When may a proof-theoretic system represent its own expressions, contexts, acts, histories, and observations, reactivate those representations, and admit them as new operative resources without changing its inherited behaviour? We develop Reflexive Logic as a typed proof theory of conservative re-entry. The framework distinguishes operative carriers from role-indexed codecs and separates representation, activation, re-entry, semantic status, and formula-boundary presentation. The main result is a local-to-global conservative-saturation theorem. Typed reduction of new boundaries, macro reduction of new acts, factorization of later observations, and coherence across saturation stages yield inherited normal forms for every saturated history with inherited boundaries. Hence saturation preserves both inherited consequence and continuation-sensitive observational equivalence. A generator lemma reduces observation factorization to primitive readouts and elementary contexts. A further well-founded theorem applies the criterion to boundary-static finite or infinite recursively dependent families of new typed act occurrences. The conservatively saturated carrier determines Minimal Reflexive Logic, whose unconditional resources are identity and exact-boundary composition. An implication in a selected formula-boundary presentation arises only when the relevant continuation presheaf is represented by a selected formula boundary; negation and structural rules require further failure, symmetry, discard, copying, and explosion resources. An adequate semantic codec induces structural denotation. Separate status-attribution and re-entry data then supply dynamics, stability, and designation. A finite strict-growth witness shows that re-entry can add genuine operative material while leaving the inherited-boundary history category unchanged. An explicit realization theorem, under an observationally isolated sentence-fragment hypothesis, realizes monotone three-valued Kripke jump data while distinguishing semantic fixedness from classical determinacy. The comparative result decomposes reflexive constructions into a common diagonal component and case-specific operative resources. Liar behaviour depends on the status domain and update dynamics; Curry collapse requires reflection, copying, implication, and composition; Gödelian incompleteness additionally requires arithmetized syntax, derivability conditions, and metatheoretic assumptions. The framework is therefore not a universal fixed-point semantics, but a package-independent sufficient criterion for conservative operative self-representation and a method for locating the resources responsible for its logical effects.
Inference with e-values is attracting increasing attention in statistics. We show that e-values can be embedded in two recent logical frameworks for statistical inference: a dynamic epistemic framework centred on learning from observation streams, and a rejection-based framework centred on non-monotonic inference. For the former we show that e-value-based plausibility maps satisfy the key learnability conditions of [3]. For the latter, we show that a natural logarithmic transform of canonical e-values yields a rejection score in the sense of [2] and relate it directly to the plausibility scores of the dynamic framework.
Modal expansionism is the view that modal space can always be expanded. On the linguistic approach, modal space expands when, upon reinterpretation, our modal vocabulary comes to range over more possibilities. Instead, this paper offers a metaphysical approach to modal expansionism. According to the proposed view, we count more possibilities after modal space expands not because our language has changed, but because there genuinely are more possibilities.
We give a possible answer to the question in the title of the conference by way of a case study of the axiom of choice and its intrinsic plausibility. The analysis employs ideas and concepts from Husserl’s phenomenology, putting into practice the Gödelian view that secure insight into mathematical axioms is achieved through a descriptive clarification of the meaning of the basic terms occuring in those axioms.
De re/de dicto distinction is a well-known concept that differentiates between properties of non-rigid designators and the properties of the objects that they refer to. The article proposes to apply this distinction to the notion of trust. The technical results are formal definitions of de re and de dicto forms of trust as modalities, the proof of the undefinability of these two modalities through each other, and a sound and complete logical system capturing the properties of the two forms of trust.
