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.

We define a natural notion of standard translation for the formulas of conditional logic which is analogous to the standard translation of modal formulas into the first-order logic. We briefly show that this translation works (modulo a lightweight first-order encoding of the conditional models) for the minimal classical conditional logic $\mathsf {CK}$ introduced by Brian Chellas in [3]; however, the main result of the article is that a classically equivalent reformulation of these notions (i.e., of standard translation plus theory of conditional models) also faithfully embeds the basic Nelsonian conditional logic $\mathsf {N4CK}$, introduced in [11] into $\mathsf {QN4}$, the paraconsistent variant of Nelson’s first-order logic of strong negation. Thus $\mathsf {N4CK}$ is the logic induced by the Nelsonian reading of the classical Chellas semantics of conditionals and can, therefore, be considered a faithful analogue of $\mathsf {CK}$ on the non-classical basis provided by the propositional fragment of $\mathsf {QN4}$. Moreover, the methods used to prove our main result can be easily adapted to the case of modal logic, which makes it possible to improve an older result [10, Proposition 7] by S. Odintsov and H. Wansing about the standard translation embedding of the Nelsonian modal logic $\mathsf {FSK}^d$ into $\mathsf {QN4}$.

We carry out a logical analysis of a convergence proof for greedy approximation schemes in uniformly smooth Banach spaces. Though the proof is by contradiction, we are able to extract computable rates of convergence that depend on the corresponding modulus of uniform smoothness for the space. While our quantitative results represent a first proof-theoretic study of greedy approximation schemes, we use this case study more generally as an opportunity to make explicit some of the high-level proof-theoretic reasoning that enables us to transform a nonconstructive convergence proof to one where computable convergence rates are apparent, representing the proof using a series of formal derivations that are designed to capture core mathematical reasoning, as opposed to low-level proof-theoretic bureaucracy. In this way we exemplify an approach to representing the process of program extraction that might, in particular, inform efforts to formalise proof mining in proof assistants.