11 problems

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…

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…

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…

Characterizations and reductions conjecture. The following characterizations and reductions hold:

Let $\operatorname{IZF}$ denote Intuitionistic Zermelo–Fraenkel set theory, and let an intermediate logic be a propositional logic extending intuitionistic propositional logic. The…

Let $\mathfrak{A}$ and $\mathfrak{B}$ be Heyting algebras such that $\mathfrak{A}\preccurlyeq\mathfrak{B}$. Let $a\in|\mathfrak{A}|$, and let $(a,a^{\ast})$ be an $\mathcal{E}$-pai…

Intermediate-extension conjecture. There is a proper normal extension $\mathbf{L}$ of $\mathbf{E}$ such that

Separation property for $\mathbf{E}$. Any formula derivable in $\mathbf{E}$ is also derivable using only the axioms in group $(\mathbf{i})$ and those groups among $(\mathbf{c})$–…

Propositional theory conjecture. The propositional theories of $\mathcal{M}_2$ and $\mathcal{M}_3$ are $\mathrm{Jan}$; in fact, there are principal factors of $\mathcal{M}_2$ and…