Logics, Volume 4, Issue 2 (June 2026) – 3 articles

Monotonicity is a prevalent feature in natural language. Notably, determiners such as every, some, and no induce monotonic (upward) or antitonic (downward) entailments. The Monotonicity Calculus is a proof system that formalizes such reasoning in natural language inferences through order statements (inequalities) involving higher-order (typed) terms. In this paper, we prove that with a slight modification of the conditions for term formation, it is decidable over ground terms, i.e., terms that contain no variables. Full article

The class of tier-based strictly 2-local (TSL2) languages has been shown to be useful in modeling patterns across different linguistic domains. This paper discusses the learnability of the intersection closure of the TSL2 languages, multi-TSL2 (MTSL2). I present two learning algorithms, one that learns a relevant subclass of MTSL in polynomial time, and one that learns MTSL proper but requires potentially exponential time. Both algorithms generalize across tree-based and string-based data representations. I show that each algorithm correctly learns its target class from a limited sample of positive data, and discuss the tradeoffs between the two. The success of these algorithms delivers a key learning result for subregular linguistics, and demonstrates the utility of subregular language classes in developing a unified learning theory that spans different linguistic domains. Full article

We undertake a thorough examination of George Boole’s claim that, as he discovered by means of his algebra, the law of idempotence is “more fundamental” than the law of non-contradiction (The Laws of Thought, Chapter III, Proposition IV). There is a paucity of sources investigating this subject (with a notable exception being (Béziau 2018)). We query Boole’s claim; we examine if and how we can make sense of it; we identify the notable Aristotelian precedent of philosophical reflections on relative fundamentality of logical principles; and we inquire as to what philosophical view of logic is consistent with Boole’s way of thinking about logical principles. Boole’s thinking is apparently burdened by a metaphysically laden view of logic. We argue in detail that it is a radically different way of thinking about logic—a formalist view that regards logic as manipulation of symbolic resources, congenial to logical positivism—which allows us to make some tentative sense of claims about relative fundamentality of logical laws, insofar as we can define such a notion in a meaningful way. However, on the other hand, entanglements in metaphysically laden phantasmagorias fail to support (or perhaps even fail to make sense of) Boole’s claim. In order to substantiate the metalogical and philosophical–logical claims, we advance and construct formal derivations within different Boolean languages with a view to showing how idempotence is primary in some formal systems, but it is derivable (from non-contradiction) in other systems. Hence, Boole’s claim, as we can make sense of it (as relative derivability), is language-dependent, and we argue that this is consistent with a certain philosophical view of what logic is. Full article