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)

Although there has been a recent swell of interest in theories of truth that attempt solutions to the liar paradox and the other paradoxes affecting our concept of truth, many of these theories have been criticized for generating new paradoxes, called revenge paradoxes. The criticism is that the theories of truth in question are inadequate because they only work for languages lacking in the resources to generate revenge paradoxes. Theorists facing these objections offer a range of replies, and the matter (...) seems now to be at a standoff. I aim, first, to bolster the revenge objections by considering a relation, internalizability, between languages and theories of truth. A theory of truth is internalizable for a language iff there is an extension of that language in which the theory is expressible and for which the theory provides an accurate and complete assignment of semantic values. There are good reasons to think that acceptable theories of truth are internalizable for any language. With this internalizability requirement in hand, I argue that properly formulated revenge objections are decisive and that the replies to them are inadequate. Second, I show that the internalizability requirement can be met by a certain theory of truth. The central claim of this theory is that truth is an inconsistent concept and should be replaced with a pair of consistent concepts that can then be used to provide a semantics for our truth predicates. This theory is compatible with classical logic, does not give rise to revenge paradoxes of any kind, and satisfies the internalizability requirement. (shrink)

Revenge arguments purport to show that any proposed solution to the semantic paradoxes generates new paradoxes that prove that solution to be inadequate. In this paper, I focus on revenge arguments that employ the model-theoretic semantics of a target theory and I argue, contra the current revenge-theoretic wisdom, that they can constitute genuine expressive limitations. I consider the anti-revenge strategy elaborated by Field and argue that it does not offer a way out of the revenge problem. More generally, I argue (...) that the difference between ‘standard’ and ‘revenge’ paradoxes is ill-conceived and should be abandoned. This will contribute to show that the theories that provide a uniform account of truth and other semantic notions are the ones best equipped to avoid the paradoxes altogether—‘standard’ and ‘revenge’ alike. (shrink)

The general thesis of this paper is that metasemantic theories can play a central role in determining the correct solution to the liar paradox. I argue for the thesis by providing a specific example. I show how Lewis’s reference-magnetic metasemantic theory may decide between two of the most influential solutions to the liar paradox: Kripke’s minimal fixed point theory of truth and Gupta and Belnap’s revision theory of truth. In particular, I suggest that Lewis’s metasemantic theory favours Kripke’s solution to (...) the paradox over Gupta and Belnap’s. I then sketch how other standard criteria for assessing solutions to the liar paradox, such as whether a solution faces a so-called revenge paradox, fit into this picture. While the discussion of the specific example is itself important, the underlying lesson is that we have an unused strategy for resolving one of the hardest problems in philosophy. (shrink)

The vast majority of putative solutions to the liar paradox face the infamous revenge problem. In recent work, however, Kevin Scharp has extensively developed an exciting and highly novel “inconsistency approach” to the paradox that, he claims, does not face revenge. If Scharp is right, then this represents a significant step forward in our attempts to solve the liar paradox. However, in this paper, I raise a revenge problem that faces Scharp's inconsistency approach.

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)

Since Saul Kripke’s influential work in the 1970s, the revisionary approach to semantic paradox—the idea that semantic paradoxes must be solved by weakening classical logic—has been increasingly popular. In this paper, we present a new revenge argument to the effect that the main revisionary approaches breed new paradoxes that they are unable to block.

The concept of information has well-known difficulties. Among the many issues that have been discussed is the alethic nature of a semantic conception of information. Floridi :197–222, 2004; Philos Phenomenol Res 70:351–370, 2005; EUJAP 3:31–41, 2007; The philosophy of information, Oxford University Press, Oxford, 2011) argued that semantic information must be truthful. In this article, arguments will be presented in favor of an alethically neutral conception of semantic information and it will be shown that such a conception can withstand Floridi’s (...) criticism. In particular, it is argued that an alethically neutral conception of semantic information can manage the so-called Bar-Hillel Carnap paradox, according to which contradictions have maximum informational content. This issue, as well as some of Floridi’s other arguments, is resolved by disentangling the property of being information from the property of being informative. The essay’s final conclusion is that although semantic information is alethically neutral, a veridical conception of semantic information can, and should, be retained as a subconcept of semantic information, as it is essential for the analysis of informativity, which, unlike the property of being information, depends on truth. (shrink)

Restrictionism is a response to the Liar and other paradoxes concerning truth. Restrictionists—as I will call proponents of the strategy—respond to these paradoxes by giving up instances of the schema -/- <p> is true iff p. -/- My aim is to show that the current unpopularity of restrictionism is undeserved. I will argue that, whilst cautious versions of the strategy may face serious problems, a radical and previously overlooked version of restrictionism provides a strong and defensible response to the paradoxes.

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)

This article explores ways in which the Revision Theory of Truth can be expressed in the object language. In particular, we investigate the extent to which semantic deficiency, stable truth, and nearly stable truth can be so expressed, and we study different axiomatic systems for the Revision Theory of Truth.

Although there is a prima facie strong case for a close connection between the meaning and inferential role of certain expressions, this connection seems seriously threatened by the semantic and logical paradoxes which rely on these inferential roles. Some philosophers have drawn radical conclusions from the paradoxes for the theory of meaning in general, and for which sentences in our language are true. I criticize these overreactions, and instead propose to distinguish two conceptions of inferential role. This distinction is closely (...) tied to two conceptions of deductive logic, and it is the key, I argue, for understanding first the connection between meaning and inferential role, and second what the paradoxes show more generally. (shrink)

A method of supervaluation for Kripke’s theory of truth is presented. It differs from Kripke’s own method in that it employs trees; results in a compositional semantics; assigns the intuitively correct truth values to the sentences of a particularly tricky example of Gupta’s; and – it is argued – is acceptable as an explication of the correspondence theory of truth.

In his Lectures on the History of Philosophy, Hegel develops a subtle analysis of Megarian paradoxes: the Liar, the Veiled Man and the Sorites. In this paper, we focus on Hegel's interpretation of...

This article is a preliminary presentation of conjunctive paraconsistency, the claim that there might be non-explosive true contradictions, but contradictory propositions cannot be considered separately true. In case of true ‘p and not p’, the conjuncts must be held untrue, Simplification fails. The conjunctive approach is dual to non-adjunctive conceptions of inconsistency, informed by the idea that there might be cases in which a proposition is true and its negation is true too, but the conjunction is untrue, Adjunction fails. While (...) non-adjunctivism is a well-known option, the other view is not so much studied nowadays, but it was not unknown in the tradition, and there are some positive suggestions, in recent literature, that the position is plausible and deserves to be developed. The article compares conjunctivism, non-adjunctivism and dialetheism, then focuses on some possible justifications, costs and benefits of the conjunctive view. (shrink)

In this paper, I examine the deep theological faultline generated by divergent understandings of the divine attributes among two early antagonistic Muslim groups – the traditionalists (main...

Jenkins (2007) charges that the language advanced in Beall (2007) is either expressively impoverished, or inconsistent. We argue that Jenkins’ objections are based on unreasonably strong constraints on formal theories of truth. Our primary concern is not to defend the ‘paranormal’ framework advanced in Beall, but to respond to a common – and implausible – ‘revenge’-style charge directed at a certain class of formal theories of truth and paradox.

I provide an interpretation of Wittgenstein's much criticized remarks on Gödel's First Incompleteness Theorem in the light of paraconsistent arithmetics: in taking Gödel's proof as a paradoxical derivation, Wittgenstein was right, given his deliberate rejection of the standard distinction between theory and metatheory. The reasoning behind the proof of the truth of the Gödel sentence is then performed within the formal system itself, which turns out to be inconsistent. I show that the models of paraconsistent arithmetics (obtained via the Meyer-Mortensen (...) collapsing filter on the standard model) match with many intuitions underlying Wittgensteins philosophy of mathematics, such as its strict finitism and the insistence on the decidability of any meaningful mathematical question. (shrink)

Liar-like paradoxes are typically arguments that, by using very intuitive resources of natural language, end up in contradiction. Consistent solutions to those paradoxes usually have difficulties either because they restrict the expressive power of the language, or else because they fall prey to extended versions of the paradox. Dialetheists, like Graham Priest, propose that we should take the Liar at face value and accept the contradictory conclusion as true. A logical treatment of such contradictions is also put forward, with the (...) Logic of Paradox, which should account for the manifestations of the Liar. In this paper we shall argue that such a formal approach, as advanced by Priest, is unsatisfactory. In order to make contradictions acceptable, Priest has to distinguish between two kinds of contradictions, internal and external, corresponding, respectively, to the conclusions of the simple and of the extended Liar. Given that, we argue that while the natural interpretation of LP was intended to account for true and false sentences, dealing with internal contradictions, it lacks the resources to tame external contradictions. Also, the negation sign of LP is unable to represent internal contradictions adequately, precisely because of its allowance of sentences that may be true and false. As a result, the formal account suffers from severe limitations, which make it unable to represent the contradiction obtained in the conclusion of each of the paradoxes. (shrink)

Logic: the Basics is an accessible introduction to the core philosophy topic of standard logic. Focussing on traditional Classical Logic the book deals with topics such as mathematical preliminaries, propositional logic, monadic quantified logic, polyadic quantified logic, and English and standard ‘symbolic transitions’. With exercises and sample answers throughout this thoroughly revised new edition not only comprehensively covers the core topics at introductory level but also gives the reader an idea of how they can take their knowledge further and the (...) philosophical questions around logic. -/- Logic: the Basics is essential reading for first-year undergraduate philosophy students on standard introductory logic courses. (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)

Supervaluational theories of vagueness have achieved considerable popularity in the past decades, as seen in eg [5], [12]. This popularity is only natural; supervaluations let us retain much of the power and simplicity of classical logic, while avoiding the commitment to strict bivalence that strikes many as implausible. Like many nonclassical logics, the supervaluationist system SP has a natural dual, the subvaluationist system SB, explored in eg [6], [28].1 As is usual for such dual systems, the classical features of SP (...) (typically viewed as benefits) appear in SB in ‘mirror-image’ form, and the nonclassical features of SP (typically viewed as costs) also appear in SB in ‘mirror-image’ form. Given this circumstance, it can be difficult to decide which of two dual systems is better suited for an approach to vagueness.2 The present paper starts from a consideration of these two approaches— the supervaluational and the subvaluational—and argues that neither of them is well-positioned to give a sensible logic for vague language. §2 presents the systems SP and SB and argues against their usefulness. Even if we suppose that the general picture of vague language they are often taken to embody is accurate, we ought not arrive at systems like SP and SB. Instead, such a picture should lead us to truth-functional systems like strong Kleene logic (K3) or its dual LP. §3 presents these systems, and argues that supervaluationist and subvaluationist understandings of language are better captured there; in particular, that a dialetheic approach to vagueness based on the logic LP is a more sensible approach. §4 goes on to consider the phenomenon of higher-order vagueness within an LP-based approach, and §5 closes with a consideration of the sorites argument itself. (shrink)

The present paper is concerned with an exposé of the basic general-semantic theses presented in the tract called “De insolubilibus” written by the 14th century British logician Roger Swyneshed. Swyneshed‘s semantics is analysed as a highly specific theory of truth, correspondence and facts . Swyneshed’s theory revises the correspondence theory of truth and rejects the principle of bivalence, while offering the solution to two different types of paradoxes . As it is usual for theories of this kind, Swyneshed’s semantics has (...) to face the specific forms of revenge-arguments, which lead to a specific conception of truthmaking. (shrink)

To borrow a colorful phrase from Kant, this dissertation offers a prolegomenon to any future semantic theory. The dissertation investigates Yablo's omega-liar paradox and draws the following consequence. Any semantic theory that accepts the existence of semantic objects must face Yablo's paradox. The dissertation endeavors to position Yablo's omega-liar in a role analogous to that which Russell's paradox has for the foundations of mathematics. Russell's paradox showed that if we wed mathematics to sets, then because of the many different possible (...) restrictions available for blocking the paradox, mathematics fractionates. There would be different mathematics. This is intolerable. It is similarly intolerable to have restrictions on the `objects' of Intentionality. Hence, in the light of Yablo's omega-liar, Intentionality cannot be wed to any theory of semantic objects. We ought, therefore, to think of Yablo's paradox as a natural language paradox, and as such we must accept its implications for the semantics of natural language, namely that those entities which are `meanings' must not be construed as objects. To establish our result, Yablo's paradox is examined in light of the criticisms of Priest. Priest maintains that Yablo's original omega-liar is flawed in its employment of a Tarski-style T-schema for its truth-predicate. Priest argues that the paradox is not formulable unless it employs a "satisfaction" predicate in place of its truth-predicate. Priest is mistaken. However, it will be shown that the omega-liar paradox depends essentially on the assumption of semantic objects. No formulation of the paradox is possible without this assumption. Given this, the dissertation looks at three different sorts of theories of propositions, and argues that two fail to specify a complete syntax for the Yablo sentences. Purely intensional propositions, however, are able to complete the syntax and thus generate the paradox. In the end, however, the restrictions normally associated with purely intensional propositions begin to look surprisingly like the hierarchies that Yablo sought to avoid with his paradox. The result is that while Yablo's paradox is syntactically formable within systems with formal hierarchies, it is not semantically so. (shrink)