We observe that Jc Beall’s shrieking and shrugging strategy gives us an opportunity to reflect on the Australian plan for negation in FDE, a basic subclassical logic that is used in Beall’s argument for subclassical logics. An implication of our observation is applied to a recent defense of the Australian plan for negation by Francesco Berto and Greg Restall.

We show that for every countable recursively saturated model M of Peano arithmetic and every subset $A\subseteq M$, there exists a full satisfaction class $S_A\subseteq M^2$ such that A is definable in $(M,S_A)$ without parameters. It follows that in every such model, there exists a full satisfaction class which makes every element definable, and thus the expanded model is minimal and rigid. On the other hand, as observed by Roman Kossak, for every full satisfaction class S there are two elements which have the same arithmetical type, but exactly one of them is in S. In particular, the automorphism group of a model expanded with a satisfaction class is never equal to the automorphism group of the original model. The analogue of the first result proved here for full satisfaction classes was obtained also by Roman Kossak for partial inductive satisfaction classes. However, the proof relied on the induction scheme in a crucial way, so recapturing the result in the setting of full satisfaction classes requires quite different arguments.

A bijection $(f_1,f_2)$ between $M^2$ and M is said to be a pairing function with no cycles if any composition of its coordinate functions has no fixed point. We compute here the Grothendieck ring of the pairing function without cycles to be isomorphic to $\mathbb{Z}[X]∕(X-X^2)\simeq {\mathbb{Z}}^2$.

We adjust the notion of typicality that originated with Russell, which was introduced and studied in a previous paper for general first-order structures, to make it expressible in the language of set theory. The adopted definition of the class $\mathrm{NT}$ of nontypical sets comes out as a natural strengthening of Russell’s initial definition, which employs properties of small (minority) extensions, when the latter are restricted to the various levels $V_{\mathit{\zeta }}$ of V. This strengthening leads to defining $\mathrm{NT}$ as the class of sets that belong to some countable ordinal definable set. It follows that $\mathrm{OD}\subseteq \mathrm{NT}$, and hence $\mathrm{HOD}\subseteq \mathrm{HNT}$. It is proved that the class $\mathrm{HNT}$ of hereditarily nontypical sets is an inner model of $\mathrm{ZF}$. Moreover, the (relative) consistency of $V\ne \mathrm{NT}$ is established, by showing that in many forcing extensions $M[G]$ the generic set G is a typical element of $M[G]$, a fact which is fully in accord with the intuitive meaning of typicality. In particular, it is consistent that there exist continuum many typical reals. In addition, it follows from a result of Kanovei and Lyubetsky that $\mathrm{HOD}\ne \mathrm{HNT}$ is also relatively consistent. In particular, it is consistent that $\mathcal{P}(\mathit{\omega })\cap \mathrm{OD}⊊\mathcal{P}(\mathit{\omega })\cap \mathrm{NT}$. However, many questions remain open, among them the consistency of $\mathrm{HOD}\ne \mathrm{HNT}\ne V$, $\mathrm{HOD}=\mathrm{HNT}\ne V$, and $\mathrm{HOD}\ne \mathrm{HNT}=V$.

The decision time of an infinite time algorithm is the supremum of its halting times over all real inputs. The decision time of a set of reals is the least decision time of an algorithm that decides the set; semidecision times of semidecidable sets are defined similarly. It is not hard to see that ${\mathit{\omega }}_1$ is the maximal decision time of sets of reals. Our main results determine the supremum of countable decision times as σ and that of countable semidecision times as τ, where σ and τ denote the suprema of ${\mathrm{\Sigma }}_1$- and ${\mathrm{\Sigma }}_2$-definable ordinals, respectively, over $L_{{\mathit{\omega }}_1}$. We further compute analogous suprema for singletons.

We consider subintuitionistic logics as an extension of positive propositional logic with a binary modality, interpreted over ordered and unordered monotone neighborhood frames, with a range of frame conditions. This change in perspective allows us to apply tools and techniques from the modal setting to subintuitionistic logics. We provide a Priestley-style duality, and transfer results from the (classical) logic of monotone neighborhood frames to obtain completeness, conservativity, and a finite model property for the basic logic, extended with a number of axioms.

In this paper, we study general propositional connectives characterized by truth functions and provide a necessary and sufficient condition for intuitionistic logic that coincides with classical logic.