In this article we show that bi-intuitionistic predicate logic lacks the Craig Interpolation Property. We proceed by adapting the counterexample given by Mints, Olkhovikov and Urquhart for intuitionistic predicate logic with constant domains [13]. More precisely, we show that there is a valid implication $\phi \rightarrow \psi $ with no interpolant. Importantly, this result does not contradict the unfortunately named ‘Craig interpolation’ theorem established by Rauszer in [24] since that article is about the property more correctly named ‘deductive interpolation’ (see Galatos, Jipsen, Kowalski and Ono’s use of this term in [5]) for global consequence. Given that the deduction theorem fails for bi-intuitionistic logic with global consequence, the two formulations of the property are not equivalent.

We present a new frame semantics for positive relevant and substructural propositional logics. This frame semantics is both a generalisation of Routley–Meyer ternary frames and a simplification of them. The key innovation of this semantics is the use of a single accessibility relation to relate collections of points to points. Different logics are modeled by varying the kinds of collections used: they can be sets, multisets, lists or trees. We show that collection frames on trees are sound and complete for the basic positive distributive substructural logic $\mathsf {B}^+$, that collection frames on multisets are sound and complete for $\mathsf {RW}^+$ (the relevant logic $\mathsf {R}^+$, without contraction, or equivalently, positive multiplicative and additive linear logic with distribution for the additive connectives), and that collection frames on sets are sound for the positive relevant logic $\mathsf {R}^+$. The completeness of set frames for $\mathsf {R}^+$ is, currently, an open question.

In this essay we introduce a new tool for studying the patterns of sentential reference within the framework introduced in [2] and known as the language of paradox $\mathcal {L}_{\mathsf {P}}$: strong $\mathcal {L}_{\mathsf {P}}$-homomorphisms. In particular, we show that (i) strong $\mathcal {L}_{\mathsf {P}}$-homomorphisms between $\mathcal {L}_{\mathsf {P}}$ constructions preserve paradoxicality, (ii) many (but not all) earlier results regarding the paradoxicality of $\mathcal {L}_{\mathsf {P}}$ constructions can be recast as special cases of our central result regarding strong $\mathcal {L}_{\mathsf {P}}$-homomorphisms, and (iii) that we can use strong $\mathcal {L}_{\mathsf { P}}$-homomorphisms to provide a simple demonstration of the paradoxical nature of a well-known paradox that has not received much attention in this context: the McGee paradox. In addition, along the way we will highlight how strong $\mathcal {L}_{\mathsf {P}}$-homomorphisms highlight novel connections between the graph-theoretic analyses of paradoxes mobilized in the $\mathcal {L}_{\mathsf {P}}$ framework and the methods and tools of category theory.

We present a series of proof systems for exact entailment (i.e., relevant truthmaker preservation from premises to conclusion) and prove soundness and completeness. Using the proof systems, we observe that exact entailment is hyperintensional not only in the sense of Cresswell, but also in the sense recently proposed by Odintsov and Wansing.

Hyperlogic is a hyperintensional system designed to regiment metalogical claims (e.g., “Intuitionistic logic is correct” or “The law of excluded middle holds”) into the object language, including within embedded environments such as attitude reports and counterfactuals. This paper is the first of a two-part series exploring the logic of hyperlogic. This part presents a minimal logic of hyperlogic and proves its completeness. It consists of two interdefined axiomatic systems: one for classical consequence (truth preservation under a classical interpretation of the connectives) and one for “universal” consequence (truth preservation under any interpretation). The sequel to this paper explores stronger logics that are sound and complete over various restricted classes of models as well as languages with hyperintensional operators.

In providing a good foundation for mathematics, set theorists often aim to develop the strongest theories possible and avoid those theories that place undue restrictions on the capacity to possess strength. For example, adding a measurable cardinal to $ZFC$ is thought to give a stronger theory than adding $V=L$ and the latter is thought to be more restrictive than the former. The two main proponents of this style of account are Penelope Maddy and John Steel. In this paper, I’ll offer a third account that is intended to provide a simple analysis of restrictiveness based on the algebraic concept of retraction in the category of theories. I will also deliver some results and arguments that suggest some plausible alternative approaches to analyzing restrictiveness do not live up to their intuitive motivation.

Neo-Fregean logicists claim that Hume’s Principle (HP) may be taken as an implicit definition of cardinal number, true simply by fiat. A long-standing problem for neo-Fregean logicism is that HP is not deductively conservative over pure axiomatic second-order logic. This seems to preclude HP from being true by fiat. In this paper, we study Richard Kimberly Heck’s Two-Sorted Frege Arithmetic (2FA), a variation on HP which has been thought to be deductively conservative over second-order logic. We show that it isn’t. In fact, 2FA is not conservative over n-th order logic, for all $n \geq 2$. It follows that in the usual one-sorted setting, HP is not deductively Field-conservative over second- or higher-order logic.

Minimally inconsistent LP (MiLP) is a nonmonotonic paraconsistent logic based on Graham Priest’s logic of paradox (LP). Unlike LP, MiLP purports to recover, in consistent situations, all of classical reasoning. The present paper conducts a proof-theoretic analysis of MiLP. I highlight certain properties of this logic, introduce a simple sequent system for it, and establish soundness and completeness results. In addition, I show how to use my proof system in response to a criticism of this logic put forward by J. C. Beall.

This paper generalises an argument for probabilism due to Lindley [9]. I extend the argument to a number of non-classical logical settings whose truth-values, seen here as ideal aims for belief, are in the set $\{0,1\}$, and where logical consequence $\models $ is given the “no-drop” characterization. First I will show that, in each of these settings, an agent’s credence can only avoid accuracy-domination if its canonical transform is a (possibly non-classical) probability function. In other words, if an agent values accuracy as the fundamental epistemic virtue, it is a necessary requirement for rationality that her credence have some probabilistic structure. Then I show that for a certain class of reasonable measures of inaccuracy, having such a probabilistic structure is sufficient to avoid accuracy-domination in these non-classical settings.

In “Some Remarks on Extending and Interpreting Theories with a Partial Truth Predicate”, Reinhardt [21] famously proposed an instrumentalist interpretation of the truth theory Kripke–Feferman ( $\mathrm {KF}$ ) in analogy to Hilbert’s program. Reinhardt suggested to view $\mathrm {KF}$ as a tool for generating “the significant part of $\mathrm {KF}$ ”, that is, as a tool for deriving sentences of the form $\mathrm{Tr}\ulcorner {\varphi }\urcorner $ . The constitutive question of Reinhardt’s program was whether it was possible “to justify the use of nonsignificant sentences entirely within the framework of significant sentences”. This question was answered negatively by Halbach & Horsten [10] but we argue that under a more careful interpretation the question may receive a positive answer. To this end, we propose to shift attention from $\mathrm {KF}$ -provably true sentences to $\mathrm {KF}$ -provably true inferences, that is, we shall identify the significant part of $\mathrm {KF}$ with the set of pairs $\langle {\Gamma , \Delta }\rangle $ , such that $\mathrm {KF}$ proves that if all members of $\Gamma $ are true, at least one member of $\Delta $ is true. In way of addressing Reinhardt’s question we show that the provably true inferences of suitable $\mathrm {KF}$ -like theories coincide with the provable sequents of matching versions of the theory Partial Kripke–Feferman ( $\mathrm {PKF}$ ).

Is it possible to maintain classical logic, stay close to classical semantics, and yet accept that language might be semantically indeterminate? The article gives an affirmative answer by Ramsifying classical semantics, which yields a new semantic theory that remains much closer to classical semantics than supervaluationism but which at the same time avoids the problematic classical presupposition of semantic determinacy. The resulting Ramsey semantics is developed in detail, it is shown to supply a classical concept of truth and to fully support the rules and metarules of classical logic, and it is applied to vague terms as well as to theoretical or open-ended terms from mathematics and science. The theory also demonstrates how diachronic or synchronic interpretational continuity across languages is compatible with semantic indeterminacy.

The use of the symbol $\mathbin {\boldsymbol {\vee }}$ for disjunction in formal logic is ubiquitous. Where did it come from? The paper details the evolution of the symbol $\mathbin {\boldsymbol {\vee }}$ in its historical and logical context. Some sources say that disjunction in its use as connecting propositions or formulas was introduced by Peano; others suggest that it originated as an abbreviation of the Latin word for “or,” vel. We show that the origin of the symbol $\mathbin {\boldsymbol {\vee }}$ for disjunction can be traced to Whitehead and Russell’s pre-Principia work in formal logic. Because of Principia’s influence, its notation was widely adopted by philosophers working in logic (the logical empiricists in the 1920s and 1930s, especially Carnap and early Quine). Hilbert’s adoption of $\mathbin {\boldsymbol {\vee }}$ in his Grundzüge der theoretischen Logik guaranteed its widespread use by mathematical logicians. The origins of other logical symbols are also discussed.

Francesco Berto proposed a logic for imaginative episodes. The logic establishes certain (in)validities concerning episodic imagination. They are not all equally plausible as principles of episodic imagination. The logic also does not model that the initial input of an imaginative episode is deliberately chosen. Stit-imagination logic models the imagining agent’s deliberate choice of the content of their imagining. However, the logic does not model the episodic nature of imagination. The present paper combines the two logics, thereby modelling imaginative episodes with deliberately chosen initial input. We use a combination of stit-imagination logic and a content-sensitive variably strict conditional à la Berto, for which we give a Chellas–Segerberg semantics. The proposed semantics has the following advantages over Berto’s: (i) we model the deliberate choice of initial input of imaginative episodes (in a multi-agent setting), (ii) we show frame correspondences for axiomatic analogues of Berto’s validities, which (iii) allows the possibility to disregard axioms that might be considered not plausible as principles concerning imaginative episodes. We do not take a definite stance on whether these should be disregarded but give reasons for why one might want to disregard them. Finally, we compare our semantics briefly with recent work, which aims to model voluntary imagination, and argue that our semantics models different aspects.

The way Leibniz applied his philosophy to mathematics has been the subject of longstanding debates. A key piece of evidence is his letter to Masson on bodies. We offer an interpretation of this often misunderstood text, dealing with the status of infinite divisibility in nature, rather than in mathematics. In line with this distinction, we offer a reading of the fictionality of infinitesimals. The letter has been claimed to support a reading of infinitesimals according to which they are logical fictions, contradictory in their definition, and thus absolutely impossible. The advocates of such a reading have lumped infinitesimals with infinite wholes, which are rejected by Leibniz as contradicting the part–whole principle. Far from supporting this reading, the letter is arguably consistent with the view that infinitesimals, as inassignable quantities, are mentis fictiones, i.e., (well-founded) fictions usable in mathematics, but possibly contrary to the Leibnizian principle of the harmony of things and not necessarily idealizing anything in rerum natura. Unlike infinite wholes, infinitesimals—as well as imaginary roots and other well-founded fictions—may involve accidental (as opposed to absolute) impossibilities, in accordance with the Leibnizian theories of knowledge and modality.

According to conciliatory views on the significance of disagreement, it’s rational for you to become less confident in your take on an issue in case your epistemic peer’s take on it is different. These views are intuitively appealing, but they also face a powerful objection: in scenarios that involve disagreements over their own correctness, conciliatory views appear to self-defeat and, thereby, issue inconsistent recommendations. This paper provides a response to this objection. Drawing on the work from defeasible logics paradigm and abstract argumentation, it develops a formal model of conciliatory reasoning and explores its behavior in the troubling scenarios. The model suggests that the recommendations that conciliatory views issue in such scenarios are perfectly reasonable—even if outwardly they may look odd.

In the literature, predicativism is connected not only with the Vicious Circle Principle but also with the idea that certain totalities are inherently potential. To explain the connection between these two aspects of predicativism, we explore some approaches to predicativity within the modal framework for potentiality developed in Linnebo (2013) and Linnebo and Shapiro (2019). This puts predicativism into a more general framework and helps to sharpen some of its key theses.

The purpose of this paper is to compare the notion of a Grzegorczyk point introduced in [19] (and thoroughly investigated in [3, 14, 16, 18]) to the standard notions of a filter in Boolean algebras and round filter in Boolean contact algebras. In particular, we compare Grzegorczyk points to filters and ultrafilters of atomic and atomless algebras. We also prove how a certain extra axiom influences topological spaces for Grzegorczyk contact algebras. Last but not least, we do not refrain from a philosophical interpretation of the results from the paper.

Standard Type Theory, ${\textrm {STT}}$, tells us that $b^n(a^m)$ is well-formed iff $n=m+1$. However, Linnebo and Rayo [23] have advocated the use of Cumulative Type Theory, $\textrm {CTT}$, which has more relaxed type-restrictions: according to $\textrm {CTT}$, $b^\beta (a^\alpha )$ is well-formed iff $\beta>\alpha $. In this paper, we set ourselves against $\textrm {CTT}$. We begin our case by arguing against Linnebo and Rayo’s claim that $\textrm {CTT}$ sheds new philosophical light on set theory. We then argue that, while $\textrm {CTT}$’s type-restrictions are unjustifiable, the type-restrictions imposed by ${\textrm {STT}}$ are justified by a Fregean semantics. What is more, this Fregean semantics provides us with a principled way to resist Linnebo and Rayo’s Semantic Argument for $\textrm {CTT}$. We end by examining an alternative approach to cumulative types due to Florio and Jones [10]; we argue that their theory is best seen as a misleadingly formulated version of ${\textrm {STT}}$.

In this paper we examine various requirements on the formalisation choices under which self-reference can be adequately formalised in arithmetic. In particular, we study self-referential numberings, which immediately provide a strong notion of self-reference even for expressively weak languages. The results of this paper suggest that the question whether truly self-referential reasoning can be formalised in arithmetic is more sensitive to the underlying coding apparatus than usually believed. As a case study, we show how this sensitivity affects the formal study of certain principles of self-referential truth.

Can conjunctive propositions be identical without their conjuncts being identical? Can universally quantified propositions be identical without their instances being identical? On a common conception of propositions, on which they inherit the logical structure of the sentences which express them, the answer is negative both times. Here, it will be shown that such a negative answer to both questions is inconsistent, assuming a standard type-theoretic formalization of theorizing about propositions. The result is not specific to conjunction and universal quantification, but applies to any binary operator and propositional quantifier. It is also shown that the result essentially arises out of giving a negative answer to both questions, as each negative answer is consistent by itself.