semantics and probabilistic team semantics, adapting a relational framework introduced in [1]. This also leads to new developments in independence logic and probabilistic independence logic.

In this paper, a proof-theoretic perspective on counterfactual inference is proposed. On this perspective, proof-theoretic structure is fundamental. We start from a certain primacy of inferential practice and structural proof theory. Models are required neither for the explanation of the meaning of counterfactuals, nor for that of counterfactual inference. Taking a proof-theoretic perspective and an intuitionistic stance on meaning (cf. BHK), we define modal intuitionistic natural deduction systems for drawing conclusions from counterfactual assumptions. These proof systems are modal insofar as derivations in them make use of assumption modes which are sensitive to the factuality status (e.g., factual, counterfactual) of the formula that is to be assumed. This status is determined by a reference proof system on top of which a modal proof system is defined. The rules of a modal system draw on this status.

The main results obtained are preservation, normalization, subexpression (incl. subformula) property, and internal completeness. The systems are applied to the analysis of reasoning with natural language constructions such as ‘If A were the case, B would [might] be the case’, ‘Since A is the case, B is [might be] the case’. A proof-theoretic semantics is provided for them.