Grelling’s Paradox

Grelling’s Paradox is the paradox which results from considering whether heterologicality, the word-property which a designator has when and only when the designator does not bear the word-property it designates, is had by ‘ ȁ8heterologicality’. Although there has been some philosophical debate over its solution, Grelling’s Paradox is nearly uniformly treated as a variant of either the Liar Paradox or Russell’s Paradox, a paradox which does not present any philosophical challenges not already presented by the two better known paradoxes. The aims of this paper are, first, to offer a precise formulation of Grelling’s Paradox which is clearly distinguished from both the Liar Paradox and Russell’s Paradox; second, to offer a solution to Grelling’s Paradox which both resolves the paradoxical reasoning and accounts for unproblematic predications of heterologicality; and, third, to argue that there are two lessons to be drawn from Grelling’s Paradox which have not yet been drawn from the Liar or Russell’s Paradox. The first lesson is that it is possible for the semantic content of a predicate to be sensitive to the semantic context; i.e., it is possible for a predicate to be an indexical expression. The second lesson is that the semantic content of an indexical predicate, though unproblematic for many cases, can nevertheless be problematic in some cases.     

Truthlikeness and the Lottery Paradox via the Preface Paradox

In a 2017 AJP paper, Cevolani and Schurz (C&S) propose a novel solution to the Preface Paradox that appeals to the notion of expected truthlikeness. This discussion note extends and analyses their approach by applying it to the related Lottery Paradox.     

The No-No Paradox Is a Paradox

The No-No Paradox consists of a pair of statements, each of which ‘says’ the other is false. Roy Sorensen claims that the No-No Paradox provides an example of a true statement that has no truthmaker: Given the relevant instances of the T-schema, one of the two statements comprising the ‘paradox’ must be true (and the other false), but symmetry constraints prevent us from determining which, and thus prevent there being a truthmaker grounding the relevant assignment of truth values. Sorensen's view is mistaken: situated within an appropriate background theory of truth, the statements comprising the No-No Paradox are genuinely paradoxical in the same sense as is the Liar (and thus, on Sorensen's view, must fail to have truth values). This result has consequences beyond Sorensen's semantic framework. In particular, the No-No Paradox, properly understood, is not only a new paradox, but also provides us with a new type of paradox, one which depends upon a general background theory of the truth predicate in a way that the Liar Paradox and similar constructions do not.     

Do constitutive norms on belief explain Moore’s Paradox?

Philosophical Studies - In this article I assess the prospects for a particular kind of resolution to Moore’s Paradox. It is that Moore’s Paradox is explained by the existence of a...     

The Byzantine Liar

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.     

Gwiazda on the Bayesian Argument for God

Jeremy Gwiazda made two criticisms of my formulation in terms of Bayes’s theorem of my probabilistic argument for the existence of God. The first criticism depends on his assumption that I claim that the intrinsic probabilities of all propositions depend almost entirely on their simplicity; however, my claim is that that holds only insofar as those propositions are explanatory hypotheses. The second criticism depends on a claim that the intrinsic probabilities of exclusive and exhaustive explanatory hypotheses of a phenomenon must sum to 1; however it is only those probabilities plus the intrinsic probability of the non-occurrence of the phenomenon which must sum to 1.     

Operators in the paradox of the knower

Predicates are term-to-sentence devices, and operators are sentence-to-sentence devices. What Kaplan and Montague's Paradox of the Knower demonstrates is that necessity and other modalities cannot be treated as predicates, consistent with arithmetic; they must be treated as operators instead. Such is the current wisdom.A number of previous pieces have challenged such a view by showing that a predicative treatment of modalities neednot raise the Paradox of the Knower. This paper attempts to challenge the current wisdom in another way as well: to show that mere appeal to modal operators in the sense of sentence-to-sentence devices is insufficient toescape the Paradox of the Knower. A family of systems is outlined in which closed formulae can encode other formulae and in which the diagonal lemma and Paradox of the Knower are thereby demonstrable for operators in this sense.I am deeply indebted to Robert F. Barnes and Evan W. Conyers, without whom these ideas might not have germinated and certainly would not have grown. Many of the results offered here evolved in the course of mutual discussion and correspondence. I am also grateful to an anonymous reviewer forSynthese for many very helpful suggestions.The current paper contains the technical results promised in Footnote 25 of Grim (1988) and Footnote 26, Chapter 3, of Grim (1991).     

Russell＇s Paradox of Predicates

Russell＇s letter to Frege of June 16, 1902 contains the famous paradox of the class of all classes which are not members of themselves as well as a second paradox of the predicates that cannot be predicated of themselves. The latter paradox arises out of Russell＇s theory of classes and class concepts in Principles of Mathematics.     

Alethic fictionalism,alethic nihilism,and the Liar Paradox

Recently, several philosophers have proposed fictionalist accounts of truth-talk, as a means for resolving the semantic pathology that the Liar Paradox appears to present. These alethic fictionalists aim to vindicate truth-talk as a kind of as if discourse, while rejecting that the talk attributes any real property of truth. Liggins (Analysis 74:566–574, 2014) has recently critically assessed one such proposal, Beall’s (The law of non-contradiction: new philosophical essays. Oxford University Press, New York, pp 197–216, 2004) constructive methodological deflationist (henceforth, ‘CMD’), offering objections to Beall’s proposed alethic fictionalism that potentially generalize to other alethic fictionalist accounts. Liggins further argues that CMD supports a classically consistent response to the Liar Paradox—one that can be extracted from CMD, while leaving its putatively problematic fictionalist elements behind in favor of alethic nihilism. In this paper, after establishing that Liggins’s criticisms of CMD are off base, we show that the classical resolution of the Liar Paradox that he proposes is unworkable. Since his resistance to alethic fictionalism turns out to be unmotivated, we conclude that this approach is still worth considering as a framework for a resolution of the Liar Paradox.     

An implicit technology of generalization

Traditionally, discrimination has been understood as an active process, and a technology of its procedures has been developed and practiced extensively. Generalization, by contrast, has been considered the natural result of failing to practice a discrimination technology adequately, and thus has remained a passive concept almost devoid of a technology. But, generalization is equally deserving of an active conceptualization and technology. This review summarizes the structure of the generalization literature and its implicit embryonic technology, categorizing studies designed to assess or program generalization according to nine general headings: Train and Hope; Sequential Modification; Introduce to Natural Maintaining Contingencies; Train Sufficient Exemplars; Train Loosely; Use Indiscriminable Contingencies; Program Common Stimuli; Mediate Generalization; and Train "To Generalize".     

Perceptual identification thresholds for 150 fragmented pictures from the Snodgrass and Vanderwart picture set

This paper reports perceptual identification thresholds for 150 pictures from the 1980 Snodgrass and Vanderwart picture set. These pictures were fragmented and presented on the Apple Macintosh microcomputer in a picture-fragment completion task in which identification thresholds were obtained at three phases of learning: Train (initial presentation), New (initial presentation after training on a different set), and Old (repeated presentation of the Train set). Pictures were divided into five sets of two subsets of 15 pictures each, which served alternately as the Train and New sets. A total of 100 subjects participated in the task, with 10 subjects assigned to each subset. Individual thresholds for each picture at each phase of learning are presented, along with the fragmented pictures identified by 35% of the subjects across the Train and New learning phases. This set of fragmented pictures is provided for use in experiments in which a single level of fragmented image is presented for identification after a priming phase. Correlations between the Snodgrass and Vanderwart norms and identification thresholds at the three phases of learning are also reported.     

Curry’s Paradox,Generalized Modus Ponens Axiom and Depth Relevance

“Weak relevant model structures” (wr-ms) are defined on “weak relevant matrices” by generalizing Brady’s model structure ${\mathcal{M}_{\rm CL}}$ built upon Meyer’s Crystal matrix CL. It is shown how to falsify in any wr-ms the Generalized Modus Ponens axiom and similar schemes used to derive Curry’s Paradox. In the last section of the paper we discuss how to extend this method of falsification to more general schemes that could also be used in deriving Curry’s Paradox.     

New surprises for the Ramsey Test

In contemporary discussions of the Ramsey Test for conditionals, it is commonly held that (i) supposing the antecedent of a conditional is adopting a potential state of full belief, and (ii) Modus Ponens is a valid rule of inference. I argue on the basis of Thomason Conditionals (such as ‘If Sally is deceiving, I do not believe it’) and Moore’s Paradox that both claims are wrong. I then develop a double-indexed Update Semantics for conditionals which takes these two results into account while doing justice to the key intuitions underlying the Ramsey Test. The semantics is extended to cover some further phenomena, including the recent observation that epistemic modal operators give rise to something very like, but also very unlike, Moore’s Paradox.     

Probability,Approximate Truth,and Truthlikeness: More Ways out of the Preface Paradox

The so-called Preface Paradox seems to show that one can rationally believe two logically incompatible propositions. We address this puzzle, relying on the notions of truthlikeness and approximate truth as studied within the post-Popperian research programme on verisimilitude. In particular, we show that adequately combining probability, approximate truth, and truthlikeness leads to an explanation of how rational belief is possible in the face of the Preface Paradox. We argue that our account is superior to other solutions of the paradox, including a recent one advanced by Hannes Leitgeb (Analysis 74.1).     

The Solution to the Surprise Exam Paradox

The Surprise Exam Paradox continues to perplex and torment despite the many solutions that have been offered. This paper proposes to end the intrigue once and for all by refuting one of the central pillars of the Surprise Exam Paradox, the “No Friday Argument,” which concludes that an exam given on the last day of the testing period cannot be a surprise. This refutation consists of three arguments, all of which are borrowed from the literature: the “Unprojectible Announcement Argument,” the “Wright & Sudbury Argument,” and the “Epistemic Blindspot Argument.” The reason that the Surprise Exam Paradox has persisted this long is not because any of these arguments is problematic. On the contrary, each of them is correct. The reason that it has persisted so long is because each argument is only part of the solution. The correct solution requires all three of them to be combined together. Once they are, we may see exactly why the No Friday Argument fails and therefore why we have a solution to the Surprise Exam Paradox that should stick.     

The Timing of Conscious Experience: A Critical Review and Reinterpretation of Libet''s Research

An extended examination of Libet's works led to a comprehensive reinterpretation of his results. According to this reinterpretation, the Minimum Train Duration of electrical brain stimulation should be considered as the time needed to create a brain stimulus efficient for producing conscious sensation and not as a basis for inferring the latency for conscious sensation of peripheral origin. Latency for conscious sensation with brain stimulation may occurafterthe Minimum Train Duration. Backward masking with cortical stimuli suggests a 125–300 ms minimum value for the latency for conscious sensation of threshold skin stimuli. Backward enhancement is not suitable for inferring this latency. For determining temporal relations between stimuli that correspond to subjects' reports, theendof cerebral Minimum Train Duration should be used as reference, rather than its onset. Results of coupling peripheral and cortical stimuli are explained by a latency after the cortical Minimum Train Duration, having roughly the same duration as the latency for supraliminal skin stimuli. Results of coupling peripheral stimuli and stimuli to medial lemniscus (LM) are explained by a shorter LM latency and/or a longer peripheral latency. This interpretation suggests a 230 ms minimum value for the latency for conscious sensation of somatosensory near-threshold stimuli. The backward referral hypothesis, as formulated by Libet, should not be retained. Long readiness potentials preceding spontaneous conscious or nonconscious movements suggest that both kinds of movement are nonconsciously initiated. The validity of Libet's measures of W and M moments (Libet et al., 1983a) is questionable due to problems involving latencies, training, and introspective distinction of W and M. Veto of intended actions may be initially nonconscious but dependent on conscious awareness.     

The Knower Paradox and Epistemic Closure

The Knower Paradox has had a brief but eventful history, and principles of epistemic closure (which say that a subject automatically knows any proposition she knows to be materially implied, or logically entailed, by a proposition she already knows) have been the subject of tremendous debate in epistemic logic and epistemology more generally, especially because the fate of standard arguments for and against skepticism seems to turn on the fate of closure. As far as I can tell, however, no one working in either area has emphasized the result I emphasize in this paper: the Knower Paradox just falsifies even the most widely accepted general principles of epistemic closure. After establishing that result, I discuss five of its more important consequences.     

From the Knowability Paradox to the existence of proofs

The Knowability Paradox purports to show that the controversial but not patently absurd hypothesis that all truths are knowable entails the implausible conclusion that all truths are known. The notoriety of this argument owes to the negative light it appears to cast on the view that there can be no verification-transcendent truths. We argue that it is overly simplistic to formalize the views of contemporary verificationists like Dummett, Prawitz or Martin-Löf using the sort of propositional modal operators which are employed in the original derivation of the Paradox. Instead we propose that the central tenet of verificationism is most accurately formulated as follows: if \({\varphi}\) is true, then there exists a proof of \({\varphi}\). Building on the work of Artemov (Bull Symb Log 7(1): 1–36, 2001), a system of explicit modal logic with proof quantifiers is introduced to reason about such statements. When the original reasoning of the Paradox is developed in this setting, we reach not a contradiction, but rather the conclusion that there must exist non-constructed proofs. This outcome is evaluated relative to the controversy between Dummett and Prawitz about proof existence and bivalence.     

The Principal Paradox of Time Travel

Most arguments against the possibility of time travel use the same old, familiar objection: If I could travel back in time, then I could kill my earlier (i.e. younger) self. Since I do exist such an action would result in a contradiction. Therefore time travel is impossible. This is a statement of the Principal Paradox of Time Travel. Some philosophers have argued that such actions as attempting to kill one's earlier self would always fail and that there is nothing especially strange about such failures. Despite these arguments, the problem generally is not viewed as being solved in favour of time travel. The above objection to time travel is also used to dismiss a particular class of cosmological models as being unphysical. This paper provides a solution to the Principal Paradox by exploring both the logical and causal implications of time travel.     

Hempel's Paradox, Law-likeness and Causal Relations

It is widely thought that Bayesian confirmation theory has provided a solution to Hempel's Paradox (the Ravens Paradox). I discuss one well-known example of this approach, by John Mackie, and argue that it is unconvincing. I then suggest an alternative solution, which shows that the Bayesian approach is altogether mistaken. Nicod's Condition should be rejected because a generalisation is not confirmed by any of its instances if it is not law-like. And even law-like non-basic empirical generalisations, which are expressions of assumed underlying causal regularities, are not so confirmed if they are absurd in the light of our causal background knowledge or if their instances are not also possible instances of the relevant causal claim.