There appears to be few, if any, limits on what sorts of logical connectives can be added to a given logic. One source of potential limitations is the motivating ideology associated with a logic. While extraneous to the logic, the motivating ideology is often important for the development of formal and philosophical work on that logic, as is the case with intuitionistic logic. One family of logics for which the philosophical ideology is important is the family of relevant logics. In (...) this paper, I explore the limits of what a relevant connective is, showing how some basic criteria motivated by the ideology of relevant logicians provide robust limits on potential connectives. These criteria provide some plausible necessary conditions on being a relevant connective. (shrink)

There is a standard objection against purported explanations of how a language L can express the notion of being a true sentence of L. According to this objection, such explanations avoid one paradox (the Liar) only to succumb to another of the same kind. Even if L can contain its own truth predicate, we can identify another notion it cannot express, on pain of contradiction via Liar-like reasoning. This paper seeks to undermine such ‘revenge’ by arguing that it presupposes a (...) dubious assumption about the linguistic expression of concepts. Successful revenge would require that there be a notion other than truth that plays the same role with respect to concept-expression that truth is naturally thought to play before we are confronted with the Liar paradox. (shrink)

The recent, short debate over the alethic undecidability of a Liar Sentence between Stephen Barker and Mark Jago is revisited. It is argued that Jago’s objections succeed in refuting Barker’s alethic undecidability solution to the Liar Paradox, but that, nevertheless, this approach may be revived as the alethic indeterminacy solution to the Liar Paradox. According to the alethic indeterminacy solution, there is genuine metaphysical indeterminacy as to whether a Liar Sentence bears an alethic property, whether truth or falsity. While the (...) alethic indeterminacy solution is presented here, and some revenge cases are considered and addressed, the primary aim of this paper is to revive and defend this underexplored and auspicious approach to solving the Liar Paradox. (shrink)

The coherence of omniscience is sometimes challenged using self-referential sentences like, “No omniscient entity knows that which this very sentence expresses,” which suggest that there are truths which no omniscient entity knows. In this paper, I consider two strategies for addressing these challenges: The Common Strategy, which dismisses such self-referential sentences as meaningless, and The Conciliatory Strategy, which discounts them as quirky outliers with no impact on one’s status as being omniscient. I argue that neither strategy succeeds. The Common Strategy (...) fails because it is both unmotivated and impotent. The Conciliatory Strategy fails because it leads to embarrassing situations in which omniscient entities are epistemically inferior to non-omniscient entities: we can, for example, devise trivia-based drinking games that force omniscient entities into an intoxicated state; and, given plausible closure principles for belief, such entities are unable to have the sorts of beliefs that give them reason to refuse to play. (shrink)

This article introduces, studies, and applies a new system of logic which is called ‘HYPE’. In HYPE, formulas are evaluated at states that may exhibit truth value gaps and truth value gluts. Simple and natural semantic rules for negation and the conditional operator are formulated based on an incompatibility relation and a partial fusion operation on states. The semantics is worked out in formal and philosophical detail, and a sound and complete axiomatization is provided both for the propositional and the (...) predicate logic of the system. The propositional logic of HYPE is shown to contain first-degree entailment, to have the Finite Model Property, to be decidable, to have the Disjunction Property, and to extend intuitionistic propositional logic conservatively when intuitionistic negation is defined appropriately by HYPE’s logical connectives. Furthermore, HYPE’s first-order logic is a conservative extension of intuitionistic logic with the Constant Domain Axiom, when intuitionistic negation is again defined appropriately. The system allows for simple model constructions and intuitive Euler-Venn-like diagrams, and its logical structure matches structures well-known from ordinary mathematics, such as from optimization theory, combinatorics, and graph theory. HYPE may also be used as a general logical framework in which different systems of logic can be studied, compared, and combined. In particular, HYPE is found to relate in interesting ways to classical logic and various systems of relevance and paraconsistent logic, many-valued logic, and truthmaker semantics. On the philosophical side, if used as a logic for theories of type-free truth, HYPE is shown to address semantic paradoxes such as the Liar Paradox by extending non-classical fixed-point interpretations of truth by a conditional as well-behaved as that of intuitionistic logic. Finally, HYPE may be used as a background system for modal operators that create hyperintensional contexts, though the details of this application need to be left to follow-up work. (shrink)

The paper discusses the Inconsistency Theory of Truth (IT), the view that “true” is inconsistent in the sense that its meaning-constitutive principles include all instances of the truth-schema (T). It argues that (IT) entails that anyone using “true” in its ordinary sense is committed to all the (T)-instances and that any theory in which “true” is used in that sense entails the (T)-instances (which, given classical logic, entail contradictions). More specifically, I argue that theorists are committed to the meaning-constitutive principles (...) of logical constants, relative to the interpretation they intend thereof (e.g., classical), and that theories containing logical constants entail those principles. Further, I argue, since there is no relevant difference from the case of “true”, inconsistency theorists’ uses of “true” commit them to the (T)-instances. Adherents of (IT) are recommended, as a consequence, to eschew the truth-predicate. I also criticise Matti Eklund’s account of how the semantic value of “true” is determined, which can be taken as an attempt to show how “true” can be consistently used, despite being inconsistent. (shrink)

Dialetheic accounts of the liar paradox hold that liar sentences are both true and false. One problem that besets accounts of the liar paradox is that of “revenge liars”. A revenge liar is a liar sentence which, while being of the same kind as the liar sentences an account attempts to handle, cannot be handled in the same way they are without generating contradictions that the account in question is powerless to resolve. It might be thought that dialetheic accounts are (...) immune to revenge problems: If one can intelligibly hold that standard liars are both true and false, why not revenge liars as well? However, in this paper it is argued that dialetheism faces a dilemma: Either it cannot express the distinction between those sentences which are both true and false and those which are not, or else it too suffers from revenge problems. I explore a few different ways in which a dialetheist might try to avoid this dilemma. First, I present a variant of the logic called LP, and show both that it is subject to revenge problems and that it is not well suited to a dialetheic interpretation. Second, I develop a means of expressing the exclusive truth or falsity of sentences which can be utilized by any language that has certain features. Unfortunately, it leads straight to trivialism. Finally, I examine the claim that dialetheists can express the exclusive truth or falsity of sentences in the same way a non-dialetheist can, and conclude that they cannot do so. In the end, it seems that dialetheism’s dilemma is inescapable. (shrink)

ABSTRACT: This chapter offers a revenge-free solution to the liar paradox (at the centre of which is the notion of Gestalt shift) and presents a formal representation of truth in, or for, a natural language like English, which proposes to show both why -- and how -- truth is coherent and how it appears to be incoherent, while preserving classical logic and most principles that some philosophers have taken to be central to the concept of truth and our use of (...) that notion. The chapter argues that, by using a truth operator rather than truth predicate, it is possible to provide a coherent, model-theoretic representation of truth with various desirable features. After investigating what features of liar sentences are responsible for their paradoxicality, the chapter identifies the logic as the normal modal logic KT4M (= S4M). Drawing on the structure of KT4M (=S4M), the author proposes that, pace deflationism, truth has content, that the content of truth is bivalence, and that the notions of both truth and bivalence are semideterminable. (shrink)