Let T be Kripke–Platek set theory with infinity extended by the axiom $(\mathsf{Beta})$ plus the schema that claims that every set-bounded Σ-definable monotone operator from the collection of all sets to $\mathit{Pow}(a)$ for some set a has a fixed point. Then T proves that every such operator has a least fixed point. This result is obtained by following the proof of an analogous result for von Neumann–Bernays–Gödel set theory in an earlier work by Sato, with some minor modifications.

Motivated by the Ax–Kochen/Ershov principle, a large number of questions about Henselian valued fields have been shown to reduce to analogous questions about the value group and residue field. In this article, we investigate the burden of Henselian valued fields in the three-sorted Denef–Pas language. If T is a theory of Henselian valued fields admitting relative quantifier elimination (in any characteristic), we show that the burden of T is equal to the sum of the burdens of its value group and residue field. As a consequence, T is NTP₂ if and only if its residue field and value group are; the same is true for the statements “T is strong” and “T has finite burden.”

We examine the reverse mathematical strength of a variation of Hindman’s Theorem (HT) constructed by essentially combining HT with the Thin Set Theorem to obtain a principle that we call thin-HT. This principle states that every coloring $c:\mathbb{N}\to \mathbb{N}$ has an infinite set $S\subseteq \mathbb{N}$ whose finite sums are thin for c, meaning that there is an i with $c(s)\ne i$ for all nonempty sums s of finitely many distinct elements of S. We show that there is a computable instance of thin-HT such that every solution computes ${\mathrm{\varnothing }}^{\prime }$, as is the case with HT, as shown by Blass, Hirst, and Simpson (1987). In analyzing this proof, we deduce that thin-HT implies ${\mathrm{ACA}}_0$ over ${\mathrm{RCA}}_0+\mathrm{I}{\mathrm{\Sigma }}_2^0$. On the other hand, using Rumyantsev and Shen’s computable version of the Lovász Local Lemma, we show that there is a computable instance of the restriction of thin-HT to sums of exactly 2 elements such that any solution has diagonally noncomputable degree relative to ${\mathrm{\varnothing }}^{\prime }$. Hence there is a computable instance of this restriction of thin-HT with no ${\mathrm{\Sigma }}_2^0$ solution.

Fix a countable nonstandard model $\mathcal{M}$ of Peano arithmetic. Even with some rather severe restrictions placed on the types of minimal cofinal extensions $\mathcal{N}\succ \mathcal{M}$ that are allowed, we still find that there are $2^{{\mathrm{\aleph }}_0}$ possible theories of $(\mathcal{N},M)$ for such $\mathcal{N}$’s.

A notable early result of David Makinson establishes that every monotone modal logic can be extended to $L_{\mathbf{I}}$, $L_{\mathbf{V}}$, or $L_{\mathbf{F}}$, and every antitone logic can be extended to $L_{\mathbf{N}}$, $L_{\mathbf{V}}$, or $L_{\mathbf{F}}$, where $L_{\mathbf{I}}$, $L_{\mathbf{N}}$, $L_{\mathbf{V}}$, and $L_{\mathbf{F}}$ are logics axiomatized, respectively, by the schemas $\square \mathit{\alpha }\leftrightarrow \mathit{\alpha }$, $\square \mathit{\alpha }\leftrightarrow \neg \mathit{\alpha }$, $\square \mathit{\alpha }\leftrightarrow \top$, and $\square \mathit{\alpha }\leftrightarrow \perp$. We investigate logics that are both monotone and antitone (hereafter amphitone). There are exactly three: $L_{\mathbf{V}}$, $L_{\mathbf{F}}$, and the minimum amphitone logic $\mathsf{AM}$ axiomatized by the schema $\square \mathit{\alpha }\to \square \mathit{\beta }$. These logics, along with $L_{\mathbf{I}}$, $L_{\mathbf{N}}$, and a wider class of “extensional” logics, bear close affinities to classical propositional logic. Characterizing those affinities reveals differences among several accounts of equivalence between logics. Some results about amphitone logics do not carry over when logics are construed as consequence or generalized (“multiple-conclusion”) consequence relations on languages that may lack some or all of the nonmodal connectives. We close by discussing these divergences and conditions under which our results do carry over.

Since McCall (1966), the heterodox principle of propositional logic that it is impossible for a proposition to be entailed by its own negation—in symbols, $\neg (\neg \mathit{\phi }\to \mathit{\phi })$—has gone by the name of Aristotle’s thesis, since Aristotle apparently endorses it in Prior Analytics 2.4, 57b3–14. Scholars have contested whether Aristotle did endorse his eponymous thesis, whether he could do so consistently, and for what purpose he endorsed it if he did. In this article, I reconstruct Aristotle’s argument from this passage and show that he accepts this thesis. Further, I show that the argument he gives is, making plausible assumptions, a correct proof in a consistent fragmentary nonclassical metalogic for a metatheorem he previously states concerning his assertoric syllogistic. In this way, Aristotle’s argument emerges as a fascinating case study in the use of a nonclassical metalogic to prove a result about a nonclassical object system.