Carnap’s (Categoricity) Problem concerns the relationship between (rules of) inference and model-theoretic values. In particular, it asks whether proof-theoretic constraints are ‘strong enough’ to uniquely determine intended semantic values. Carnap [20] demonstrated that already in the classical bivalent setting this is not the case for the majority of the usual logical constants. To remedy this underdetermination of ‘semantics by syntax’ a variety of solution strategies has been explored in the literature. This article is a philosophical-logical survey of these attempts, comparing them with respect to scope, motivation, and success. Besides the mathematical interest held by Carnap’s Problem, the underdetermination it uncovers has significant consequences for a variety of philosophical projects and positions, warranting a systematic study of attempts at resolving it.

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.

A Constructive Logic of Evidence and Truth ($\mathsf {LET_C}$) is introduced. This logic is both paraconsistent and paracomplete, providing connectives for consistency and determinedness that enable the independent recovery of explosiveness and the law of excluded middle for specific propositions. Dual connectives for inconsistency and undeterminedness are also defined in $\mathsf {LET_C}$. Evidence is explicitly formalised by integrating lambda calculus terms into $\mathsf {LET_C}$, resulting in the type system $\mathsf {LET_C^{\lambda }}$. In this system, lambda calculus terms represent procedures for constructing evidence for compound formulas based on the evidence of their constituent parts. A realisability interpretation is provided for $\mathsf {LET_C}$, establishing a strong connection between deductions in this system and recursive functions.

There are known characterisations of several fragments of hybrid logic by means of invariance under bisimulations of some kind. The fragments include $\{\mathord {\downarrow }, \mathord {@}\}$ with or without nominals (Areces, Blackburn, Marx), $\mathord {@}$ with or without nominals (ten Cate), and $\mathord {\downarrow }$ without nominals (Hodkinson, Tahiri). Some pairs of these characterisations, however, are incompatible with one another. For other fragments of hybrid logic no such characterisations were known so far. We prove a generic bisimulation characterisation theorem for all standard fragments of hybrid logic, in particular for the case with $\mathord {\downarrow }$ and nominals, left open by Hodkinson and Tahiri. Our characterisation is built on a common base and for each feature extension adds a specific condition, so it is modular in an engineering sense.

The lattice problem for models of Peano Arithmetic ($\mathsf {PA}$) is to determine which lattices can be represented as lattices of elementary submodels of a model of $\mathsf {PA}$, or, in greater generality, for a given model $\mathcal {M}$, which lattices can be represented as interstructure lattices of elementary submodels $\mathcal {K}$ of an elementary extension $\mathcal {N}$ such that $\mathcal {M}\preccurlyeq \mathcal {K}\preccurlyeq \mathcal {N}$. The problem has been studied for the last 60 years and the results and their proofs show an interesting interplay between the model theory of PA, Ramsey style combinatorics, lattice representation theory, and elementary number theory. We present a survey of the most important results together with a detailed analysis of some special cases to explain and motivate a technique developed by James Schmerl for constructing elementary extensions with prescribed interstructure lattices. The last section is devoted to a discussion of lesser-known results about lattices of elementary submodels of countable recursively saturated models of PA.