Cicero often challenged Epicureanism on the grounds of inconsistency. Cicero personifies the charge through his character Torquatus, who defends Epicureanism in De finibus 1–2. Cicero highlights the discrepancies among Torquatus’ beliefs and between them and his behaviour. Torquatus holds that the senses incontestably verify the tenets of Epicureanism, and that logic is superfluous. Yet he is sensitive to the fact that Epicurus’ teachings are not intuitive and require a fair amount of logical argumentation in its defence. Therefore, he defends his school against Cicero's criticisms. But by engaging in a defence of the system, Torquatus has already spoken against his commitment to the obviousness of Epicureanism and his disavowal of logic.

According to Charles Travis, Frege’s principle to “always to sharply separate the psychological from the logical, the subjective from the objective” involves a move called “the fundamental abstraction.” I try to explain what this abstraction is and why it is interesting. I then raise a problem for it, and describe what I think is a better way to understand Frege’s principle.

I argue that Charles Travis’s interpretation of Frege, in Frege: The Pure Business of Being True, as consistent with Travis’s conception of occasion-sensitivity does not in fact require any modal notions, and so is consistent with the amodalist interpretation of Frege I elaborate in Necessity Lost.

We give explicit formulas witnessing IP, IP$_{\!n}$, or TP2 in fields with Artin–Schreier extensions. We use them to control p-extensions of mixed characteristic henselian valued fields, allowing us most notably to generalize to the NIP$_{\!n}$ context one way of Anscombe–Jahnke’s classification of NIP henselian valued fields. As a corollary, we obtain that NIP$_{\!n}$ henselian valued fields with NIP residue field are NIP. We also discuss tameness results for NTP2 henselian valued fields.

Jc Beall's Divine Contradiction is a fascinating defence of the idea that contradictions are true of the tri-personal God. This project requires a logic that avoids the consequence that every proposition follows from a contradiction. Beall presents such a logic. This ‘gap/glut’ logic is the topic of this article. A gap/glut logic presupposes that falsity is not simply the absence of truth – for a proposition that is true may also be false. This article is essentially an examination of the idea that falsity is not simply untruth. The author rejects this position but does not claim to have an argument against it. In lieu of an argument, he presents three ‘considerations’. First, it is possible to give an intuitive semantics for the language of sentential logic that yields ‘classical’ sentential logic (including ‘p, ¬ p ⊢ q’) and which makes no mention of truth-values. Second, it is possible to imagine a race who manage their affairs very well without having the concept ‘falsity’. Third, it is possible to construct a semantics that yields a logic identical with the dialetheist logic and which makes no mention of truth-values – and which, far from being plausible, seems pointless.

Nancy Cartwright's 1983 book How the Laws of Physics Lie argued that theories of physics often make use of idealisations, and that as a result many of these theories were not true. The present paper looks at idealisation in logic and argues that, at least sometimes, the laws of logic fail to be true. That might be taken as a kind of skepticism, but I argue rather that idealisation is a legitimate tool in logic, just as in physics, and recognising this frees logicians up to use false laws where these are helpful.

We define $\Psi $-autoreducible sets given an autoreduction procedure $\Psi $. Then, we show that for any $\Psi $, a measurable class of $\Psi $-autoreducible sets has measure zero. Using this, we show that classes of cototal, uniformly introenumerable, introenumerable, and hyper-cototal enumeration degrees all have measure zero.

By analyzing the arithmetical complexity of the classes of cototal sets and cototal enumeration degrees, we show that weakly 2-random sets cannot be cototal and weakly 3-random sets cannot be of cototal enumeration degree. Then, we see that this result is optimal by showing that there exists a 1-random cototal set and a 2-random set of cototal enumeration degree. For uniformly introenumerable degrees and introenumerable degrees, we utilize $\Psi $-autoreducibility again to show the optimal result that no weakly 3-random sets can have introenumerable enumeration degree. We also show that no 1-random set can be introenumerable.

A key theme throughout Maimon’s works is a circularity he diagnoses at the heart of Kant’s response to Hume. The objective validity of Kant’s category of causality ultimately rests, Maimon argues, upon the logical status of the hypothetical judgement – on its inclusion among the forms of pure general logic. In turn, however, the inclusion of the hypothetical within pure general logic itself rests upon the objective validity of causal judgements. This article examines Maimon’s diagnosis and traces it back to a debate that has its origins in Wolff’s German Logic, concerning the relationship between categorical and hypothetical judgements.

David Lewis is widely regarded as the philosopher who introduced the concept of common knowledge. His account of common knowledge differs greatly from most later accounts in philosophy and economy, with the central notion of his theory being ‘having reason to believe’ rather than ‘knowledge’. Unfortunately, Lewis’s account is rather informal, and the argument has a few gaps. This paper assesses two major attempts to formalise Lewis’s account and argues that these formalisations are missing a crucial aspect of this account. Therefore, a new reconstruction is proposed, which explicitly discusses ‘reasons’ and uses a logic inspired by justification logic.

Reasoning with quantifier expressions in natural language combines logical and arithmetical features, transcending strict divides between qualitative and quantitative. Our topic is this cooperation of styles as it occurs in common linguistic usage and its extension into the broader practice of natural language plus ‘grassroots mathematics’.

We begin with a brief review of $\mathsf {FO}(\#)$, first-order logic with counting operators and cardinality comparisons. This system is known to be of very high complexity, and drowns out finer aspects of the combination of logic and counting. We therefore move to a small fragment that can represent numerical syllogisms and basic reasoning about comparative size: monadic first-order logic with counting, $\mathsf {MFO}(\#)$. We provide normal forms that allow for axiomatization, determine which arithmetical notions can be defined on finite and on infinite models, and conversely, we discuss which logical notions can be defined out of purely arithmetical ones, and what sort of (non-)classical logics can be induced.

Next, we investigate a series of strengthenings of $\mathsf {MFO}(\#)$, again using normal form methods. The monadic second-order version is close, in a precise sense, to additive Presburger Arithmetic, while versions with the natural device of tuple counting take us to Diophantine equations, making the logic undecidable. We also define a system $\mathsf {ML}(\#)$ that combines basic modal logic over binary accessibility relations with counting, needed to formulate ubiquitous reasoning patterns such as the Pigeonhole Principle. We prove decidability of $\mathsf {ML}(\#)$, and provide a new kind of bisimulation matching the expressive power of the language.

As a complement to the fragment approach pursued here, we also discuss two other ways of lowering the complexity of $\mathsf {FO}(\#)$ by changing the semantics of counting in natural ways. A first approach replaces cardinalities by abstract but well-motivated values of ‘mass’ or other mereological aggregating notions. A second approach keeps the cardinalities but generalizes the meaning of counting to work in models that allow dependencies between variables.

Finally, we return to our starting point in natural language, confronting the architecture of our formal systems with linguistic quantifier vocabulary and syntax, as well as with natural reasoning modules such as the monotonicity calculus. In addition to these encounters with formal semantics, we discuss the role of counting in semantic evaluation procedures for quantifier expressions and determine, for instance, which binary quantifiers are computable by finite ‘semantic automata’. We conclude with some general thoughts on yet further entanglements of logic and counting in formal systems, on rethinking the qualitative/quantitative divide, and on connecting our analysis to empirical findings in cognitive science.

Fairness in machine learning is of considerable interest in recent years owing to the propensity of algorithms trained on historical data to amplify and perpetuate historical biases. In this paper, we argue for a formal reconstruction of fairness definitions, not so much to replace existing definitions but to ground their application in an epistemic setting and allow for rich environmental modeling. Consequently we look into three notions: fairness through unawareness, demographic parity and counterfactual fairness, and formalize these in the epistemic situation calculus.

In this article I present some modal-epistemic arguments for the existence of God, based on the possibility of omniscience. For this, I provide modal formal systems that allow obtaining the existence of God as a theorem. Moreover, based on what I assume as reasonable premises, they show that the strong agnostic position is contradictory, since it allows the conclusion both that God exists and that God does not exist.