Abstract Recent years have witnessed extensive logical studies of bundled operators. A known difficulty of such studies is how to axiomatize bundled operators. In this paper, we propose a uniform approach to axiomatizing such operators, which we call ‘the method based on almost-definability schemas’. The approach is useful in finding axioms and inference rules, and in defining a suitable canonical relation, thus in the completeness proof of logical systems. This is, hopefully, good news for modal logicians who are interested in axiomatizing bundled operators. To explicate this approach, we choose four bundled operators—the operator N of purely physical necessity, the operator $N'$ called ‘All $_1$ and Only $_2$ ’, the operator $N"$ , and the operator $N"'$ , in the literature, where $N\varphi := \Box _1\varphi \land \neg \Box _2\varphi $ , $N'\varphi := \Box _1\varphi \land \Box _2\neg \varphi $ , $N"\varphi := \Box _1\varphi \vee \neg \Box _2\varphi $ , $N"'\varphi := \Box _1\varphi \vee \Box _2\neg \varphi $ . This approach can uniformly deal with axiomatizing these bundled operators. Among other contributions, we also answer several open questions, and obtain alternative axiomatizations which are deductively equivalent to the existing ones in the literature.

Abstract The paper proposes and studies new classical, type-free theories of truth and determinateness with unprecedented features. The theories are fully compositional, strongly classical (namely, their internal and external logics are both classical), and feature a defined determinateness predicate satisfying desirable and widely agreed principles. The theories capture a conception of truth and determinateness according to which the generalizing power associated with the classicality and full compositionality of truth is combined with the identification of a natural class of sentences—the determinate ones—for which clear-cut semantic rules are available. Our theories can also be seen as the classical closures of Kripke–Feferman truth: their $\omega $ -models, which we precisely pin down, result from including in the extension of the truth predicate the sentences that are satisfied by a Kripkean closed-off fixed-point model. The theories compare to recent theories proposed by Fujimoto and Halbach, featuring a primitive determinateness predicate. In the paper we show that our theories entail all principles of Fujimoto and Halbach’s theories, and are proof-theoretically equivalent to Fujimoto and Halbach’s $\mathsf {CD}^{+}$ . We also show establish some negative results on Fujimoto and Halbach’s theories: such results show that, unlike what happens in our theories, the primitive determinateness predicate prevents one from establishing clear and unrestricted semantic rules for the language with type-free truth.

Abstract Logical inferentialists have expected identity to be susceptible of harmonious introduction and elimination rules in natural deduction. While Read and Klev have proposed rules they argue are harmonious, Griffiths and Ahmed have criticized these rules as insufficient for harmony. These critics, moreover, suggest that no harmonious rules are forthcoming. We argue that these critics are correct: the logical inferentialist should abandon hope for harmonious rules for identity. The paper analyzes the three major uses of identity in presumed-logical languages: variable coordination, definitional substitution, and co-reference. We show that identity qua variable coordination is not logical by providing a harmonious natural-deduction system that captures this use through the quantifiers. We then argue that identity qua definitional substitution or co-reference faces a dilemma: either its rules are harmonious but they obscure its actual use in inference, or its rules are not harmonious but they make its actual use in inference plain. We conclude that the inferentialist may have harmonious rules for identity only by disrespecting its inferential use.

Abstract This paper offers a substantial improvement in the revision-theoretic approach to conditionals in theories of transparent truth. The main modifications are (i) a new limit rule; (ii) a modification of the extension to the continuum-valued case; and (iii) the suggestion of a variation on how universal quantification is handled, leading to more satisfactory laws of restricted quantification.

Abstract Two salient notions of sameness of theories are synonymy, aka definitional equivalence, and bi-interpretability. Of these two definitional equivalence is the strictest notion. In which cases can we infer synonymy from bi-interpretability? We study this question for the case of sequential theories. Our result is as follows. Suppose that two sequential theories are bi-interpretable and that the interpretations involved in the bi-interpretation are one-dimensional and identity preserving. Then, the theories are synonymous. The crucial ingredient of our proof is a version of the Schröder–Bernstein theorem under very weak conditions. We think this last result has some independent interest. We provide an example to show that this result is optimal. There are two finitely axiomatized sequential theories that are bi-interpretable but not synonymous, where precisely one of the interpretations involved in the bi-interpretation is not identity preserving.