Logical inferentialists claim that the meanings of the logical constants are given by their inference rules. To rule out tonk-like expressions, it is often demanded that pairs of inference rules must be harmonious. The usual inference rules for the identity predicate are not harmonious, but most inferentialists want identity to be logical. Stephen Read has tried to formulate alternative, harmonious inference rules for identity. It will be proved, however, that his rules are precisely as strong as the old rules and that, because the old rules are not harmonious (as Read argues), nor are his. Further, it will be shown that no sound rules will be any improvement. Identity remains in need of satisfactory inferentialist treatment.

Though the study of grounding is still in the early stages, Kit Fine, in ”The Pure Logic of Ground”, has made a seminal attempt at formalization. Formalization of this sort is supposed to bring clarity and precision to our theorizing, as it has to the study of other metaphysically important phenomena, like modality and vagueness. Unfortunately, as I will argue, Fine ties the formal treatment of grounding to the obscure notion of a weak ground. The obscurity of weak ground, together with its centrality in Fine’s system, threatens to undermine the extent to which this formalization offers clarity and precision. In this paper, I show how to overcome this problem. I describe a system, the logic of strict ground (LSG) and demonstrate its adequacy; I specify a translation scheme for interpreting Fine’s weak grounding claims; I show that the interpretation verifies all of the principles of Fine’s system; and I show that derivability in Fine’s system can be exactly characterized in terms of derivability in LSG. I conclude that Fine’s system is reducible to LSG.

I develop a formal logic in which quantified arguments occur in argument positions of predicates. This logic also incorporates negative predication, anaphora and converse relation terms, namely, additional syntactic features of natural language. In these and additional respects, it represents the logic of natural language more adequately than does any version of Frege’s Predicate Calculus. I first introduce the system’s main ideas and familiarize it by means of translations of natural language sentences. I then develop a formal system built on these principles, the Quantified Argument Calculus or Quarc. I provide a truth-value assignment semantics and a proof system for the Quarc. I next demonstrate the system’s power by a variety of proofs; I prove its soundness; and I comment on its completeness. I then extend the system to modal logic, again providing a proof system and a truth-value assignment semantics. I proceed to show how the Quarc versions of the Barcan formulas, of their converses and of necessary existence come out straightforwardly invalid, which I argue is an advantage of the modal Quarc over modal Predicate Logic as a system intended to capture the logic of natural language.

In this second and last paper of the two part investigation on “Modality and Axiomatic Theories of Truth” we apply a general strategy for constructing modal theories over axiomatic theories of truth to the theory Kripke-Feferman. This general strategy was developed in the first part of our investigation. Applying the strategy to Kripke-Feferman leads to the theory Modal Kripke-Feferman which we discuss from the three perspectives that we had already considered in the first paper, where we discussed the theory Modal Friedman-Sheard. That is, we first show that Modal Kripke-Feferman preserves theoremhood modulo translation with respect to modal operator logic. Second, we develop a modal semantics fitting the newly developed theory. Third, we investigate whether the modal predicate of Modal Kripke-Feferman can be understood along the lines of a proposal of Kripke, namely as a truth predicate modified by a modal operator.

This paper scrutinizes the relationship between inconsistency and incoherence with a special focus on probabilistic measures of coherence. As is shown, while the majority of extant coherence measures face problems regarding the assessment of inconsistent sets of propositions, it is possible to adapt the measures in order to improve their performance. Furthermore, different intuitions regarding the degree of incoherence of inconsistent sets of propositions are surveyed and assessed with respect to extant measures. In this context, a refined approach to measuring coherence is introduced. As is argued, by means of this approach one can account for the diverging coherence intuitions regarding inconsistent sets independently of the discussion on the adequacy of different probabilistic explications of coherence. The last part of the paper is devoted to the question of whether there is a covariation between degrees of inconsistency and degrees of incoherence in the sense that the higher the degree of inconsistency of a set of propositions, the higher its degree of incoherence. Focusing on two straightforward measures of the degree of inconsistency, this latter question is answered in the negative.

In this paper, we investigate the phenomenon of speed-up in the context of theories of truth. We focus on axiomatic theories of truth extending Peano arithmetic. We are particularly interested on whether conservative extensions of PA have speed-up and on how this relates to a deflationist account. We show that disquotational theories have no significant speed-up, in contrast to some compositional theories, and we briefly assess the philosophical implications of these results.

Restricted quantification poses a serious and under-appreciated challenge for nonclassical approaches to both vagueness and the semantic paradoxes. It is tempting to explain “All A are B” as “For all x, if x is A then x is B”; but in the nonclassical logics typically used in dealing with vagueness and the semantic paradoxes (even those where ‘if … then’ is a special conditional not definable in terms of negation and disjunction or conjunction), this definition of restricted quantification fails to deliver important principles of restricted quantification that we’d expect. If we’re going to use a nonclassical logic, we need one that handles restricted quantification better.

The challenge is especially acute for naive theories of truth—roughly, theories that take True(〈A〉) to be intersubstitutable with A, even when A is a “paradoxical” sentence such as a Liar-sentence. A naive truth theory inevitably involves a somewhat nonclassical logic; the challenge is to get a logic that’s compatible with naive truth and also validates intuitively obvious claims involving restricted quantification (for instance, “If S is a truth stated by Jones, and every truth stated by Jones was also stated by Smith, then S is a truth stated by Smith”). No extant naive truth theory even comes close to meeting this challenge, including the theory I put forth in Saving Truth from Paradox. After reviewing the motivations for naive truth, and elaborating on some of the problems posed by restricted quantification, I will show how to do better. (I take the resulting logic to be appropriate for vagueness too, though that goes beyond the present paper.)

In showing that the resulting logic is adequate to naive truth, I will employ a somewhat novel fixed point construction that may prove useful in other contexts.

A typical first stab at explicating the thesis of physicalism is this: physicalism is true iff every fact about the world is entailed by the conjunction of physical facts. The same holds, mutatis mutandis, for other hypotheses about the fundamental nature of our world. But it has been recognized that this would leave such hypotheses without the fighting chance that they deserve: certain negative truths, like the truth (if it is one) that there are no angels, are not entailed by the physical facts, but nonetheless do not threaten physicalism. A plausible remedy that has been suggested by Jackson and Chalmers is that physicalism boils down to the thesis that every truth is entailed by the conjunction of the physical facts prefixed by a “that’s it” or “totality” operator. To evaluate this suggestion, we need to know what that operator means, and—since the truth of physicalism hinges on what is entailed by a totality claim—what its logic is. That is, we need to understand the logic of totality, or total logic. In this paper, I add a totality operator to the language of propositional logic and present a model theory for it, building on a suggestion by Chalmers and Jackson. I then prove determination results for a number of different systems.

We give some negative results on the expressiveness of naïve set theory (NS) in LP and in the four variants of minimally inconsistent LP defined in Crabbé (2011): ${\rm{L}}{{\rm{P}}_m},{\rm{L}}{{\rm{P}}_ = },{\rm{L}}{{\rm{P}}_ \subseteq }$, and ${\rm{L}}{{\rm{P}}_ \supseteq }$. We show that NS in LP cannot prove the existence of sets that behave like singleton sets, Cartesian pairs, or infinitely ascending linear orders. We show that NS is close to trivial in ${\rm{L}}{{\rm{P}}_m}$ and ${\rm{L}}{{\rm{P}}_ \subseteq }$, in the sense that its only minimally inconsistent model is a one-element model. We show that NS in ${\rm{L}}{{\rm{P}}_ = }$ and ${\rm{L}}{{\rm{P}}_ \supseteq }$ has the same limitations we give for NS in LP.

This paper investigates the question of characterizing first-order LFIs (logics of formal inconsistency) by means of two-valued semantics. LFIs are powerful paraconsistent logics that encode classical logic and permit a finer distinction between contradictions and inconsistencies, with a deep involvement in philosophical and foundational questions. Although focused on just one particular case, namely, the quantified logic QmbC, the method proposed here is completely general for this kind of logics, and can be easily extended to a large family of quantified paraconsistent logics, supplying a sound and complete semantical interpretation for such logics. However, certain subtleties involving term substitution and replacement, that are hidden in classical structures, have to be taken into account when one ventures into the realm of nonclassical reasoning. This paper shows how such difficulties can be overcome, and offers detailed proofs showing that a smooth treatment of semantical characterization can be given to all such logics. Although the paper is well-endowed in technical details and results, it has a significant philosophical aside: it shows how slight extensions of classical methods can be used to construct the basic model theory of logics that are weaker than traditional logic due to the absence of certain rules present in classical logic. Several such logics, however, as in the case of the LFIs treated here, are notorious for their wealth of models precisely because they do not make indiscriminate use of certain rules; these models thus require new methods. In the case of this paper, by just appealing to a refined version of the Principle of Explosion, or Pseudo-Scotus, some new constructions and crafty solutions to certain nonobvious subtleties are proposed. The result is that a richer extension of model theory can be inaugurated, with interest not only for paraconsistency, but hopefully to other enlargements of traditional logic.

A unified answer is offered to two distinct fundamental questions: whether a nonclassical solution to the semantic paradoxes should be extended to other apparently similar paradoxes (in particular, to the paradoxes of logical properties) and whether a nonclassical logic should be expressed in a nonclassical metalanguage. The paper starts by reviewing a budget of paradoxes involving the logical properties of validity, inconsistency, and compatibility. The author’s favored substructural approach to naive truth is then presented and it is explained how that approach can be extended in a very natural way so as to solve a certain paradox of validity. However, three individually decisive reasons are later provided for thinking that no approach adopting a classical metalanguage can adequately account for all the features involved in the paradoxes of logical properties. Consequently, the paper undertakes the task to do better, and, building on the system already developed, introduces a theory in a nonclassical metalanguage that expresses an adequate logic of naive truth and of some naive logical properties.

This paper evaluates four different qualitative (probabilistic) accounts to coherence with a focus on structural properties (symmetries, asymmetries, and transitivity). It is shown that while coherence is not transitive on any of these accounts, there are screening-off conditions that render coherence transitive. In a second step, an array of quantitative (probabilistic) accounts to coherence is considered. The upshot is that extant measures differ considerably with respect to a number of symmetry constraints.