We investigate what happens when ‘truth’ is replaced with ‘provability’ in Yablo’s paradox. By diagonalization, appropriate sequences of sentences can be constructed. Such sequences contain no sentence decided by the background consistent and sufficiently strong arithmetical theory. If the provability predicate satisfies the derivability conditions, each such sentence is provably equivalent to the consistency statement and to the Gödel sentence. Thus each two such sentences are provably equivalent to each other. The same holds for the arithmetization of the existential Yablo (...) paradox. We also look at a formulation which employs Rosser’s provability predicate. (shrink)
We investigate the properties of Yablo sentences and for- mulas in theories of truth. Questions concerning provability of Yablo sentences in various truth systems, their provable equivalence, and their equivalence to the statements of their own untruth are discussed and answered.
We investigate the properties of Yablo sentences and formulas in theories of truth. Questions concerning provability of Yablo sentences in various truth systems, their provable equivalence, and their equivalence to the statements of their own untruth are discussed and answered.
Charlie Pelling presents an impropriety paradox for the truth account of assertion. After solving his paradox I show that it is a version of the liar paradox. I then show that for any account of truth there is a strengthened liar-like paradox, and that for any solution to the strengthened liar paradox, there is a parallel solution to each of these "new" paradoxes.
The purpose of this note is to present a strong form of the liar paradox. It is strong because the logical resources needed to generate the paradox are weak, in each of two senses. First, few expressive resources required: conjunction, negation, and identity. In particular, this form of the liar does not need to make any use of the conditional. Second, few inferential resources are required. These are: (i) conjunction introduction; (ii) substitution of identicals; and (iii) the inference: From ¬(p (...) ∧ p), infer ¬ p. It is, interestingly enough, also essential to the argument that the ‘strong’ form of the diagonal lemma be used: the one that delivers a term λ such that we can prove: λ = ¬ T(⌈λ⌉); rather than just a sentence Λ for which we can prove: Λ ≡ ¬T(⌈Λ⌉). -/- The truth-theoretic principles used to generate the paradox are these: ¬(S ∧ T(⌈¬S⌉); and ¬(¬S ∧ ¬T(⌈¬S⌉). These are classically equivalent to the two directions of the T-scheme, but they are intuitively weaker. -/- The lesson I would like to draw is: There can be no consistent solution to the Liar paradox that does not involve abandoning truth-theoretic principles that should be every bit as dear to our hearts as the T-scheme. So we shall have to learn to live with the Liar, one way or another. (shrink)
The Liar paradox raises foundational questions about logic, language, and truth (and semantic notions in general). A simple Liar sentence like 'This sentence is false' appears to be both true and false if it is either true or false. For if the sentence is true, then what it says is the case; but what it says is that it is false, hence it must be false. On the other hand, if the statement is false, then it is true, since it (...) says (only) that it is false. -/- How, then, should we classify Liar sentences? Are they true or false? A natural suggestion would be that Liars are neither true nor false; that is, they fall into a category beyond truth and falsity. This solution might resolve the initial problem, but it beckons the Liar's revenge. A sentence that says of itself only that it is false or beyond truth and falsity will, in effect, bring back the initial problem. The Liar's revenge is a witness to the hydra-like nature of Liars: in dealing with one Liar you often bring about another. -/- JC Beall presents fourteen new essays and an extensive introduction, which examine the nature of the Liar paradox and its resistance to any attempt to solve it. Written by some of the world's leading experts in the field, the papers in this volume will be an important resource for those working in truth studies, philosophical logic, and philosophy of language, as well as those with an interest in formal semantics and metaphysics. (shrink)
Bringing together powerful new tools from set theory and the philosophy of language, this book proposes a solution to one of the few unresolved paradoxes from antiquity, the Paradox of the Liar. Treating truth as a property of propositions, not sentences, the authors model two distinct conceptions of propositions: one based on the standard notion used by Bertrand Russell, among others, and the other based on J.L. Austin's work on truth. Comparing these two accounts, the authors show that while the (...) Russellian conception of the relation between sentences, propositions, and truth is crucially flawed in limiting cases, the Austinian perspective has fruitful applications to the analysis of semantic paradox. In the course of their study of a language admitting circular reference and containing its own truth predicate, Barwise and Etchemendy also develop a wide range of model-theoretic techniques--based on a new set-theoretic tool, Peter Aczel's theory of hypersets--that open up new avenues in logical and formal semantics. (shrink)
The Liar Paradox is an argument that arrives at a contradiction by reasoning about a Liar Sentence. The classical Liar Sentence is the self-referential sentence “This sentence is false.”.
This book is about one of the most baffling of all paradoxes--the famous Liar paradox. Suppose we say: "We are lying now." Then if we are lying, we are telling the truth; and if we are telling the truth we are lying. This paradox is more than an intriguing puzzle, since it involves the concept of truth. Thus any coherent theory of truth must deal with the Liar. Keith Simmons discusses the solutions proposed by medieval philosophers and offers his own (...) solutions and in the process assesses other contemporary attempts to solve the paradox. Unlike such attempts, Simmons' "singularity" solution does not abandon classical semantics and does not appeal to the kind of hierarchical view found in Barwise's and Etchemendy's The Liar. Moreover, Simmons' solution resolves the vexing problem of semantic universality--the problem of whether there are semantic concepts beyond the expressive reach of a natural language such as English. (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 this book, Yaqub describes a simple conception of truth and shows that it yields a semantical theory that accommodates the whole range of our seemingly conflicting intuitions about truth. This conception takes the Tarskian biconditionals (such as "The sentence 'Johannes loved Clara' is true if and only if Johannes loved Clara") as correctly and completely defining the notion of truth. The semantical theory, which is called the revision theory, that emerges from this conception paints a metaphysical picture of truth (...) as a property whose applicability is given by a revision process rather than by a fixed extension. The main advantage of this revision process is its ability to explain why truth seems in many cases almost redundant, in others substantial, and yet in others paradoxical (as in the famous Liar). Yaub offers a comprehensive defense of the revision theory of truth by developing consistent and adequate formal semantics for languages in which all sorts of problematic sentences (Liar and company) can be constructed. Yaqub concludes by introducing a logic of truth that further demonstrates the adequacy of the revision theory. (shrink)
It is proved that Yablo’s paradox and the Liar paradox are equiparadoxical, in the sense that their paradoxicality is based upon exactly the same circularity condition—for any frame ${\mathcal{K}}$ , the following are equivalent: (1) Yablo’s sequence leads to a paradox in ${\mathcal{K}}$ ; (2) the Liar sentence leads to a paradox in ${\mathcal{K}}$ ; (3) ${\mathcal{K}}$ contains odd cycles. This result does not conflict with Yablo’s claim that his sequence is non-self-referential. Rather, it gives Yablo’s paradox a new significance: (...) his construction contributes a method by which we can eliminate the self-reference of a paradox without changing its circularity condition. (shrink)
Richard Heck has recently drawn attention on a new version of the Liar Paradox, one which relies on logical resources that are so weak as to suggest that it may not admit of any “truly satisfying, consistent solution”. I argue that this conclusion is too strong. Heck's Liar reduces to absurdity principles that are already rejected by consistent paracomplete theories of truth, such as Kripke's and Field's. Moreover, the new Liar gives us no reasons to think that (versions of) these (...) principles cannot be consistently retained once the structural rule of contraction is restricted. I suggest that revisionary logicians have independent reasons for restricting such a rule. (shrink)
The paper argues that the liar paradox teaches us these lessons about English. First, the paradox-yielding sentence is a sentence of English that is neither true nor false in English. Second, there is no English name for any such thing as a set of all and only true sentences of English. Third, ‘is true in English’ does not satisfy the axiom of comprehension.
Here is the liar paradox. We have a sentence, (L), which somehow says of itself that it is false. Suppose (L) is true. Then things are as (L) says they are. (For it would appear to be a mere platitude that if a sentence is true, then things are as the sentence says they are.) (L) says that (L) is false. So, (L) is false. Since the supposition that (L) is true leads to contradiction, we can assert that (L) is (...) false. But since this is just what (L) says, (L) is then true. (For it would appear to be a mere platitude that if things are as a given sentence says they are, the sentence is true.) So (L) is true. So (L) is both true and false. Contradiction. (shrink)
(Beall ed. The Revenge of the Liar, forthcoming from Oxford University Press) > The main presentation of my approach to the semantic paradoxes. I take them to show that understanding a natural language is sharing a cognitive relation to a logically false semantic theory with other speakers.
In recent years there has been a revitalised interest in non-classical solutions to the semantic paradoxes. In this paper I show that a number of logics are susceptible to a strengthened version of Curry's paradox. This can be adapted to provide a proof theoretic analysis of the omega-inconsistency in Lukasiewicz's continuum valued logic, allowing us to better evaluate which logics are suitable for a naïve truth theory. On this basis I identify two natural subsystems of Lukasiewicz logic which individually, but (...) not jointly, lack the problematic feature. (shrink)
The Gödelian symphony -- Foundations and paradoxes -- This sentence is false -- The liar and Gödel -- Language and metalanguage -- The axiomatic method or how to get the non-obvious out of the obvious -- Peano's axioms -- And the unsatisfied logicists, Frege and Russell -- Bits of set theory -- The abstraction principle -- Bytes of set theory -- Properties, relations, functions, that is, sets again -- Calculating, computing, enumerating, that is, the notion of algorithm -- Taking numbers (...) as sets of sets -- It's raining paradoxes -- Cantor's diagonal argument -- Self-reference and paradoxes -- Hilbert -- Strings of symbols -- In mathematics there is no ignorabimus -- Gödel on stage -- Our first encounter with the incompleteness theorem -- And some provisos -- Gödelization, or say it with numbers! -- TNT -- The arithmetical axioms of tnt and the standard model N -- The fundamental property of formal systems -- The Gödel numbering -- And the arithmetization of syntax -- Bits of recursive arithmetic -- Making algorithms precise -- Bits of recursion theory -- Church's thesis -- The recursiveness of predicates, sets, properties, and relations -- And how it is represented in typographical number theory -- Introspection and representation -- The representability of properties, relations, and functions -- And the Gödelian loop -- I am not provable -- Proof pairs -- The property of being a theorem of TNI (is not recursive!) -- Arithmetizing substitution -- How can a TNT sentence refer to itself? -- Fixed point -- Consistency and omega-consistency -- Proving G1 -- Rosser's proof -- The unprovability of consistency and the immediate consequences of G1 and -- G2 -- Technical interlude -- Immediate consequences of G1 and G2 -- Undecidable1 and undecidable 2 -- Essential incompleteness, or the syndicate of mathematicians -- Robinson arithmetic -- How general are Gödel's results? -- Bits of turing machine -- G1 and G2 in general -- Unexpected fish in the formal net -- Supernatural numbers -- The culpability of the induction scheme -- Bits of truth (not too much of it, though) -- The world after Gödel -- Bourgeois mathematicians! : the postmodern interpretations -- What is postmodernism? -- From Gödel to Lenin -- Is biblical proof decidable? -- Speaking of the totality -- Bourgeois teachers! -- (un)interesting bifurcations -- A footnote to Plato -- Explorers in the realm of numbers -- The essence of a life -- The philosophical prejudices of our times -- From Gödel to Tarski -- Human, too human -- Mathematical faith -- I'm not crazy! -- Qualified doubts -- From gentzen to the dialectica interpretation -- Mathematicians are people of faith -- Mind versus computer : Gödel and artificial intelligence -- Is mind (just) a program? -- Seeing the truth and going outside the system -- The basic mistake -- In the haze of the transfinite -- Know thyself : Socrates and the inexhaustibility of mathematics -- Gödel versus wittgenstein and the paraconsistent interpretation -- When geniuses meet -- The implausible Wittgenstein -- There is no metamathematics -- Proof and prose -- The single argument -- But how can arithmetic be inconsistent? -- The costs and benefits of making Wittgenstein plausible. (shrink)
We describe the earliest occurrences of the Liar Paradox in the Arabic tradition. e early Mutakallimūn claim the Liar Sentence is both true and false; they also associate the Liar with problems concerning plural subjects, which is somewhat puzzling. Abharī (1200-1265) ascribes an unsatisfiable truth condition to the Liar Sentence—as he puts it, its being true is the conjunction of its being true and false—and so concludes that the sentence is not true. Tūsī (1201-1274) argues that self-referential sentences, like the (...) Liar, are not truth-apt, and defends this claim by appealing to a correspondence theory of truth. Translations of the texts are provided as an appendix. (shrink)
Dialetheism is the view that some contradictions are true. This is a view which runs against orthodoxy in logic and metaphysics since Aristotle, and has implications for many of the core notions of philosophy. Doubt Truth to Be a Liar explores these implications for truth, rationality, negation, and the nature of logic, and develops further the defense of dialetheism first mounted in Priest's In Contradiction, a second edition of which is also available.
Cramer's Transactional Interpretation (TI) is applied to the ``Quantum Liar Experiment'' (QLE). It is shown how some apparently paradoxical features can be explained naturally, albeit nonlocally (since TI is an explicitly nonlocal interpretation). At the same time, it is proposed that in order to preserve the elegance and economy of the interpretation, it may be necessary to consider offer and confirmation waves as propagating in a ``higher space'' of possibilities.
Some fourteenth-century treatises on paradoxes of the liar family offer a promising starting-point for the formulation of full-fledged theories of truth with systematic relevance in their own right. In particular, Bradwardine's thesis that sentences typically say more than one thing gives rise to a quantificational approach to truth, and Buridan's theory of truth based on the notion of suppositio allows for remarkable metaphysical parsimony. Bradwardine's and Buridan's theories both have theoretical advantages, but fail to provide a satisfactory account of truth (...) because both are committed to the thesis, fatal for both, that every sentence signifies/implies its own truth. I close with remarks on Greg Restall's recent model-theoretic formalization of Bradwardine's theory of truth. (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 presents an approach to truth and the Liar paradox which combines elements of context dependence and hierarchy. This approach is developed formally, using the techniques of model theory in admissible sets. Special attention is paid to showing how starting with some ideas about context drawn from linguistics and philosophy of language, we can see the Liar sentence to be context dependent. Once this context dependence is properly understood, it is argued, a hierarchical structure emerges which is neither ad (...) hoc nor unnatural. (shrink)
A new solution to the liar paradox is developed using the insight that it is illegitimate to even suppose (let alone assert) that a liar sentence has a truth-status (true or not) on the grounds that supposing this sentence to be true/not-true essentially defeats the telos of supposition in a readily identifiable way. On that basis, the paradox is blocked by restricting the Rule of Assumptions in Gentzen-style presentations of the sequent-calculus. The lesson of the liar is that not all (...) assumptions are for free. One merit of this proposal is that it is free from the revenge problem. (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)
In defense of the minimalist conception of truth, Paul Horwich(2001) has recently argued that our acceptance of the instances of the schema,`the proposition that p is true if and only if p', suffices to explain our acceptanceof truth generalizations, that is, of general claims formulated using the truth predicate.In this paper, I consider the strategy Horwich develops for explaining our acceptance of truth generalizations. As I show, while perhaps workable on its own, the strategy is in conflictwith his response to (...) the liar paradox. Something must give. I consider and reject variousalternatives and emendations to the strategy. In order to resolve the conflict,I propose an alternative approach to the liar, one that supports Horwich's strategywhile leaving minimalism maximally uncompromised. (shrink)
This essay attempts to give substance to the claim that the liar''sparadox shows the truth predicate to be context sensitive. The aim ismodest: to provide an account of the truth predicate''s contextsensitivity (1) that derives from a more general understanding ofcontext sensitivity, (2) that does not depend upon a hierarchy ofpredicates and (3) that is able to address the liar''s paradox. Theconsequences of achieving this goal are not modest, though. Perhapssurprisingly, for reasons that will be discussed in the last section (...) ofthis essay, a natural account of the truth predicate''s contextsensitivity appears to lead naturally to a version of the correspondencetheory of truth according to which the truth predicate can be understoodas a relation holding between a sentence and a salient set of contexts.The plan of this essay is as follows. Section 1 contains a generalaccount of context sensitivity. The purpose of this section is toisolate certain features of context sensitivity and formal methods oftreating them, which we will then apply to the truth predicate. Section 2then outlines two minimal conditions to be satisfied by a truthpredicate. In Section 3, I present a version of the liar paradoxthat results from these conditions and the assumption that the truthpredicate is not context sensitive in the sense described in sectionone. Finally, in section four, I provide what appear to be naturalconsequences of a truth predicate''s context sensitivity. Section 4 isadmittedly speculative and points in the direction for future research. (shrink)
A solution to the Liar must do two things. First, it should say exactly which step in the Liar reasoning - the reasoning which leads to a contradiction - is invalid. Secondly, it should explains why this step is invalid.
The medieval philosopher Jean Buridan says that at one time, he favored a solution to Liar−type paradoxes that relied on the claim that "every proposition, by its very form, signifies or asserts itself to be true."1 (I shall refer to this as Buridan's view, though he came to reject it when he wrote his Sophismata , in which he reports the view.) C.S. Peirce also suggested something like this in response to the Liar, and in a classic discussion of Buridan, (...) Arthur Prior evinces great sympathy for the view (in contrast to his rejection of Buridan's official solution).2 But what exactly does it mean for an arbitrary proposition to assert itself to be true? And is it really a plausible view to hold that every proposition does assert itself to be true? (shrink)
A view often expressed is that to classify the liar sentence as neither true nor false is satisfactory for the simple liar but not for the strengthened liar. I argue that in fact it is equally unsatisfactory for both liars. I go on to discuss whether, nevertheless, Kripke''s theory of truth represents an advance on that of Tarski.
About twenty-five years ago, Charles Parsons published a paper that began by asking why we still discuss the Liar Paradox. Today, the question seems all the more apt. In the ensuing years we have seen not only Parsons’ work (1974), but seminal work of Saul Kripke (1975), and a huge number of other important papers. Too many to list. Surely, one of them must have solved it! In a way, most of them have. Most papers on the Liar Paradox offer (...) some explanation of the behavior of paradoxical sentences, and most also offer some extension for the predicate ‘true’ that they think is adequate, at least in some restricted setting. But if this is a solution, then the problem we face is far from a lack of solutions; rather, we have an overabundance of conflicting ones. Kripke’s work alone provides us with uncountably many different extensions for the truth predicate. Even if it so happens that one of these is a conclusive solution, we do not seem to know which one, or why. We should also ask, given that we are faced with many technically elegant but contradictory views, if they are really all addressing the same problem. What we lack is not solutions, but a way to compare and evaluate the many ones we have. (shrink)
In this paper we concentrate on the nature of the liar paradox asa cognitive entity; a consistently testable configuration of properties. We elaborate further on a quantum mechanical model (Aerts, Broekaert and Smets, 1999) that has been proposed to analyze the dynamics involved, and we focus on the interpretation and concomitant philosophical picture. Some conclusions we draw from our model favor an effective realistic interpretation of cognitive reality.
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)
The only passage from Aristotle's works that seemsto discuss the paradox of the liar is within chapter 25 of Sophistici Elenchi (180a34–b7). This passage raises several questions: Is it really about the paradox of the liar? If it is, is it addressing a strong version of the paradox or some weak strain of it? If it is addressing a strong version of the paradox, what solution does it propose? The conciseness of the passage does not enable one to answer these (...) questions beyond doubt, and commentators have offered very different replies. However, a reasonable case can be made for claiming, first, that the passage in question is about the paradox of the liar, second, that it addresses a strong version of the paradox, and, third, that it attempts to solve it by assuming that someone uttering 'I am speaking falsely' (or whatever sentence-type the paradox turns on) is neither speaking truly nor speaking falsely absolutely. (shrink)
An Omega Point Theory says that reality is making progress from some initial state to some final state. It moves from some Alpha Point (the initial state) to some Omega Point (the final state). The progress is an increase in some quality. For example, reality is making progress from the chaotic to the orderly; or it is making progress from the simple to the complex; or from the mindless to the mental; or from evil to good. Here we focus on (...) the Omega Point theory of Peirce. An Omega Point Theory is an evolutionary cosmology – it says that the universe is evolving from the Alpha Point to the Omega Point. Peirce refers to his evolutionary cosmology as a Cosmogonic Philosophy would suppose that in the beginning – infinitely remote – there was a chaos of unpersonalized feeling, which being without connection or regularity would properly be without existence. This feeling, sporting here and there in pure arbitrariness, would have started the germ of a generalizing tendency. Its other sportings would be evanescent, but this would have a growing virtue. Thus the tendency to habit would be started; and from this, with the other principles of evolution, all the regularities of the universe would be evolved. At any time, however, an element of pure chance survives and will remain until the world becomes an absolutely perfect, rational, and symmetrical system, in which mind is at last crystallized in the infinitely distant future. (Peirce, Collected Papers , 6.33). (shrink)
This paper develops a classical model for our ordinary use of the truth predicate (1) that is able to address the liar's paradox and (2) that satisfies a very strong version of deflationism. Since the model is a classical in the sense that it has no truth value gaps, the model is able to address Tarski's indictment of our ordinary use of the predicate as inconsistent. Moreover, since it is able to address the liar's paradox, it responds to arguments against (...) deflationism based upon that paradox alone. The model is based upon a notion of the complexity of propositions that a fixed set of speakers might express. A context-sensitive definition of the truth predicate is then provided based upon a class of possible worlds defined in terms of these speakers. Reasonable constraints on the memories and lifetimes of ordinary speakers are used to limit the set of propositions that they might express so that deflationist requirements are satisfied. (shrink)
Charles Sanders Peirce proposed two different solutions to the Liar Paradox. He proposed the first in 1865 and the second in 1869. However, no one has yet noted in the literature that Peirce rejected his 1869 solution in 1903. Peirce never explicitly proposed a third solution to the Liar Paradox. Nonetheless, I shall argue he developed the resources for a third and novel solution to the Liar Paradox.In what follows, I will first explain the Liar Paradox. Second, I will briefly (...) rehearse Peirce's 1865 solution and his reasons for rejecting it.1 Third, I will review his 1869 solution and his reasons for rejecting it in 1903. Lastly, I will propose a novel solution to the Liar Paradox by drawing upon Peirce's later .. (shrink)
In this paper, we do two things. First, we provide some support for adopting a version of the meaningless strategy with respect to the liar paradox, and, second, we extend that strategy, by providing, albeit tentatively, a solution to that paradox—one that is semantic , rather than logical.
The Stoic philosopher Chrysippus wrote extensively on the liar paradox, but unfortunately the extant testimony on his response to the paradox is meager and mainly hostile. Modern scholars, beginning with Alexander Rüstow in the first decade of the twentieth century, have attempted to reconstruct Chrysippus? solution. Rüstow argued that Chrysippus advanced a cassationist solution, that is, one in which sentences such as ?I am speaking falsely? do not express propositions. Two more recent scholars, Walter Cavini and Mario Mignucci, have rejected (...) Rüstow's thesis that Chrysippus used a cassationist approach. Each has proposed his own thesis about Chrysippus? solution. I argue that Rüstow's view is fundamentally correct, and that the cassationist thesis gains greater plausibility when viewed in light of a passage in Sextus Empiricus? Adversus mathematicos that the previous commentators have ignored, and when understood within the broader context of Stoic logical theory and philosophy of language. I close with a brief remark on the significance of Chrysippus? work for the modern debate on the semantic paradoxes. (shrink)
The solution John Buridan offers for the Paradox of the Liar has not been correctly placed within the framework of his philosophy of language. More precisely, there are two important points of the Buridanian philosophy of language that are crucial to the correct understanding of his solution to the Liar paradox that are either misrepresented or ignored in some important accounts of his theory. The first point is that the Aristotelian formula, ` propositio est vera quia qualitercumque significat in rebus (...) significatis ita est ', once amended, is a correct way to talk about the truth of a sentence. The second one is that he has a double indexing theory of truth: a sentence is true in a time about a time, and such times should be distinguished in the account of the truth-conditions of sentences. These two claims are connected in an important way: the Aristotelian formula indicates the time about which a sentence is true. Some interpreters of the Buridanian solution to the paradox, following the lead of Herzberger, have missed these points and have been led to postulate truth-values gaps, or surrogates of truth-value gaps, when there is nothing of this sort in his theory. I argue against this tradition of interpretation of Buridan and propose an interpretation of his solution to the Liar. (shrink)
In the Tarskian theory of truth, the strengthened liar sentence is a theorem. More generally, any formalized truth theory which proves the full, self-applicative scheme True(“f”) f will prove the strengthened liar sentence. (This scheme is sometimes called (T-Out).).
Zeno''s paradoxes of motion and the semantic paradoxes of the Liar have long been thought to have metaphorical affinities. There are, in fact, isomorphisms between variations of Zeno''s paradoxes and variations of the Liar paradox in infinite-valued logic. Representing these paradoxes in dynamical systems theory reveals fractal images and provides other geometric ways of visualizing and conceptualizing the paradoxes.
The formalism of abstracted quantum mechanics is applied in a model of the generalized Liar Paradox. Here, the Liar Paradox, a consistently testable configuration of logical truth properties, is considered a dynamic conceptual entity in the cognitive sphere (Aerts, Broekaert, & Smets, [Foundations of Science 1999, 4, 115–132; International Journal of Theoretical Physics, 2000, 38, 3231–3239]; Aerts and colleagues[Dialogue in Psychology, 1999, 10; Proceedings of Fundamental Approachs to Consciousness, Tokyo ’99; Mind in Interaction]. Basically, the intrinsic contextuality of the truth-value (...) of the Liar Paradox is appropriately covered by the abstracted quantum mechanical approach. The formal details of the model are explicited here for the generalized case. We prove the possibility of constructing a quantum model of the m-sentence generalizations of the Liar Paradox. This includes (i) the truth–falsehood state of the m-Liar Paradox can be represented by an embedded 2m-dimensional quantum vector in a (2m) m -dimensional complex Hilbert space, with cognitive interactions corresponding to projections, (ii) the construction of a continuous ‘time’ dynamics is possible: typical truth and falsehood value oscillations are described by Schrödinger evolution, (iii) Kirchoff and von Neumann axioms are satisfied by introduction of ‘truth-value by inference’ projectors, (iv) time invariance of unmeasured state. (shrink)
There are some seemingly small points to be made, first of all, about usemention confusions in Stephen Read’s paper ‘The Truth Schema and the Liar’. But underlying them is a grammatical point that has much wider repercussions. For it generates, on its own, a more straightforward way of understanding what gets people into a tangle with Liar and Strengthened Liar sentences, and that leads to a much fuller, critical assessment of the line of approach to these matters that Read derives (...) from Bradwardine. (shrink)
This paper is a contribution to the reconstruction of Tarski’s semantic background in the light of the ideas of his master, Stanislaw Lesniewski. Although in his 1933 monograph Tarski credits Lesniewski with crucial negative results on the semantics of natural language, the conceptual relationship between the two logicians has never been investigated in a thorough manner. This paper shows that it was not Tarski, but Lesniewski who first avowed the impossibility of giving a satisfactory theory of truth for ordinary language, (...) and the necessity of sanitation of the latter for scientific purposes. In an early article (1913) Lesniewski gave an interesting solution to the Liar Paradox, which, although different from Tarski’s in detail, is nevertheless important to Tarski’s semantic background. To illustrate this I give an analysis of Lesniewski’s solution and of some related aspects of Lesniewski’s later thought. (shrink)
An eleventh-century Greek text, in which a fourth-century patristic text is discussed, gives an outline of a solution to the Liar Paradox. The eleventh-century text is probably the first medieval treatment of the Liar. Long passages from both texts are translated in this article. The solution to the Liar Paradox, which they entail, is analysed and compared with the results of modern scholarship on several Latin solutions to this paradox. It is found to be a solution, which bears some analogies (...) to contemporary game semantics. Further, an overview of other Byzantine scholia on the Liar Paradox is provided. The findings and the originality of the discussed solution to the Liar Paradox suggest a change in the way in which Byzantine Logic is traditionally regarded in contemporary scholarship. (shrink)
In this paper, I examine a solution to the Liar paradox found in the work of Ockham, Burley, and Pseudo-Sherwood. I reject the accounts of this solution offered by modern commentators. I argue that this medieval line suggests a non-hierarchical solution to the Liar, according to which ?true? is analysed as an indexical term, and paradox is avoided by minimal restrictions on tokens of ?true?. In certain respects, this solution resembles the recent approaches of Charles Parsons and Tyler Burge; in (...) other respects, it is related to a suggestion of Gödel. But, as a whole, it suggests an original solution to the Liar paradox, quite unlike any current proposals. (shrink)
Abstract: A Liar would express a proposition that is true and not true. A Liar Paradox would, per impossibile, demonstrate the reality of a Liar. To resolve a Liar Paradox it is sufficient to make out of its demonstration a reductio of the existence of the proposition that would be true and not true, and to "explain away" the charm of the paradoxical contrary demonstration. Persuasive demonstrations of the Liar Paradox in this paper trade on allusive scope-ambiguities of English definite (...) descriptions, and can seem confirmed by symbolizations in a Fregean theory in which scopes of definite descriptions are determinate. Symbolizing instead in a Russellian description theory in which alternative scopes are possible reveals that however the scope-ambiguities of the demonstration are settled the result is unsound. (shrink)
A Liar sentence is a sentence that, paradoxically, we cannot evaluate for truth in accordance with classical logic and semantics without arriving at a contradiction. For example, consider L If we assume that L is true, then given that what L says is ‘L is false,’ it follows that L is false. On the other hand, if we assume that L is false, then given that what L says is ‘L is false,’ it follows that L is true. Thus, L (...) is an example of a Liar sentence. Several philosophers have proposed that the Liar paradox, and related paradoxes, can be solved by accepting the contradictions that these paradoxes seem to imply (including Priest 2006, Rescher and Brandom 1980). The theory that there are true .. (shrink)
We extend the ordinary logic of knowledge based on the operator K and the system of axioms S5 by adding a new operator U, standing for the agent utters , and certain axioms and a rule for U, forming thus a new system KU. The main advantage of KU is that we can express in it intentions of the speaker concerning the truth or falsehood of the claims he utters and analyze them logically. Specifically we can express in the new (...) language various notions of lying, as well as of telling the truth. Consequently, as long as lying or telling the truth about a fact is an intentional mode of the speaker, we can resolve the Liar paradox, or at least some of its variants, turning it into an ordinary (false or true) sentence. Also, using Kripke structures analogous to those employed by S. Kraus and D. Lehmann in [3] for modelling the logic of knowledge and belief, we offer a sound and complete semantics for KU. (shrink)
Two periods in the history of logic and philosophy are characterized notably by vivid interest in self-referential paradoxical sentences in general, and Liar sentences in particular: the later medieval period (roughly from the 12th to the 15th century) and the last 100 years. In this paper, I undertake a comparative taxonomy of these two traditions. I outline and discuss eight main approaches to Liar sentences in the medieval tradition, and compare them to the most influential modern approaches to such sentences. (...) I also emphasize the aspects of each tradition that find no counterpart in the other one. It is expected that such a comparison may point in new directions for future research on the paradoxes; indeed, the present analysis allows me to draw a few conclusions about the general nature of Liar sentences, and to identify aspects that would require further investigation. (shrink)
First, language and axioms of Church's paper 'Comparison of Russell's Resolution of the Semantical Antinomies with that of Tarski' are slightly modified and a version of the Liar paradox tentatively reconstructed. An obvious natural solution of the paradox leads to a hierarchy of truth predicates which is of a different kind from the one defined by Church: it depends on the enlargement of the semantical vocabulary and its levels do not differ in the ramified-type-theoretical sense. Second, two attempts are made (...) in order to justify the Russellian, and perhaps Churchian, idea that language should not be fragmented beyond what is required by type distinctions. After all, because of reducibility, which seems to allow a semantics without propositions, this comes out to be possible only at the cost of resorting to two disputable theses. (shrink)
I gratefully acknowledge and respond here to four reviews of my recent book, Cosmology from Alpha to Omega. Nancey Murphy stresses the importance of showing consistency between Christian theology and natural science through a detailed examination of my recent model of their creative interaction. She suggests how this model can be enhanced by adopting Alasdair MacIntyre's understanding of tradition in order to adjudicate between competing ways of incorporating science into a wider worldview. She urges the inclusion of ethics in my (...) model and predicts that this would successfully challenge the competing naturalist tradition in contemporary society. John F. Haught weighs the alternatives of viewing divine action as objective versus subjective and of divine action at one level in nature or at all levels. He asks whether physics is fundamental to nature, arguing instead that metaphysics should be considered as fundamental. Michael Ruse assesses occasional versus universal divine action, the problems raised to divine action when it is related to quantum mechanics, and the way these relations exacerbate the challenge of natural theodicy. As an alternative he suggests viewing God as outside time and acting through unbroken natural law. Willem B. Drees discusses my use of the bridge metaphor for the relation between theology and science, the implications when science is inspired by theology, the role of contingency and necessity in the anthropic principle/many-worlds debate, and the challenge of cosmology to eschatology with the ensuing problem of theodicy. (shrink)
The liar paradox is standardly supposed to arise from three conditions: classical bivalent truth value semantics, the Tarskian truth schema, and the formal constructability of a sentence that says of itself that it is not true. Standard solutions to the paradox, beginning most notably with Tarski, try to forestall the paradox by rejecting or weakening one or more of these three conditions. It is argued that all efforts to avoid the liar paradox by watering down any of the three assumptions (...) suffers serious disadvantages that are at least as undesirable as the liar paradox itself. Instead, a new solution is proposed that admits that if the liar sentence is true then it is false, in the first paradox dilemma horn, but denies that the liar sentence is true, but asserting instead that it is false, and refuting the second paradox dilemma horn according to which it is supposed to follow that if the liar sentence is false then it is true. The reasoning for the second paradox dilemma horn is flawed, in that is not only not supported by but actually contradicted by the Tarskian truth schema. We could only infer the second dilemma horn if it were to clasically follow from the assumption that the liar sentence is false, and from the three liar paradox conditions, that therefore it is false that the liar sentence is false. This entire sentence can be shown to be false on the basis of the standard truth schema, thus blocking the paradox. Alternative formulations of the liar sentence are discussed, and the formal proofs and counterproofs for the two paradox dilemma horns, are considered along with the further philosophical implications of maintaining a resolute stance that the liar sentence is simply false. (shrink)
This paper uses the resources of illocutionary logic to provide a new understanding of the Liar Paradox. In the system of illocutionary logic of the paper, denials are irreducible counterparts of assertions; denial does not in every case amount to the same as the assertion of the negation of the statement that is denied. Both a Liar statement, (a) Statement (a) is not true, and the statement which it negates can correctly be denied; neither can correctly be asserted. A Liar (...) statement, more precisely, an attempted Liar statement, fails to fulfill conditions essential to statements, but no linguistic rules are violated by the attempt. Ordinary language, our ordinary practice of using language, is not inconsistent or incoherent because of the Liar. We are committed to deny Liars, but not to accept or assert them. This understanding of the Liar Paradox and its sources cannot be fully accommodated in a conventional logical system, which fails to mark the distinction between sentences/statements and illocutionary acts of accepting, rejecting, and supposing statements. (shrink)
In his recent paper in History and Philosophy of Logic, John Kearns argues for a solution of the Liar paradox using an illocutionary logic (Kearns 2007 ). Paraconsistent approaches, especially dialetheism, which accepts the Liar as being both true and false, are rejected by Kearns as making no ?clear sense? (p. 51). In this critical note, I want to highlight some shortcomings of Kearns' approach that concern a general difficulty for supposed solutions to (semantic) antinomies like the Liar. It is (...) not controversial that there are languages which avoid the Liar. For example, the language which consists of the single sentence ?Benedict XVI was born in Germany? lacks the resources to talk about semantics at all and thus avoids the Liar. Similarly, more interesting languages such as the propositional calculus avoid the Liar by lacking the power to express semantic concepts or to quantify over propositions. Kearns also agrees with the dialetheist claim that natural languages are semantically closed (i.e. are able to talk about their sentences and the semantic concepts and distinctions they employ). Without semantic closure, the Liar would be no real problem for us (speakers of natural languages). But given the claim, the expressive power of natural languages may lead to the semantic antinomies. The dialetheist argues for his position by proposing a general hypothesis (cf. Bremer 2005 , pp. 27?28): ?(Dilemma) A linguistic framework that solves some antinomies and is able to express its linguistic resources is confronted with strengthened versions of the antinomies?. Thus, the dialetheist claims that either some semantic concepts used in a supposed solution to a semantic antinomy are inexpressible in the framework used (and so, in view of the claim, violate the aim of being a model of natural language), or else old antinomies are exchanged for new ones. One horn of the dilemma is having inexpressible semantic properties. The other is having strengthened versions of the antinomies, once all semantic properties used are expressible. This dilemma applies, I claim, to Kearns' approach as well. (shrink)
Each science has its own domain of investigation, but one and the same science can be formalized in different languages with different universes of discourse. The concept of the domain of a science and the concept of the universe of discourse of a formalization of a science are distinct, although they often coincide in extension. In order to analyse the presuppositions and implications of choices of domain and universe, this article discusses the treatment of omega arguments in three very different (...) formalizations of arithmetic. In Peano's formalization the domain is a restricted class of individuals, while the universe of discourse is the unrestricted class of all individuals. In Gödel's formalization the domain is a restricted class of individuals as in Peano's formalization, but the universe of discourse coincides with the domain. In Whitehead-Russell's formalization the domain is a class of logical notions in Tarski's sense, that are necessarily not individuals, whereas the universe of discourse is the unrestricted class of individuals as in Peano's formalization. The present approach emphasizes the viewpoint that the universe of discourse of a given discourse is important in determining which propositions are expressed by which sentences. (shrink)
Given an ideal $I$ , let $\mathbb{P}_{I}$ denote the forcing with $I$ -positive sets. We consider models of forcing axioms $MA(\Gamma)$ which also have a normal ideal $I$ with completeness $\omega_{2}$ such that $\mathbb{P}_{I}\in \Gamma$ . Using a bit more than a superhuge cardinal, we produce a model of PFA (proper forcing axiom) which has many ideals on $\omega_{2}$ whose associated forcings are proper; a similar phenomenon is also observed in the standard model of $MA^{+\omega_{1}}(\sigma\mbox{-closed})$ obtained from a supercompact cardinal. (...) Our model of PFA also exhibits weaker versions of ideal properties, which were shown by Foreman and Magidor to be inconsistent with PFA. Along the way, we also show (1) the diagonal reflection principle for internally club sets ( $\mathit{DRP}(IC_{\omega_{1}})$ ) introduced by the author in earlier work is equivalent to a natural weakening of “there is an ideal $I$ such that $\mathbb{P}_{I}$ is proper”; and (2) for many natural classes $\Gamma$ of posets, $MA^{+\omega_{1}}(\Gamma)$ is equivalent to an apparently stronger version which we call $MA^{+\operatorname{Diag}}(\Gamma)$. (shrink)
La construcción de un lenguaje formal en el que sea posible llevar a cabo fonnulaciones sobre la verdad de los enunciados deI propio lenguaje se ha revelado en extremo problemático, puesto que los llamados enunciados deI mentiroso conducen a paradojas. En su libro The Liar, Barwise y Etchemendy afirman haber solucionado el problema mediante su semántica russelliana y semantica austiniana. Sin embargo, en este articulo va a ser demostrado que la semántica russelliana fracasa en solucionar el problema por las mismas (...) razones que planteamientos clásicos suelen fracasar, y que la semantica austiniana fracasa totalmente puesto que esta semantica no contiene ningún predicado veritativo.Formal languages with truth predicates are seriously affected by paradoxes in the form of Liar sentences. In their best-seller The Liar, Barwise and Etchemendy achieved to convince a respectable part of the philosophical world that they have solved this problem by means of their Russellian- and Austinian semantics. The aim of this paper is to stop the rumour that the Liar paradox is solved. lt will be shown that Russellian semantics fails because of the same reasons classical approaches use to fail, and that Austinian semantics fails totally since it contains no truth predicate, i.e. in Austinian semantics it is generally impossible to express the truth or falsehood of a proposition. (shrink)
Da Costa's C systems are surveyed and motivated, and significant failings of the systems are indicated. Variations are then made on these systems in an attempt to surmount their defects and limitations. The main system to emerge from this effort, system $CC_{\omega}$ , is investigated in some detail, and "dual-intuitionistic" semantical analyses are developed for it and surrounding systems. These semantics are then adapted for the original C systems, first in a rather unilluminating relational fashion, subsequently in a more illuminating (...) way through the introduction of impossible situations where and and or change roles. Finally other attempts to break out of impasses for the original and expanded C systems, by going inside them, are looked at, and further research directions suggested. (shrink)
In this book Yaqub describes a simple conception of truth and shows that it yields a semantical theory that accommodates the whole range of our seemingly conflicting intuitions about truth. This conception takes the Tarskian biconditionals (such as "The sentence `Johannes loved Clara' is true if and only if Johannes loved Clara") as correctly and completely defining the notion of truth. The semantical theory, which is called the revision theory, that emerges from this conception paints a metaphysical picture of truth (...) as a property whose applicability is given by a revision process rather than by a fixed extension. The main advantage of this revision process is its ability to explain why truth seems in many cases almost redundant, in others substantial, and yet in others paradoxical (as in the famous Liar). Yaqub offers a comprehensive defense of the revision theory of truth by developing consistent and adequate formal semantics for languages in which all sorts of problematic sentences (Liar and company) can be constructed. He also offers a detailed critical exposition of the proposals of Herzberger, Gupta, and Belnap. Yaqub concludes by introducing a logic of truth that further demonstrates the adequacy of the revision theory. -/- The book starts with a basic and intuitive understanding of the notion of truth and ends with a complex logic of truth. This book will interest students of logic, truth theory, formal semantics, and philosophy of language. (shrink)
In this paper I respond to Jacquette’s criticisms, in (Jacquette, 2008), of my (Barker, 2008). In so doing, I argue that the Liar paradox is in fact a problem about the disquotational schema, and that nothing in Jacquette’s paper undermines this diagnosis.
John Barker, in two recent essays, raises a variety of intriguing criticisms to challenge my interpretation of the liar paradox and the type of solution I proposein ‘Denying the Liar’ and ‘Denying the Liar Reaffirmed.’ Barker continues to believe that I have misunderstood the logical structure of the liar sentence and itsexpression, and that as a result my solution misfires. I shall try to show that on the contrary my analysis is correct, and that Barker does not properly grasp what (...) mysolution to the liar paradox involves. Additionally, I argue that Barker makes fundamental errors in the explanation of liar sentence formulations in intensional contexts and in the classical metatheory he invokes to support his criticisms. (shrink)
forthcoming in American Philosophical Quarterly. We argue that it would seem to be a mistake to blame Liar-like paradox on certain features of the object language, since the effect can be created with very minimal object languages that contain none of the usual suspects (truth-like predicates, reference to their own truth-bearers, negation, etc.).
We study the reverse mathematics of pigeonhole principles for finite powers of the ordinal $\omega$ . Four natural formulations are presented, and their relative strengths are compared. In the analysis of the pigeonhole principle for $\omega^{2}$ , we uncover two weak variants of Ramsey’s theorem for pairs.
Both Origen and Basil of Caesarea report that some people saw Ps. 115,2 LXX – “ I said in my alarm, ' Every human being is a liar ' ” -- as an expression of the Liar Paradox and formulated a version of the paradox based upon it. But Ps. 115,2 is actually not susceptible to the Liar paradox, despite Origen and Basil believing it to be so. Not realizing this, both sought to undermine the possibility that Ps. 115,2 did (...) express the Liar paradox by offering a contextual exegesis, in which they argue that the speaker of the verse, David, can be considered a god, not a human being. (shrink)
In this article the author argues that the 'Liar' Paradox sentence: "This sentence is false" is neither true nor false because it does not express any proposition or "Satz" in the sense of Bernard Bolzano. The difficulty left open is that by a similar line of reasoning also the sentence "This sentence is true" would not express any proposition, yet it is sometimes taken to be true (on the strength of a theorem by Loewe).
For sentences $\phi$ of $L_{\omega_{1},\omega}$, we investigate the question of absoluteness of $\phi$ having models in uncountable cardinalities. We first observe that having a model in $\aleph_{1}$ is an absolute property, but having a model in $\aleph_{2}$ is not as it may depend on the validity of the continuum hypothesis. We then consider the generalized continuum hypothesis (GCH) context and provide sentences for any $\alpha\in\omega_{1}\setminus\{0,1,\omega\}$ for which the existence of a model in $\aleph_{\alpha}$ is nonabsolute (relative to large cardinal hypotheses). (...) Finally, we present a complete sentence for which model existence in $\aleph_{3}$ is nonabsolute. (shrink)
The paper "Partitions of Products" [DiPH] investigated the polarized partition relation $\begin{pmatrix}\omega\\\omega\\\omega\\\vdots\end{pmatrix} \rightarrow \begin{pmatrix}\alpha_1\\\alpha_1\\\alpha_2\\\vdots \end{pmatrix}$ The relation is consistent relative to an inaccessible cardinal if every α i is finite, but inconsistent if two are infinite. We show here that it consistent (relative to an inaccessible) for one to be infinite. Along the way, we prove an interesting proposition from ZFC concerning partitions of the finite subsets of ω.
There is a prevalent view against the disquotational and the minimal theories of truth, that the most sensible solution to the Liar—that is, the gappy solution—is not available to them. I would like to argue that, though this solution is unavailable to the two theories, the prevailing argument and the reasoning behind this view are wrong. This paper mainly focuses on Simmons’ “Deflationary Truth and the Liar” (1999), within which the idea that disquotationalism can take the Liar in its stridein (...) terms of the gappy option is thoroughly criticised. Albeit Simmons’ account is about disquotationalism, it is in fact about truth theories with the disquotational feature. For Horwich’s minimal theory of truth to be feasible, it is in need of providing an account of which the primary truth bearers are utterances or sentences. The reasoning behind Simmons’ account and his argument is a widely accepted but in my opinion mistaken reading of deflationary theories, reading the deflationary axiom schemata as emphasising the redundant feature of the true predicate only. By analysing and criticising this reasoning the mistakes of this interpretation of the deflationary theories of truth are revealed. Simmons bases his argument on two premises: taking disquotational theory of truth asdefinitional theory and considering the main feature of the disquotational truth predicate as eliminability. In terms of the notion of parasitic liar, I will argue that Simmons fails to show the plausibility of one crucial premise of his argument—that is, the paradoxical or the pathological feature is missing from the disquotational mirrors of the Liar. I will further show what deflationary feature is misunderstood by those accounts similar to Simmons’. (shrink)
This article examines the various Liar paradoxes and their near kin, Grelling’s paradox and Gödel’s Incompleteness Theorem with its self-referential Gödel sentence. It finds the family of paradoxes to be generated by circular definition–whether of statements, predicates, or sentences–a manoeuvre that generates the fatal disorders of the Liar syndrome: semantic vacuity, semantic incoherence, and predicative catalepsy. Afflicted statements, such as the self-referential Liar statement, fail to be genuine statements. Hence they say nothing, a point that invalidates the reasoning on which (...) the various paradoxes rest. The seeming plausibility of the paradoxes is due to the fact that the same sentence may be used to make both the pseudo-statement and a genuine statement about the pseudo-statement. Hence, if a formal system is to avoid ambiguity and consequent seeming paradox, it requires some sort of disambiguator to distinguish the two statements. Gödel’s Theorem presents a further complication in that the self-reference involved is sentential rather than statemental. Nevertheless, on the intended interpretation of the system as a formalization of arithmetic, the self-referential Gödel sentence can only be an ambiguous statement, one that is both a pseudo-statement and its genuine double. Consequently, the conclusions commonly drawn from Gödel’s theorem must be deemed unwarranted. Arithmetic might well be formalized in a proper system that either excludes circular definition or introduces disambiguators. (shrink)
The story goes that Epimenides, a Cretan, used to claim that all Cretans are always liars. Whether he knew it or not, this claim is odd. It is easy to see it is odd by asking if it is true or false. If it is true, then all Cretans, including Epimenides, are always liars, in which case what he said must be false. Thus, if what he says is true, it is false. Conversely, suppose what Epimenides said is false. Then (...) some Cretan at some time speaks truly. This might not tell us anything about Epimenides. But if, to make the story simple, he were the only Cretan ever to speak, and this was the only thing he ever said, then indeed, he would have to speak truly. And we would then have shown that if what he said was false, it must be true. (shrink)
I distinguish paradoxes and hypodoxes among the conundrums of time travel. I introduce ‘hypodoxes’ as a term for seemingly consistent conundrums that seem to be related to various paradoxes, as the Truth-teller is related to the Liar. In this article, I briefly compare paradoxes and hypodoxes of time travel with Liar paradoxes and Truth-teller hypodoxes. I also discuss Lewis’ treatment of time travel paradoxes, which I characterise as a Laissez Faire theory of time travel. Time travel paradoxes are impossible according (...) to Laissez Faire theories, while it seems hypodoxes are possible. (shrink)
There are many reasons why one might be tempted to reject certain instances of the law of excluded middle. And it is initially natural to take ‘reject’ to mean ‘deny’, that is, ‘assert the negation of’. But if we assert the negation of a disjunction, we certainly ought to assert the negation of each disjunct (since the disjunction is weaker1 than the disjuncts). So asserting..
The first sentence in this essay is a lie. There is something odd about saying so, as has been known since ancient times. To see why, remember that all lies are untrue. Is the first sentence true? If it is, then it is a lie, and so it is not true. Conversely, suppose that it is not true. As we (viz., the authors) have said it, presumably with the intention of you believing it when it is not true, it is (...) a lie. But then it is true! (shrink)
Badici [2008] criticizes views of Priest [2002] concerning the Inclosure Schema and the paradoxes of self-reference. This article explains why his criticisms are to be rejected.
In this paper a logic for reasoning disquotationally about truth is presented and shown to have a standard model. This work improves on Hartry Field's recent results establishing consistency and omega-consistency of truth-theories with strong conditional logics. A novel method utilising the Banach fixed point theorem for contracting functions on complete metric spaces is invoked, and the resulting logic is shown to validate a number of principles which existing revision theoretic methods have heretofore failed to provide.
Argument that Q∃ expresses more than one proposition: (1) Q∃ expresses the proposition that Q∃ expresses some proposition that isn’t true. ((E)) (2) If Q ∃ expresses only true propositions, then the proposition that Q ∃ expresses some proposition that isn’t true is true. ((1)) (3) If Q∃ expresses only true propositions, then some proposition expressed by Q∃ is not true. (2, T) (4) Some proposition expressed by Q ∃ is not true. ((3)) (5) The proposition that Q ∃ expresses (...) some proposition that isn’t true is true. (4, T) (6) Q∃ expresses at least one true proposition. (1,5) (7) Q∃ expresses at least two propositions. (3, 6) (A parallel argument shows that Q∀ expresses both true and false propositions. (shrink)
In this essay (for undergraduates) I introduce three of the famous semantic paradoxes: the Liar, Grelling’s, and the No-No. Collectively, they seem to show that the notion of truth is highly paradoxical, perhaps even contradictory. They seem to show that the concept of truth is a bit akin to the concept of a married bachelor—it just makes no sense at all. But in order to really understand those paradoxes one needs to be very comfortable thinking about how lots of interesting (...) sentences talk about not dogs or cats or elections or baseball but sentences. That is, we need to get familiar analyzing sentences that talk about sentences. (shrink)
The Pinocchio paradox, devised by Veronique Eldridge-Smith in February 2001, is a counter-example to solutions to the Liar that restrict the use or definition of semantic predicates. Pinocchio’s nose grows if and only if what he is stating is false, and Pinocchio says ‘My nose is growing’. In this statement, ‘is growing’ has its normal meaning and is not a semantic predicate. If Pinocchio’s nose is growing it is because he is saying something false; otherwise, it is not growing. ‘Because’ (...) stands here for a non-semantic relation; it might be supposed to be causal or of some other nature, but it is not semantic. The paradox is discussed in relation to Tarski’s and Kripke’s theories of truth. Although the paradox is not necessarily a counter-example to a theory of a truth predicate, it is a problem for a theory of truth of the kind preserved by validity. (shrink)
This paper offers an analysis of a hitherto neglected text on insoluble propositions dating from the late XiVth century and puts it into perspective within the context of the contemporary debate concerning semantic paradoxes. The author of the text is the italian logician Peter of Mantua (d. 1399/1400). The treatise is relevant both from a theoretical and from a historical standpoint. By appealing to a distinction between two senses in which propositions are said to be true, it offers an unusual (...) solution to the paradox, but in a traditional spirit that contrasts a number of trends prevailing in the XiVth century. It also counts as a remarkable piece of evidence for the reconstruction of the reception of English logic in italy, as it is inspired by the views of John Wyclif. Three approaches addressing the Liar paradox (Albert of Saxony, William Heytesbury and a version of strong restrictionism) are first criticised by Peter of Mantua, before he presents his own alternative solution. The latter seems to have a prima facie intuitive justification, but is in fact acceptable only on a very restricted understanding, since its generalisation is subject to the so-called revenge problem. (shrink)
There is a certain approach to the semantic paradoxes that is highly intuitive and for that reason alone never seems to go away. Roughly put, it's the idea that the paradoxical sentences just don't really have any truth conditions at all, no matter how grammatically sound and meaningful they and their parts are. I suppose that just about anyone who spends even a relatively modest amount of time thinking about the paradoxes comes up with this idea eventually. There is a (...) great deal to recommend this approach, especially when it carefully distinguishes sentence tokens from sentence types. For one thing, it requires no significant alteration in commonsensical views about language or logic. Let us call it the Token Approach, as it trades on distinguishing linguistic tokens from types. The approach does not contain any of the flashy logical moves that characterize most other current responses to the semantic paradoxes. Many contemporary philosophers of language and logic ignore the Token Approach in part because, it seems, they cannot display their logical chops when investigating it. Despite this devastating drawback, the approach strikes me as good as any. -/- It faces two obstacles: it apparently lacks a plausible explanation of how certain type-identical sentence tokens can differ in truth conditions, and it may fail to adequately deal with certain paradoxical sentences of the liar family. However, I don't take the obstacles to be insurmountable: in each case the advocate of the Token Approach can appeal to a traditional and highly credentialed-if controversial and obscure-contemporary view of linguistic meaning that promises to supply suitable ways around both obstacles. (shrink)
There is no consensus as to whether a Liar sentence is meaningful or not. Still, a widespread conviction with respect to Liar sentences (and other ungrounded sentences) is that, whether or not they are meaningful, they are useless . The philosophical contribution of this paper is to put this conviction into question. Using the framework of assertoric semantics , which is a semantic valuation method for languages of self-referential truth that has been developed by the author, we show that certain (...) computational problems, called query structures , can be solved more efficiently by an agent who has self-referential resources (amongst which are Liar sentences) than by an agent who has only classical resources; we establish the computational power of self-referential truth . The paper concludes with some thoughts on the implications of the established result for deflationary accounts of truth. (shrink)
I propose a solution to the aletheic paradoxes on which truth predicates are assessment-sensitive. Truth is not an antecedently plausible topic for a semantic relativist treatment; nevertheless, the aletheic paradoxes give us good reason to think that truth is an inconsistent concept, and there are good reasons to think that semantic relativism is appropriate for inconsistent concepts, especially those that display what I call empirical inconsistency. Thus, I show that a promising version of the best approach to the paradoxes is (...) an application of semantic relativism to truth itself—arguing from results about the paradoxes and general considerations about language use to aletheic assessment-sensitivity. The paper is divided into three parts, the first on the aletheic paradoxes, the second on semantic relativism, and the third on assessment-sensitivity with respect to truth predicates. The first contains an overview of my preferred approach to the paradoxes, which entails that truth is an inconsistent concept that that should be replaced (for certain purposes) by a team of consistent concepts that can do its work without causing troubling paradoxes. The second part provides an overview of semantic relativism and its rivals. The third considers which treatment is most appropriate for inconsistent concepts in general and truth in particular. In it, I propose an assessment-sensitivity view of truth, discuss some prominent objections to semantic relativism, and review some issues that arise for approaches to the aletheic paradoxes. (shrink)
