In this article, we revisit a number of disputes regarding significance logics—that is, inferential frameworks capable of handling meaningless, although grammatical, sentences—that took place in a series of articles most of which appeared in the Australasian Journal of Philosophy between 1966 and 1978. These debates concern (i) the way in which logical consequence ought to be approached in the context of a significance logic, and (ii) the way in which the logical vocabulary has to be modified (either by restricting some notions, or by adding some vocabulary) to keep as much of classical logic as possible. Our aim is to show that the divisions arising from these disputes can be dissolved in the context of a novel and intuitive proposal that we put forward.

Assuming that there is no inner model with a Woodin cardinal, we obtain a characterization of λ-tall cardinals in extender models that are iterable. In particular, we prove that in such extender models, a cardinal κ is a tall cardinal if and only if it is either a strong cardinal or a measurable limit of strong cardinals.

We analyze the dual Ramsey theorem for k partitions and ℓ colors (${\mathsf{DRT}}_{\ell }^k$) in the context of reverse math, effective analysis, and strong reductions. Over $\mathsf{RCA}0$, the dual Ramsey theorem stated for Baire colorings ${\mathsf{Baire}\mathsf{\text{-}}\mathsf{DRT}}_{\ell }^k$ is equivalent to the statement for clopen colorings ${\mathsf{ODRT}}_{\ell }^k$ and to a purely combinatorial theorem ${\mathsf{CDRT}}_{\ell }^k$. When the theorem is stated for Borel colorings and $k\ge 3$, the resulting principles are essentially relativizations of ${\mathsf{CDRT}}_{\ell }^k$. For each α, there is a computable Borel code for a ${\mathrm{\Delta }}_{\mathit{\alpha }}^0$-coloring such that any partition homogeneous for it computes ${\mathrm{\varnothing }}^{(\mathit{\alpha })}$ or ${\mathrm{\varnothing }}^{(\mathit{\alpha }-1)}$ depending on whether α is infinite or finite. For $k=2$, we present partial results giving bounds on the effective content of the principle. A weaker version for ${\mathrm{\Delta }}_n^0$-reduced colorings is equivalent to ${\mathsf{D}}_2^n$ over $\mathsf{RCA}0+\mathsf{I}{\mathrm{\Sigma }}_{n-1}^0$ and in the sense of strong Weihrauch reductions.

By slightly adapting two equivalent semantics of noncontingency operator, we obtain two variants, ⊡ and ⊞, with nonequivalent semantics. We show that on the class of models satisfying any of five basic properties (i.e., seriality, reflexivity, transitivity, symmetry, Euclideanness), the logic $\mathcal{L}(⊡)$, which has ⊡ as the sole modal primitive, is less expressive than the logic $\mathcal{L}(⊞)$, which has ⊞ as the sole modal primitive. We investigate the frame definability of both languages. We then axiomatize $\mathcal{L}(⊞)$ and $\mathcal{L}(⊡)$ over various classes of bimodal frames. Among other results, a notion of morphisms, called ⊞-morphisms, are provided to show the completeness of axiomatizations of $\mathcal{L}(⊡)$ over serial frames and also over symmetric frames.

Christine Ladd-Franklin is often hailed as a guiding star in the history of women in logic—not only did she study under C. S. Peirce and was one of the first women to receive a PhD from Johns Hopkins, she also, according to many modern commentators, solved a logical problem which had plagued the field of syllogisms since Aristotle. In this paper, we revisit this claim, posing and answering two distinct questions: Which logical problem did Ladd-Franklin solve in her thesis, and which problem did she think she solved? We show that in neither case is the answer “a long-standing problem due to Aristotle.” Instead, what Ladd-Franklin solved was a problem due to Jevons that was first articulated in the nineteenth century.

I study several kinds of generalizations of Montague’s theorem on the method of definition of a function by recursion on a well-founded relation. In particular, I extend Montague’s theorem to cover the following situations: recursion by a valuation system, non-well-founded recursion, and recursion on a dependence operator. By means of these extensions, several constructions employed in formal theories of truth and paradox can be recast as special applications of the generalized version of Montague’s theorem.

For a cardinal $\mathfrak{a}$, let $\mathrm{fin}(\mathfrak{a})$ be the cardinality of the set of all finite subsets of a set which is of cardinality $\mathfrak{a}$. It is proved without the aid of the axiom of choice that, for all infinite cardinals $\mathfrak{a}$ and all natural numbers n,

$$2^{\mathrm{fin}{(\mathfrak{a})}^n}=2^{{[\mathrm{fin}(\mathfrak{a})]}^n}.$$

On the other hand, it is proved consistent with $\mathsf{ZF}$ that there exists an infinite cardinal $\mathfrak{a}$ such that

$$2^{\mathrm{fin}(\mathfrak{a})}<2^{\mathrm{fin}{(\mathfrak{a})}^2}<2^{\mathrm{fin}{(\mathfrak{a})}^3}<\cdots <2^{\mathrm{fin}(\mathrm{fin}(\mathfrak{a}))}.$$

We show that the set of algebraic extensions F of $\mathbb{Q}$ in which $\mathbb{Z}$ or the ring of integers ${\mathcal{O}}_F$ are definable is meager in the set of all algebraic extensions.