47 problems

Prawitz's semantics concerns valid arguments in first-order intuitionistic logic: an inference rule is interpreted as logically valid when it satisfies the relevant semantic condit…

Solovay's conjecture. ${\bf FIM}$ has the same consistency strength as ${\bf BI}$.

Let a Grothendieck topos be a topos serving as a model of intuitionistic mathematics, and consider the $lambda-Pi$ calculus with universes and inductive objects described in the so…

Intuitionistic computability logic conjecture. The set of valid formulas of the resulting fragment of computability logic is described by Heyting's intuitionistic calculus…

Intuitionistic characterization conjecture. The valid formulas of this language are exactly those provable in Heyting's intuitionistic calculus.

Let $\mathbf{ICKT}$ and $\mathbf{ICKS4}$ be the intuitionistic common knowledge logics considered in the paper, and let their validity problems ask whether a formula is valid in th…

Let $\mathsf{L}$ be a system of proof rules, let $\mathbf{J}$ be a procedure on proof structures, let $\mathscr{B}$ be a base, and let $\Phi$ be a proof structure. In Prawitz's set…

Choice-separation conjecture. 1. There exists an elementary topos in which $\mathrm{AC}_{\mathbb{N}\text{-bounded}}$ holds but $\mathrm{AC}_{\mathbb{N}\mathbb{N}}$ does not. 2. The…

Crisp-frame characterization conjecture. The principle $--\Diamond x \rightharpoonup \Diamond --x=\top$ characterizes crisp frames.

Let $\models$ be the logical-consequence relation defined in the paper's monotonic proof-theoretic semantics, and let $\texttt{IL}$ denote intuitionistic logic. Completeness over…

Let $\mathscr{R}$ be an inference rule, let $\mathfrak{J}$ be a justification structure, and let logical validity relative to $\mathfrak{J}$ mean validity relative to…

Consider the diagram of constructive quantum logics, whose nodes include classical logic, intuitionistic logic, orthologic, Ex-logic, fundamental logic and the intermediate logics…

Orthologic is the logic of quantum propositions, while intuitionistic logic is constructive logic; consider the implication-free fragment of intuitionistic logic in the signature…

Consider the $\square$-free fragments and the $\lozenge$-free fragments of the intuitionistic modal logics studied in the paper. Non-finite-axiomatizability conjecture. Either some…

Let $\mathbf{L}_{\min}$ be the minimal intuitionistic modal logic in the language with $\square$ and $\lozenge$, and let $\mathbf{WK}$ be the intuitionistic modal logic in the lang…

For each of the intuitionistic modal logics $\mathbf{L}_{\mathbf{bc}}$, $\mathbf{L}_{\mathbf{dc}}$, $\mathbf{L}_{\mathbf{bdc}}$, $\mathbf{L}_{\mathbf{fbdc}}$,…

Let $\mathbf{L}_{\min}$ be the minimal intuitionistic modal logic, and let $\mathtt{Log}(\mathcal C_{\mathbf{uc}})$ and $\mathtt{Log}(\mathcal C_{\mathbf{quc}})$ denote the logics…

Let $\Lambda$ be over the base structure $\mathfrak{B}_1$, and let $den^*_1$ be a denotation relative to $\Lambda$ such that, for some operational symbol $\phi$ of $\Lambda$,…

An operational type is a specification of the premises and conclusion of a rule together with the domains and co-domains governing its discharged assumptions. Let…

Let $\mathfrak{B}$ be an atomic base, and let an operation on grounds $phi$ have type $Gamma rhd alpha$. An operation on grounds is universal when it is an operation on grounds ove…

Let $\mathsf{WFT}_{\mathit{pred}}$ be the intuitionistically provable variant of the weak finite tree theorem in which infinite paths are represented by a predicate, and let…

Let $\mathsf{DDNS}_{\mathcal{C}}$ denote the disjunctive double-negation scheme for the class $\mathcal{C}$, and let $\mathsf{DNS}^{\forall}_{\mathcal{C}}$ denote the corresponding…

Prawitz–Dummett conjecture. No intuitionistically invalid inference is proof-theoretically valid.

Let ${\sf iPH}_{\Sigma}$ be the logic introduced in the subsection for $\Sigma_1$-preservativity, and let a $\Sigma_1$-substitution $\sigma$ assign $\Sigma_1$ arithmetical formulas…

Let ${\sf iPH}$ be the logic over the language with binary modal operator $\rhd$ introduced for arithmetical interpretations in Heyting arithmetic ${\rm HA}$. An arithmetical inter…