Substructural solutions to the semantic paradoxes have been broadly discussed in recent years. In particular, according to the non-transitive solution, we have to give up the metarule of Cut, whose role is to guarantee that the consequence relation is transitive. This concession—giving up a meta rule—allows us to maintain the entire consequence relation of classical logic. The non-transitive solution has been generalized in recent works into a hierarchy of logics where classicality is maintained at more and (...) more metainferential levels. All the logics in this hierarchy can accommodate a truth predicate, including the logic at the top of the hierarchy—known as CMω—which presumably maintains classicality at all levels. CMω has so far been accounted for exclusively in model-theoretic terms. Therefore, there remains an open question: how do we account for this logic in proof-theoretic terms? Can there be found a proof system that admits each and every classical principle—at all inferential levels—but nevertheless blocks the derivation of the liar? In the present paper, I solve this problem by providing such a proof system and establishing soundness and completeness results. Yet, I also argue that the outcome is philosophically unsatisfactory. In fact, I’m afraid that in light of my results this metainferential solution to the paradoxes can hardly be called a “solution,” let alone a good one. (shrink)

The paper offers a solution to the semantic paradoxes, one in which (1) we keep the unrestricted truth schema “True(A)↔A”, and (2) the object language can include its own metalanguage. Because of the first feature, classical logic must be restricted, but full classical reasoning applies in “ordinary” contexts, including standard set theory. The more general logic that replaces classical logic includes a principle of substitutivity of equivalents, which with the truth schema leads to the general intersubstitutivity of True(A) (...) with A within the language.The logic is also shown to have the resources required to represent the way in which sentences (like the Liar sentence and the Curry sentence) that lead to paradox in classical logic are “defective”. We can in fact define a hierarchy of “defectiveness” predicates within the language. Contrary to claims that any solution to the paradoxes just breeds further paradoxes (“revenge problems”) involving defectiveness predicates, there is a general consistency/conservativeness proof that shows that talk of truth and the various “levels of defectiveness” can all be made coherent together within a single object language. (shrink)

This paper propounds a systematic examination of the link between the Knower Paradox and provability interpretations of modal logic. The aim of the paper is threefold: to give a streamlined presentation of the Knower Paradox and related results; to clarify the notion of a syntactical treatment of modalities; finally, to discuss the kind of solution that modal provability logic provides to the Paradox. I discuss the respective strength of different versions of the Knower Paradox, both in the framework of first-order (...) arithmetic and in that of modal logic with fixed point operators. It is shown that the notion of a syntactical treatment of modalities is ambiguous between a self-referential treatment and a metalinguistic treatment of modalities, and that these two notions are independent. I survey and compare the provability interpretations of modality respectively given by Skyrms, B. (1978, The Journal of Philosophy 75: 368–387) Anderson, C.A. (1983, The Journal of Philosophy 80: 338–355) and Solovay, R. (1976, Israel Journal of Mathematics 25: 287–304). I examine how these interpretations enable us to bypass the limitations imposed by the Knower Paradox while preserving the laws of classical logic, each time by appeal to a distinct form of hierarchy. (shrink)

The Liar paradox is an obstacle to a theory of truth, but a Liar sentence need not contain a semantic predicate. The Pinocchio paradox, devised by Veronique Eldridge-Smith, was the first published paradox to show this. Pinocchio’s nose grows if, and only if, what Pinocchio is saying is untrue. What happens if Pinocchio says that his nose is growing? Eldridge-Smith and Eldridge-Smith : 212-5, 2010) posed the Pinocchio paradox against the Tarskian-Kripkean solutions to the Liar paradox that use (...) language hierarchies. Eldridge-Smith : 306-8, 2011) also set the Pinocchio paradox against semantic dialetheic solutions to the Liar. Beall argued the Pinocchio story was just an impossible story. Eldridge-Smith : 749-752, 2012b) responded that unless the T-schema is a necessary truth of some sort, the Pinocchio principle is possible. Luna : 77–86, 2016) argues that the Pinocchio contradiction proves the principle is false. D’Agostini & Ficara discuss a more plausible physical truth-tracking trait, the Blushing Liar, and argue that the Pinocchio contradiction is not a metaphysical dialetheia. I respond to Luna, and D’Agostini & Ficara, and prove that the Pinocchio paradox is a counterexample to hierarchical solutions to the Liar. (shrink)

This article advances the first semantics that is neither for nor against a default implicational link between the progressive and perfective forms, when it comes to solving the imperfective paradox. Depending on the doxastic context of its use, we contend that the progressive form sometimes allows and sometimes does not allow the inference of the corresponding simple form. In other words, the preparatory phase of an event might or might not be believed to lead to its culmination. Indeed, the context (...) can put constraints on beliefs about the time of the culmination and whether or not it allows this inference to be made. From a formal perspective, this new solution to the imperfective paradox combines a specific modal approach with an event-structure analysis originating in event semantics. Finally, this approach solves the associated difficulties that have plagued the most well-known theories in this field. (shrink)

Disagreeing with most authors on vagueness, the author proposes a solution that he calls ‘three-valued semantic pluralism’ to the age-old sorites paradox. In essence, it is a three-valued semantics for a first-order vague language with identity with the additional suggestion that a vague language has more than one correct interpretation. Unlike the traditional three-valued approach to a vague language, three-valued semantic pluralism can accommodate the phenomenon of higher-order vagueness and the phenomenon of penumbral connection when equipped with ‘suitable (...) conditionals’. The author also shows that three-valued semantic pluralism is a natural consequence of a restricted form of the Tolerance principle ) and a few related ideas, and argues that ) is well-motivated by considerations about how we learn, teach, and use vague predicates. (shrink)

We defend a set of acceptance rules that avoids the lottery paradox, that is closed under classical entailment, and that accepts uncertain propositions without ad hoc restrictions. We show that the rules we recommend provide a semantics that validates exactly Adams’ conditional logic and are exactly the rules that preserve a natural, logical structure over probabilistic credal states that we call probalogic. To motivate probalogic, we first expand classical logic to geologic, which fills the entire unit cube, and then we (...) project the upper surfaces of the geological cube onto the plane of probabilistic credal states by means of standard, linear perspective, which may be interpreted as an extension of the classical condition of indifference. Finally, we apply the geometrical/logical methods developed in the paper to prove a series of trivialization theorems against question-invariance as a constraint on acceptance rules and against rational monotonicity as an axiom of conditional logic in situations of uncertainty. (shrink)

This volume includes a target paper, taking up the challenge to revive, within a modern (formal) framework, a medieval solution to the Liar Paradox which did ...

In his papers ‘Semantic Paradox ’ and ‘The Liar Paradox: Tangles and Chains,’ Tyler Burge provides a hierarchical solution to the Liar paradox. And in his paper ‘Epistemic Paradox ’ Burge extends his hierarchy approach to the epistemic paradox of belief instability, which I shall explain shortly. Although Burge's views on the Liar paradox have been widely criticized, his views on the paradox of belief instability have not received notable attention. In this paper I shall argue that Burge's (...) proposal is inadequate as a solution to the paradox of belief instability. For this purpose, I shall criticize Burge's claim that a circular evaluation of a thought is impossible, which is crucial for his proposal. The question of whether or not a circular evaluation of belief is possible is of its own philosophical interest as well. (shrink)

Fourteenth-century treatises on paradoxes of the liar family, especially Bradwardine's and Buridan's, raise issues concerning the meaning of sentences, in particular about closure of sentential meaning under implication, semantic pluralism and the ontological status of 'meanings', which are still topical for current theories of meaning. I outline ways in which they tend to be overlooked, raising issues that must be addressed by any respectable theory of meaning as well as pointing in the direction of possible answers. I analyse (...) a Bradwardinian theory of sentential meaning as it emerges from his treatment of liar sentences, exploring where it requires more thorough elaboration if it is to be a fully developed theory of sentential meaning. (shrink)

This paper offers the Boolean many-valued solution to the Sorites Paradox. According to the precisification-based Boolean many-valued theory, from which this solution arises, sentences have not only two truth values, truth (or 1) and falsity (or 0), but many Boolean values between 0 and 1. The Boolean value of a sentence is identified with the set of precisifications in which the sentence is true. Unlike degrees fuzzy logic assigns to sentences, Boolean many values are not linearly but only partially ordered; (...) so there are values that are incomparable. Despite this fact, the sentences in a sorites series can be taken as having values in a linear order, and losing (or gaining) value as we move from sentence to sentence in the series. So there is no sharp borderline between 0 and 1. Assigning Boolean many values instead of the two truth values to sentences for their vagueness values is analogous to assigning propositions instead of the truth values to sentences for their semantic values, as propositions are, from our viewpoint, also Boolean many values. (shrink)

The author considers following question: is a consistent semantically closed language possible? The negative answer is the orthodox answer in the logic of the 20th century. It was presented in Russell's theory of types and Tarski's semantic theory of metalanguages. Nevertheless, contemporary logicians and philosophers of language return to this problem time and again, pointing to its relevance in various aspects. In particular, it is asserted that semantically closed language is a very important tool for expressing logical and philosophical (...) ideas. In logic of the 20th century, the problem of semantically closed language was discussed in connection with the problem of logical paradoxes. Russell and Tarski saw a fundamental cause of paradoxes in the phenomenon of self-reference that arises in semantically closed language. Accordingly, a solution for paradoxes was seen in eliminating the cause, that is, in prohibiting semantically closed language by means of a hierarchy of logical types of classes (Russell) or a hierarchy of metalanguages (Tarski). However, some contemporary logicians criticize the hierarchical approach, whose main argument consists in asserting that the approach was wrong in its diagnosis of the cause of paradoxes. This author does not try to correct the diagnostics of the hierarchical approach by identifying another cause of paradoxes. Instead, the author recognizes that a general solution of the problem of paradoxes cannot be given, a priori, by eliminating what fundamentally generates them. In this article, a new “ad hoc solution" of the problem is offered that rests upon an empirical method of identifying and eliminating paradoxes. A specific characteristic of the method is admitting the existence of consistent semantically closed language. (shrink)

In this paper, I focus on some intuitionistic solutions to the Paradox of Knowability. I first consider the relatively little discussed idea that, on an intuitionistic interpretation of the conditional, there is no paradox to start with. I show that this proposal only works if proofs are thought of as tokens, and suggest that anti-realists themselves have good reasons for thinking of proofs as types. In then turn to more standard intuitionistic treatments, as proposed by Timothy Williamson and, most (...) recently, Michael Dummett. Intuitionists can either point out the intuitionistc invalidity of the inference from the claim that all truths are knowable to the insane conclusion that all truths are known, or they can outright demur from asserting the existence of forever-unknown truths, perhaps questioning—as Dummett now suggests—the applicability of the Principle of Bivalence to a certain class of empirical statements. I argue that if intuitionists reject strict finitism—the view that all truths are knowable by beings just like us—the prospects for either proposal look bleak. (shrink)

Moore's paradox pits our intuitions about semantic oddness against the concept of truth-functional consistency. Most solutions to the problem proceed by explaining away our intuitions. But "consistency" is a theory-laden concept, having different contours in different semantic theories. Truth-functional consistency is appropriate only if the semantic theory we are using identifies meaning with truth-conditions. I argue that such a framework is not appropriate when it comes to analyzing epistemic modality. I show that a theory which accounts (...) for a wide variety of semantic data about epistemic modals (update semantics) buys us a solution to Moore's paradox as a corollary. It turns out that Moorean propositions, when looked at through the lense of an appropriate semantic theory, are inconsistent after all. (shrink)

Fitch’s paradox of knowability is an apparently valid reasoning from the assumption (typical of semantic anti-realism) that every true proposition is knowable to the unacceptable conclusion that every true proposition is known. The paper develops a critical dialectic wrt one of the best motivated solutions to the paradox which have been proposed on behalf of semantic anti-realism—namely, the intuitionistic solution. The solution consists, on the one hand, in accepting the intuitionistically valid part of Fitch’s reasoning while, on (...) the other hand, exploiting the characteristic weakness of intuitionistic logic in order to preserve the consistency of such acceptance with the denial of omniscience. It is first remarked how the solution still commits one to acceptance of modal claims which are unwarranted even by the lights of standard intuitionistic semantics. A novel form of the paradox is then introduced, which focuses on infallibility rather than omniscience and derives, from semantic anti-realism and a highly plausible constraint on knowledge, that every believed proposition is not untrue. Because of the logical form of this conclusion, an analogue of the intuitionistic solution for the novel form of the paradox would require drawing the characteristic intuitionistic distinctions wrt decidable propositions, which cannot be done. Semantic anti-realism still intuitionistically entails the unacceptable conclusion that every believed (decidable) proposition is true. (shrink)

In an earlier paper by the author, Åqvist , I presented an approach to the logic of historical necessity, or inevitability, in the sense of a “two-dimensional” combination of tense and modal logic for worlds, or histories, with the same time order, known as T × W logic. Distinctive features of that approach were, apart from its two-dimensionality, its being based on discrete and finite time, and its use of so-called systematic frame constants in order to enable us to indicate (...) longitudes and latitudes of any points in the co-ordinate systems under consideration. This led us to study and axiomatize an infinite hierarchy HTWxy of two-dimensional modal tense logics with the characteristic operators for historical necessity and possibility added to the original basic vocabulary. The main purpose of the present paper is then twofold: to extend the logics HTWxy to the interesting branch of philosophical logic constituted by deontic logic as combined with tense logic; and to deal with a curious puzzle known as the so-called epistemic obligation paradox – a well known stumbling-block in this area of research in philosophical logic. We argue for a solution to both these problems, which appeals to a new infinite hierarchy DHTWxym of extensions of the HTWxy in the sense of logics combining dyadic deontic modalities with temporal ones such as those for historical necessity and other two-dimensional modalities. (shrink)

Moore's paradox pits our intuitions about semantic oddnessagainst the concept of truth-functional consistency. Most solutions tothe problem proceed by explaining away our intuitions. But``consistency'' is a theory-laden concept, having different contours indifferent semantic theories. Truth-functional consistency is appropriateonly if the semantic theory we are using identifies meaning withtruth-conditions. I argue that such a framework is not appropriate whenit comes to analzying epistemic modality. I show that a theory whichaccounts for a wide variety of semantic data (...) about epistemic modals(Update Semantics) buys us a solution to Moore's paradox as a corollary.It turns out that Moorean propositions, when looked at through the lenseof an appropriate semantic theory, are inconsistent after all. (shrink)

This paper deals with the simple paradoxes of validity and with the possibility of solving them in terms of Bradwardinian-Buridanian semantics. The paradoxes of validity as conceived here are cases of semantic pathology, which result due to the use of terms signifying the validity of inference. Semantic paradoxes are a semantico-epistemological phenomenon which is a symptom of the need to revise several apparently acceptable semantic assumptions. The analysis of possible solutions to the (...) class='Hi'>paradoxes focuses on Bradwardinian-Buridanian semantics and as a result on the closed, token-based semantic theories that assume the existence of an implicit meaning of propositions. The key theses, as far as the solution to the paradoxes is concerned, are the principle of truth-implication which claims that every proposition expresses or implies its own truth and the closure principle which claims that every proposition asserts or expresses everything that follows from it logically. The present paper advances on recent research in claiming that (with certain reservations) the application of these principles can effectively solve inconsistency-paradoxes but not indeterminacy-paradoxes of validity. (shrink)

This thesis comprises three main chapters—each comprising one relatively standalone paper. The unifying theme is fragmentalism about truth, which is the view that the predicate “true” either expresses distinct concepts or expresses distinct properties. -/- In Chapter 1, I provide a formal development of alethic pluralism. Pluralism is the view that there are distinct truth properties associated with distinct domains of subject matter, where a truth property satisfies certain truth-characterizing principles. On behalf of pluralists, I propose an account of logic (...) and semantics that shows how they can answer central conceptual and logical challenges for their view. -/- In Chapter 2, I motivate and develop a modal account of propositions on the basis of an iterative conception of propositions, where the modality is logico-mathematical. The modal account of propositions takes the conception to motivate an inherently potential hierarchy of propositions. I show that the account helps provide satisfying solutions to the intensional paradoxes of Russell-Myhill, Kaplan, and Prior. -/- In Chapter 3, I propose that “true” is polysemous. I suggest that “true” is initially polysemous between correspondence truth and disquotational truth, and further polysemous between the meanings corresponding to the subconcepts of the concept truth generated by the indefinite extensibility of that concept. I show that the proposal provides satisfying solutions to the semantic paradoxes. (shrink)

The paper presents both new data and a new analysis of the semantic and pragmatic properties of the class of absolute scalar adjectives within an extension of a well-known logical framework for the analysis of gradable predicates: the delineation semantics framework . It has been long observed that the context-sensitivity, vagueness and gradability features of absolute scalar predicates give rise to certain puzzles for their analysis within most, if not all, modern formal semantic frameworks. While there exist proposals (...) for solving these puzzles within other major frameworks , it has been argued that some of their aspects are particularly challenging for the analysis of absolute scalar predicates within the delineation approach. By combining insights into the relationship between context-sensitivity and scalarity from the DelS framework with insights into the relationship between tolerance/similarity relations and the Sorites paradox from Cobreros et al.’s Tolerant, Classical, Strict framework, I propose a new logical system, called Delineation TCS , in which to set analyses of four classes of adjectival predicates. I argue that this new framework allows for an analysis of absolute scalar adjectives that answers these challenges for delineation-based frameworks, while still preserving the heart of the Klein-ian approach. (shrink)

I use the principle of truth-maker maximalism to provide a new solution to the semantic paradoxes. According to the solution, AUS, its undecidable whether paradoxical sentences are grounded or ungrounded. From this it follows that their alethic status is undecidable. We cannot assert, in principle, whether paradoxical sentences are true, false, either true or false, neither true nor false, both true and false, and so on. AUS involves no ad hoc modification of logic, denial of the T-schema's validity, (...) or obvious revenge. (shrink)

This article presents an empirical examination of the consequences of the virtual entailment principle proposed by Jean Buridan to resolve the Liar paradox. This principle states that every sentence in natural language implicitly asserts its own truth. Adopting this principle means that the Liar sentence is not paradoxical but false, because its content is contradictory to what is virtually implied. As a result, humans should perceive the Liar sentence the same way as any other false sentence. This solution to the (...) Liar paradox received criticism for making ad hoc claims about the natural language. However, thanks to modern advancements in psychophysiology, it became possible to empirically investigate if the human brain really perceives the Liar sentence like a false sentence. We designed and conducted an experiment to examine brain activity in response to true sentences, false sentences and self-referential sentences. Our results provide support for the Buridan’s hypothesis and show that the Liar sentence is processed by the human brain identically to false sentences, while the Truthteller sentence is perceived identically to true sentences. This agrees with predictions derived from the virtual entailment principle and supports the idea that humans think with the logic of truth—a logic for which the truth is a designated value of its adequate semantics. (shrink)

We introduce a framework for a graph-theoretic analysis of the semantic paradoxes. Similar frameworks have been recently developed for infinitary propositional languages by Cook and Rabern, Rabern, and Macauley. Our focus, however, will be on the language of first-order arithmetic augmented with a primitive truth predicate. Using Leitgeb’s notion of semantic dependence, we assign reference graphs (rfgs) to the sentences of this language and define a notion of paradoxicality in terms of acceptable decorations of rfgs with truth (...) values. It is shown that this notion of paradoxicality coincides with that of Kripke. In order to track down the structural components of an rfg that are responsible for paradoxicality, we show that any decoration can be obtained in a three-stage process: first, the rfg is unfolded into a tree, second, the tree is decorated with truth values (yielding a dependence tree in the sense of Yablo), and third, the decorated tree is re-collapsed onto the rfg. We show that paradoxicality enters the picture only at stage three. Due to this we can isolate two basic patterns necessary for paradoxicality. Moreover, we conjecture a solution to the characterization problem for dangerous rfgs that amounts to the claim that basically the Liar- and the Yablo graph are the only paradoxical rfgs. Furthermore, we develop signed rfgs that allow us to distinguish between ‘positive’ and ‘negative’ reference and obtain more fine-grained versions of our results for unsigned rfgs. (shrink)

This article discusses the logic of the Epimenides Paradox using a graphical representation of the logical operators involved. The graphical representation of the Logical Operators is the one used in digital electronics. This method of investigation is very effective in revealing the hidden error in the reasoning that leads to the contradiction. The article explains the vicious circle that the mind goes through when trying to analyze the Paradox. Finally, the article proposes a solution of the Paradox with final conclusions (...) that are not paradoxical but simply counterintuitive. The endless deduction is generated by subsequent evaluations that take place on different levels. In its evaluation the Mind alternates a level of logical evaluation with a level of semantic evaluation. (shrink)

The paradoxes of self-reference are genuinely paradoxical. The liar paradox, Russell’s paradox and their cousins pose enormous difficulties to anyone who seeks to give a comprehensive theory of semantics, or of sets, or of any other domain which allows a modicum of self-reference and a modest number of logical principles. One approach to the paradoxes of self-reference takes these paradoxes as motivating a non-classical theory of logical consequence. Similar logical principles are used in each of the paradoxical (...) inferences. If one or other of these problematic inferences are rejected, we may arrive at a consistent (or at least, a coherent) theory. In this paper I will show that such approaches come at a serious cost. The general approach of using the paradoxes to restrict the class of allowable inferences places severe constraints on the domain of possible propositional logics, and on the kind of metatheory that is appropriate in the study of logic itself. Proof-theoretic and model-theoretic analyses of logical consequence make provide different ways for non-classical responses to the paradoxes to be defeated by revenge problems: the redefinition of logical connectives thought to be ruled out on logical grounds. Non-classical solutions are not the “easy way out” of the paradoxes. (shrink)

It is shown that Russell's Paradox can be solved without advocating the Theory of Types, and also that the Liar's Paradox can be solved in much the same way. Neither solution requires that any of our commonsense-based beliefs be revised, let alone jettisoned. It is also shown that the Theory of Types is false.

Kevin Scharp argues that the concept of truth is defective, and is therefore unable to play its intended role in natural language truth-conditional semantics. As such, for this theoretical purpose, Scharp constructs two replacements: ascending truth and descending truth. Scharp applies the resultant theory, AD semantics, to the liar sentence, thereby obtaining a novel solution to the liar paradox. The aim of the present paper is fourfold. First, I show that, contrary to Scharp’s claims, AD semantics in fact yields an (...) inconsistency when applied to standard liar sentences. Second, I diagnose the problem: AD semantics mishandles negation. I propose an alternative treatment, resulting in what I call AD* semantics. Third, I show that AD* semantics gives Scharp the resources required to respond to an alleged revenge paradox that has been raised against his view. Finally, I argue that, these consequences notwithstanding, it remains unclear whether AD* semantics provides an adequate account of alethic paradoxes more generally. (shrink)

Semantic paradoxes pose a real threat to logics that attempt to be capable of expressing their own semantic concepts. Particularly, Curry paradoxes seem to show that many solutions must change our intuitive concepts of truth or validity or impose limits on certain inferences that are intuitively valid. In this way, the logic of a universal language would have serious problems. In this paper, we explore a different solution that tries to avoid both limitations as much (...) as possible. Thus, we argue that it is possible to capture the naive concepts of truth and validity without losing any of the valid inferences of classical logic. This approach is called the Buenos Aires plan. We present the logic of truth and validity, $$\mathsf {STTV}_{\omega }$$ based on the hierarchy of logics $$\textsf{ST}_{\omega }$$, whose validity predicate has the same semantic conditions as the material conditional. We argue that $$\mathsf {STTV}_{\omega }$$ is capable of blocking the problematic results while keeping the deductive power of classical logic as much as possible and offering an adequate semantic theory. On the other hand, one could object that it is not possible to reason with $$\mathsf {STTV}_{\omega }$$ because it is not closed under its logical principles. We respond to this objection and argue that the local characterization of validity shows how to make inferences using the logic $$\textsf{ST}_{\omega }$$. (shrink)

A requirement on any theory of vagueness is that it solve the sorites paradox. It is generally agreed that there are two aspects to such a solution: one task is to locate the error in the sorites argument; the second task is to explain why the sorites reasoning is a paradox rather than a simple mistake. I argue for a further constraint on approaches to the second task: they should conform to the standard modus operandi in formal semantics, in which (...) the semantic theory one develops is taken to be implicit in the ordinary usage of competent speakers. Thus it should not turn out that one's explanation of why ordinary speakers react to the sorites reasoning in the way they do depends on speakers not thinking that the semantics of vague language is governed by the theory one is advocating. I then argue that, out of the current main contenders for a theory of vagueness, only theories that posit degrees of truth can meet this further constraint. (shrink)

This paper argues that we should solve paradoxes for propositions (such as the Russell–Myhill paradox) in essentially the same way that we solve Russellian paradoxes for sets. That is, the standard, iterative approach to sets is extended to include properties, and then the resulting hierarchy of sets and properties is used to construct propositions. Propositions on this account are structured in the sense of mirroring the sentences that express them, and they would seem to serve the needs (...) of philosophers of language and metaphysicians who rely on such propositions. The resulting account has limitations, comparable to those faced by the iterative approach to sets, but it is argued to be in important ways preferable to others that have been proposed. (shrink)

A semantic solution to the liar paradox (“This statement is not true”) is presented in this article. Since the liar paradox seems to evince a contradiction, the principle of non-contradiction is preliminarily discussed, in order to determine whether dismissing this principle may be reason enough to stop considering the liar paradox a problem. No conclusive outcome with respect to the value of this principle is aspired to here, so that the inquiry is not concluded at this point and the (...) option to explore an alternative, semantic, solution remains open. This proposed solution is focused on what the liar paradox expresses and what it fails to express. (shrink)

An inference is standardly said to be sound just in case it is deductively valid and it has only true assumptions. The importance of a coherent concept of soundness to proof theory is obvious, in that it is only sound derivations, and not merely deductively valid arguments, that advance knowledge by providing proofs of theorems in logic and mathematics. The soundness paradox is informally albeit impredicatively formulated as argument : Argument is unsound, therefore, argument is unsound. This paper introduces and (...) explains the importance of the soundness paradox, formally demonstrates how to avoid superficial impredication via Gödelization, and compares it with the similar but significantly different liar and validity or Pseudo-Scotus paradoxes. Although there are similarities in this family of semantic diagonalizations, the soundness paradox is not just a hybrid of the liar and validity paradoxes, but is more fundamental, belonging to a special category that resists the most powerful received solutions to the liar and validity paradoxes. (shrink)

The liar paradox in its simplest form is the following argument. Consider the sentence 'this sentence is false'; call that the "liar sentence". Suppose the liar sentence is true. Then, since it says it is false, the liar sentence is false. So our supposition that it is true was mistaken, and the liar sentence must be false. But that's precisely what the liar sentence says, so it is true after all. The liar sentence is, therefore, both true and false---an absurd (...) result. ;Hans Herzberger has argued that several solutions to the Grelling paradox exhibit a common pattern: the very devices we introduce in order to avoid one version of the paradox are integral components of new versions of the paradox. Herzberger suggests that this is typical of semantic paradox in general. Chapters 1, 2, and 3 of this dissertation examine several possible solutions to the liar---especially appeals to language levels and to contexts---and attempt to locate these possible solutions in Herzberger's pattern. ;There is some reason to think that this pattern is inescapable---that is, that the liar paradox is ultimately unsolvable---so the remaining two chapters of this dissertation explore what follows from the claim that it is unsolvable. Chapter 4 develops an argument from the claim that the liar is unsolvable to the claim that we cannot understand ourselves. Chapter 5, then, attempts to make sense of this circumstance by invoking a Nietzschean view of thinking and truth---perspectivalism, which holds that all thinking concerns interpretation and that there are no uninterpreted facts. Given some perspectivalist assumptions, we can see how it is inevitable that we will be unable to understand ourselves---and how we can nevetherless attain a certain limited kind of understanding. Perspectivalism also recommends a certain intellectual humility: if everything is interpretation, then no view of mine can be a comprehensive and accurate view of the way the world really is. This by itself is not a surprising claim; the surprising claim is that the liar paradox is the basis of it. (shrink)

In this paper I shall present three solutions to the Knower Paradox which, despite important points in common, differ in several respects. The first solution, proposed by C. A. Anderson [1] is a hierarchical solution, developed in the framework of first-order arithmetic. However I will try toshow that this solution is based on an incorrect argument. The second solution, inspired by a book of R.M. Smullyan [14], is developed in the framework of modal logic and it is based on (...) the idea of interpreting one of the basic systems of the modal logic of provability in an epistemic way. Ishall give arguments in support of this solution. The third solution, proposed by P. Egrèé [8] attempts to connect the first and the second solutions. I will show that this attempt fails for philosophical and formal reasons. (shrink)

In Beyond the Limits of Thought [2002], Graham Priest argues that logical and semantic paradoxes have the same underlying structure (which he calls the Inclosure Schema ). He also argues that, in conjunction with the Principle of Uniform Solution (same kind of paradox, same kind of solution), this is sufficient to 'sink virtually all orthodox solutions to the paradoxes', because the orthodox solutions to the paradoxes are not uniform. I argue that Priest fails to (...) provide a non-question-begging method to 'sink virtually all orthodox solutions', and that the Inclosure Schema cannot be the structure that underlies the Liar paradox. Moreover, Ramsey was right in thinking that logical and semantic paradoxes are paradoxes of different kinds. (shrink)

This paper provides a philosophical analysis and a new edition of an anonymous Epitome of John Dumbleton’s solution to the semantic paradoxes. The first part of this paper briefly presents Dumbleton’s cassationist solution to the semantic paradoxes, which the English philosopher proposes in his Summa Logicae, written in the 1330s–40s. The second part investigates the solution to various types of insolubles proposed by the anonymous author of the Epitome. The third part provides a new critical edition (...) of the Latin text – a first edition was edited by Bottin in 1978 – and an English translation. (shrink)

The Liar paradox arises when we consider a sentence that says of itself that it is not true. If such self-referential sentences exist? and examples like?This sentence is not true? certainly suggest this?, then our logic and standard notion of truth allow to infer a contradiction: The Liar sentence is true and not true. What has gone wrong? Must we revise our notion of truth and our logic? Or can we dispel the common conviction that there are such self-referential sentences? (...) The present study explores the second path.0After comparing the Liar reasoning in formal and informal logic and showing that there are no Gödelian Liar sentences, the study moves on from the semantics of self-reference to the metaphysics of expressions and proposes a novel solution to the Liar paradox: Meaningful expressions are distinct from their syntactic bases and exist only relative to contexts. Detailed semantico-metaphysical arguments show that in this dynamic setting, an object can be referred to only after it has started to exist. Hence the circular reference needed in the Liar paradox cannot occur, after all. As this solution is contextualist, it evades the expressibility problems of other proposals. (shrink)

Alexius Meinong's Gegenstandstheorie is subject to a formal semantic paradox. The theory of defective objects originally developed by Meinong in response to Ernst Mally's paradox about self-referential thought is rejected as a general solution to paradox in the object theory. The intentionality thesis is also refuted by the counter-example of the unapprehended mountain. It is argued that despite these difficulties, an object theory is required in order to make intuitively correct sense of ontological commitment. ;A version of Meinong's theory (...) is formalized. The logic has non-standard truth value semantics and an objectual interpretation of quantifiers for a domain of existent and non-existent objects. Theories of intentional, referential, and extensional identity are offered, together with two theories of definite description, lambda abstraction, eight systems of alethic modal logic, and formal metatheory. The logic avoids the incompleteness and undecidability limitations of Godel and Church, and all logical, semantic, and set theoretical paradoxes, without type theory or Tarskian hierarchy of metalanguages. It is naively complete and mechanically decidable. ;Wittgenstein's private language argument is refuted as an obstacle to defining private mental objects in the logic. The argument is shown to be valid, but unsound. The object theory logic may therefore provide a mixed or impure private language, and its domain may contain private mental objects as well as ordinary public and non-existent objects. This means that it constitutes an object theory logic of intention, and can be used to represent the logical structure of intentional and specifically phenomenological philosophical theories that require the intelligible designation of private mental objects. ;Appendices include Henkin-type consistency, completeness, and compactness proofs ; criticism of Kazimierz Twardowski's arguments for the distinction between content and object in phenomenological psychology; translation of and commentary on Mally's essay, "Uber die Unabhangigkeit der Gegenstande vom Denken"; a discussion of the distinction between nuclear and extra-nuclear properties, and the eliminability of the modal moment; and a philosophical application of the logic to a solution of the paradox of analysis. (shrink)

A syntactically correct number-specification may fail to specify any number due to underspecification. For similar reasons, although each sentence in the Yablo sequence is syntactically perfect, none yields a statement with any truth-value. As is true of all members of the Liar family, the sentences in the Yablo sequence are so constructed that the specification of their truth-conditions is vacuous; the Yablo sentences fail to yield statements. The ‘revenge’ problem is easily defused. The solution to the semantical paradoxes offered (...) here revives the mediaeval cassatio approach, one that largely disappeared due to its incomprehending rejection by influential contemporary writers such as William Shyreswood and Thomas Bradwardine. The diagnosis readily extends to the set-theoretic paradoxes. (shrink)

The knower paradox states that the statement ‘We know that this statement is false’ leads to inconsistency. This article presents a fresh look at this paradox and some well-known solutions from the literature. Paul Égré discusses three possible solutions that modal provability logic provides for the paradox by surveying and comparing three different provability interpretations of modality, originally described by Skyrms, Anderson, and Solovay. In this article, some background is explained to clarify Égré’s solutions, all three of (...) which hinge on intricacies of provability logic and its arithmetical interpretations. To check whether Égré’s solutions are satisfactory, we use the criteria for solutions to paradoxes defined by Susan Haack and we propose some refinements of them. This article aims to describe to what extent the knower paradox can be solved using provability logic and to what extent the solutions proposed in the literature satisfy Haack’s criteria. Finally, the article offers some reflections on the relation between knowledge, proof, and provability, as inspired by the knower paradox and its solutions. (shrink)

Thomas Bradwardine's solution to the semantic paradoxes, presented in his Insolubilia written in Oxford in the early 1320s, turns on two main principles: that a proposition is true only if things are wholly as it signifies; and that signification is closed under consequence. After exploring the background in Walter Burley's account of the signification of propositions, the question is considered of the extent to which Bradwardine's theory is compatible with the distribution of truth over conjunction, disjunction, negation and (...) the conditional. (shrink)

A historically popular response to the liar paradox (“this sentence is false”) is to say that the liar sentence is meaningless (or semantically defective, or malfunctions, or…). Unfortunately, like all other supposed solutions to the liar, this approach faces a revenge challenge. Consider the revenge liar sentence, “this sentence is either meaningless or false”. If it is true, then it is either meaningless or false, so not true. And if it is not true, then it can’t be either meaningless (...) or false, so it must be true. Either way, we are back in a paradox. This paper provides a detailed and exhaustive discussion of the options for responding to revenge on behalf of “meaningless” theories. Though I attempt to discuss all of the options fairly, I will ultimately opt for one specific response and discuss some of its challenges. Various technical and logical matters will be discussed throughout the paper, but my focus will be philosophical, throughout. My overall conclusion is that the “meaningless” strategy is at least as well off in the face of revenge as any other approach to the liar and related paradoxes. (shrink)

This book aims to provide a solution to the semantic paradoxes. It argues for a unified solution to the paradoxes generated by our concepts of denotation, predicate extension, and truth. The solution makes two main claims. The first is that our semantic expressions 'denotes', 'extension' and 'true' are context-sensitive. The second, inspired by a brief, tantalizing remark of Godel's, is that these expressions are significant everywhere except for certain singularities, in analogy with division by zero. A (...) formal theory of singularities is presented and applied to a wide variety of versions of the definability paradoxes, Russell's paradox, and the Liar paradox. Keith Simmons argues that the singularity theory satisfies the following desiderata: it recognizes that the proper setting of the semantic paradoxes is natural language, not regimented formal languages; it minimizes any revision to our semantic concepts; it respects as far as possible Tarski's intuition that natural languages are universal; it responds adequately to the threat of revenge paradoxes; and it preserves classical logic and semantics. Simmons draws out the consequences of the singularity theory for deflationary views of our semantic concepts, and concludes that if we accept the singularity theory, we must reject deflationism. (shrink)

Semantic and soritical paradoxes display remarkable family resemblances. For one thing, several non-classical logics have been independently applied to both kinds of paradoxes. For another, revenge paradoxes and higher-order vagueness—among the most serious problems targeting solutions to semantic and soritical paradoxes—exhibit a rather similar dynamics. Some authors have taken these facts to suggest that truth and vagueness require a unified logical framework, or perhaps that the truth predicate is itself vague. However, a common (...) core of semantic and soritical paradoxes has not been identified yet, and no explanation of their relationships has been provided. Here we aim at filling this lacuna, in the framework of many-valued logics. We provide a unified diagnosis of semantic and soritical paradoxes, identifying their source in a general form of indiscernibility. We then develop our diagnosis into a theory of paradoxicality, which formalizes both semantic and soritical paradoxes as arguments involving specific instances of our generalized indiscernibility principle, and correctly predicts which logics can non-trivially solve them. (shrink)

A logical argument known as Fitch’s Paradox of Knowability, starting from the assumption that every truth is knowable, leads to the consequence that every truth is also actually known. Then, given the ordinary fact that some true propositions are not actually known, it concludes, by modus tollens, that there are unknowable truths. The main literature on the topic has been focusing on the threat the argument poses to the so called semantic anti-realist theories, which aim to epistemically characterize the (...) notion of truth; according to those theories, every true proposition must be knowable. But the paradox seems to be a problem also for epistemology and philosophy of science: the conclusion of the paradox – the claim that there are unknowable truths – seems to seriously narrow our epistemic possibilities and to constitute a limit for knowledge. This fact contrasts with certain views in philosophy of science according to which every scientific truth is in principle knowable and, at least at an ideal level, a perfected, “all-embracing”, omniscient science is possible. The main strategies proposed in order to avoid the paradoxical conclusion, given their effectiveness, are able to address only semantic problems, not epistemological ones. However, recently Bernard Linsky (2008) proposed a solution to the paradox that seems to be effective also for the epistemological problems. In particular, he suggested a possible way to block the argument employing a type-distinction of knowledge. In the present paper, firstly, we introduce the paradox and the threat it represents for a certain views in epistemology and philosophy of science; secondly, we show Linsky’s solution; thirdly, we argue that this solution, in order to be effective, needs a certain kind of justification, and we suggest a way of justifying it in the scientific field; fourthly, we show that the effectiveness of our proposal depends on the degree of reductionism adopted in science: it is available only if we do not adopt a complete reductionism. (shrink)

One way in which we might approach the challenge posed by the Sorites Paradox is considering that Sorites-susceptible predicates have several candidate extensions, or several ways in which these expressions can be made precise. For example, a candidate extension for the predicate ‘is a baby’ is the set of humans of less than two years, but also the set of those less than two years and one second, and of those less than two years and two seconds. In this chapter (...) we present and discuss two theories for vague predicates based on this idea: supervaluationism and subvaluationism. The chapter is structured in three parts. The first presents the super- and subvaluationist theories: their similarities and differences in the semantics, the resulting logics with their most characteristic features. The second reviews the super- and subvaluationist solutions to the Sorites Paradox and provides discussion on several controversies surrounding these theories. The third part introduces proof procedures for s’valuationist logics. (shrink)

This chapter contains sections titled: Tarski's Theory of Truth Tarski's Solution to the Liar Paradox Tarski's Method of Defining Truth for Formalized Languages Tarskian Semantics Three Criticisms of Tarski's Theory Kripke's Solution to the Liar Paradox A Reinterpretation of Tarski's Theory Truth Beyond Logic.

William Newcomb and Robert Nozick have provided us with the following problem in rational decision-making. There are two boxes, A and B. A contains either a million dollars or nothing. B contains a thousand dollars. I come into the room in which we have the boxes, closed. I must make one of two choices. Either I open A and take whatever money is present, M or O, or I open both and take whatever money is present, M + T or (...) O + T. What is the rational choice for me to make?Nothing unusual is meant by “rational” in this context. We assume that a person making the choice desires as much money as possible. We assume that he will take into account all relevant available information as to what decision will lead to more money. If he believes that doing X will, or is likely to, lead to more money than doing not-X, then he will do X.In the above case, it would seem that the rational choice is both. T is there for the taking, whether or not I get M, and I might as well try for both. (shrink)

AbstractṢadr al-Dīn al-Dashtakī (d. 1498) has proposed a solution to the liar paradox according to which the liar sentence is a self-referential sentence in which the predicate ‘false’ is iterated. Discussing the conditions for the truth-aptness of the sentences with nested and iterated instances of the predicates ‘true’ and/or ‘false’, Dashtakī argued that the liar sentence is not truth-apt at all. In the tradition of Arabic logic, the central elements of Dashtakī's solution—the self-referentiality of the liar sentence and the implicit (...) iteration of the predicate ‘false’—were initially highlighted in two earlier solutions proposed by Naṣīr al-Dīn al-Ṭūsī (d. 1274) and Shams al-Dīn al-Samarqandī (d. 1322), respectively. Here I investigate all three solutions and show that Dashtakī's solution can be taken as a synthesis of the other two. None of these solutions seems to be convincing at the end of the day. Nevertheless, all of them include significant logical and philosophical insights. In particular, although Dashtakī's solution is not itself compelling, it is only a few steps away from a promising solution. The appendix to this paper includes translations of the relevant passages. (shrink)