The logic of Simpson’s paradox

There are three distinct questions associated with Simpson’s paradox. (i) Why or in what sense is Simpson’s paradox a paradox? (ii) What is the proper analysis of the paradox? (iii) How one should proceed when confronted with a typical case of the paradox? We propose a “formal” answer to the first two questions which, among other things, includes deductive proofs for important theorems regarding Simpson’s paradox. Our account contrasts sharply with Pearl’s causal (and questionable) account of the first two questions. We argue that the “how to proceed question?” does not have a unique response, and that it depends on the context of the problem. We evaluate an objection to our account by comparing ours with Blyth’s account of the paradox. Our research on the paradox suggests that the “how to proceed question” needs to be divorced from what makes Simpson’s paradox “paradoxical.”     

A Berry and A Russell Without Self-Reference

In this paper I present two new paradoxes, a definability paradox (related to the paradoxes of Berry, Richard and König), and a paradox about extensions (related to Russell’s paradox). However, unlike the familiar definability paradoxes and Russell’s paradox, these new paradoxes involve no self-reference or circularity.     

How Braess’ paradox solves Newcomb's problem: Not!

In an engaging and ingenious paper, Irvine (1993) purports to show how the resolution of Braess’ paradox can be applied to Newcomb's problem. To accomplish this end, Irvine forges three links. First, he couples Braess’ paradox to the Cohen‐Kelly queuing paradox. Second, he couples the Cohen‐Kelly queuing paradox to the Prisoner's Dilemma (PD). Third, in accord with received literature, he couples the PD to Newcomb's problem itself. Claiming that the linked models are “structurally identical”, he argues that Braess solves Newcomb's problem. This paper shows that Irvine's linkage depends on structural similarities—rather than identities—between and among the models. The elucidation of functional disanalogies illuminates structural dissimilarities which sever that linkage. I claim that the Cohen‐Kelly queuing paradox cloaks a fine structure that decouples it from both Braess’ paradox and the PD (Marinoff, 1996a). I further assert that the putative reduction of the PD to a Newcomb problem (e.g. Brams, 1975; Lewis, 1979) is seriously flawed. It follows that Braess’ paradox does not solve Newcomb's problem via the foregoing and herein sundered chain. I conclude by substantiating a stronger claim, namely that Braess'paradox cannot solve Newcomb's problem at all.     

The effects of payout and probability magnitude on the Allais paradox

The Allais paradox decision bias was first offered as a challenge to the expected utility theory over 60 years ago. Although the Allais paradox is a standard challenge for normative theories of risky choice, its causes are not well understood. The present experiment uses two manipulations of the Allais paradox to investigate the commonly proposed probability-weighting explanation of the paradox. Reducing the magnitude of the outcomes did not affect the size of the Allais paradox, contradicting previous literature and supporting the probability weighting hypothesis. Reducing the probability of the nonzero outcomes to eliminate certainty reduced, but did not eliminate, the Allais paradox, a result inconsistent with probability weighting and other theories of the Allais paradox. The results suggest that the certainty effect alone cannot explain the Allais paradox.     

Conjunction-based Sorites: A Misguided Objection to Degree-Theoretic (Fuzzy) Solutions to Sorites Paradoxes

In 1987, Crispin Wright argued that degree-theoretic (fuzzy) solutions to the Sorites paradox fail because the solutions do not work when the paradox is restated using a conjunctive major premise. I show that Wright is incorrect: degree-theoretic solutions also work when the paradox is stated with a conjunctive major premise.     

On Kripke’s Dogmatism Paradox: A Logical Dynamical Analysis

As a byproduct of solving the surprise-exam paradox, Saul Kripke formulates a “dogmatism paradox” which seems to show that knowledge entails dogmatism. In this paper, the author analyzes the nature of the dogmatism paradox from a logical dynamical perspective. The author suggests that the dogmatism paradox is better understood as a paradox of knowledge attribution rather than of knowledge. Therefore, the dogmatism paradox could be solved without sacrificing the principle of epistemic closure. Based on a famous version of relevant alternatives theory, the author formalizes a logic of knowledge attribution in the style of logical dynamics, namely, public retraction logic, and analyzes how knowledge attributions are retracted with the expansion of relevant alternatives.     

Natural Deduction and Curry's Paradox

Curry's paradox, sometimes described as a general version of the better known Russell's paradox, has intrigued logicians for some time. This paper examines the paradox in a natural deduction setting and critically examines some proposed restrictions to the logic by Fitch and Prawitz. We then offer a tentative counterexample to a conjecture by Tennant proposing a criterion for what is to count as a genuine paradox.     

Is there an auditory analogue to Fechner's paradox in binaural loudness perception?

In binocular brightness perception a phenomenon called Fechner's paradox can be observed. This paradox implies non-monotonicities in the psychometric functions of binocular brightness. Lehky (1983) proposed a model that describes such non-monotonicities. He suggested that Fechner's paradox also exists in binaural loudness perception. However, until now no sufficient data have been collected to test this hypothesis. Therefore, an experiment was conducted in which 36 psychometric functions were obtained using binaural stimuli in the range of intensities in which Fechner's paradox supposedly occurs. As a result, no significant non-monotonicities were found. However, it is shown that jnds derived from the psychometric functions contradict predictions derived from the limited binaural additivity model of Gigerenzer and Strube (1983).     

The revenger's paradox

Conclusion Unlike the more commonly discussed paradoxes of deontic logic, the revenger's paradox is present not only in strong logics such as SDL, but also in much weaker logics that have been constructed to avoid the paradoxes of SDL. The paradox shows that any deontic logic with intersubstitutivity makes counter-intuitive truth assignments to some deontic expressions.However, the revenger's paradox does not pose as serious a problem as does Ross's paradox to the logics in which it occurs. A deontic logic in which Ross's paradox does not arise is preferable to one in which it does arise, even if the latter is subject to the revenger's paradox.     

The Paradox of Belief Instability and a Revision Theory of Belief

The epistemic paradox of 'belief instability' has recently received notable attention from many philosophers. In this paper I offer a new proposal, which I call a 'revision theory of belief'. This theory is in many respects an application of Gupta's and Belnap's revision theory of truth. They argue that the Liar paradox arises because our notion of truth is circular. I offer a similar proposal for handling the paradox of belief instability. In particular, I argue that our notion involved in the paradox of belief instability is circular, and this circularity of belief is the source of the paradox.     

A note on Horwich’s notion of grounding

Synthese - Horwich (Deflationism and paradox, Oxford University Press, Oxford, pp 75–84, 2005) proposes a solution to the liar paradox that relies on a particular notion of...     

PARADOXICAL PROPOSITIONS

This paper concerns two paradoxes involving propositions. The first is Russell's paradox from Appendix B of The Principles of Mathematics, a version of which was later given by Myhill. The second is a paradox in the framework of possible worlds, given by Kaplan. This paper shows a number of things about these paradoxes. First, we will see that, though the Russell/Myhill paradox and the Kaplan paradox might appear somewhat different, they are really just variants of the same phenomenon. Though they do this in different ways, the core of each paradox is to use the notion of a proposition to construct a function, f, from the power set of some set into the set itself. Next we will see how this paradox fits into the Inclosure Schema. Finally, I will provide a model of the paradox in question, showing its results to be non‐trivial, though inconsistent.     

An axiomatic version of Fitch’s paradox

A variation of Fitch’s paradox is given, where no special rules of inference are assumed, only axioms. These axioms follow from the familiar assumptions which involve rules of inference. We show (by constructing a model) that by allowing that possibly the knower doesn’t know his own soundness (while still requiring he be sound), Fitch’s paradox is avoided. Provided one is willing to admit that sound knowers may be ignorant of their own soundness, this might offer a way out of the paradox.     

On Russell's vulnerability to Russell's paradox

Influenced by G. E. Moore, Russell broke with Idealism towards the end of 1898; but in later years he characterized his meeting Peano in August 1900 as ‘the most important event’ in ‘the most important year in my intellectual life’. While Russell discovered his paradox during his post-Peano period, the question arises whether he was already committed, during his pre-Peano Moorean period, to assumptions from which his paradox may be derived. Peter Hylton has argued that the pre-Peano Russell was thus vulnerable to (at least one version of) Russell's paradox and hence that the paradox exposes a pre-existing difficulty in Russell's Moorean philosophy. Contrary to Hylton, I argue that the Moorean Russell adhered to views which insulated him against the paradox. Further, I argue that Russell became vulnerable to his paradox as a result of changes in his Moorean position occasioned, first, by his acceptance of Cantor's theory of the transfinite, and, second, by his correspondence with Frege. I conclude with some general comments regarding Russell's acceptance of naïve set theory.     

A PARADOX IN THE PERCEPTION OF LUMINANCE GRADIENTS. I

Under certain conditions subjects looking at a luminance gradient report a physically darker part of the gradient to be brighter than an adjacent area of higher luminance. This brightness paradox was studied in a series of experiments using a magnitude estimation method. The main results were that both the changing sign of the second derivative of the luminance function (Mach's hypothesis) and the higher or lower luminance of an adjacent area (McDougall's drainage theory) are critical conditions for the appearance of the paradox. In the present study none of these conditions per se resulted in a brightness paradox.     

How Braess’ paradox solves Newcomb's problem*

Newcomb's problem is regularly described as a problem arising from equally defensible yet contradictory models of rationality. Braess’ paradox is regularly described as nothing more than the existence of non‐intuitive (but ultimately non‐contradictory) equilibrium points within physical networks of various kinds. Yet it can be shown that Newcomb's problem is structurally identical to Braess’ paradox. Both are instances of a well‐known result in game theory, namely that equilibria of non‐cooperative games are generally Pareto‐inefficient. Newcomb's problem is simply a limiting case in which the number of players equals one. Braess’ paradox is another limiting case in which the ‘players’ need not be assumed to be discrete individuals. The result is that Newcomb's problem is no more difficult to solve than (the easy to solve) Braess’ paradox.     

Some Irreverent Thoughts on Paradox

There is currently great interest in the use of paradox in psychotherapy. Unfortunately, there is also considerable confusion and misunderstanding of paradox, owing, in part, to the lack of a comprehensive theory that explains the role of paradox in human problems. This paper does not address itself to such theory but explores some cultural and epistemological components of therapists' misperceptions and misattributions of “paradox.”     

The developmental paradox of false belief understanding: a dual-system solution

We explore the developmental paradox of false belief understanding. This paradox follows from the claim that young infants already have an understanding of false belief, despite the fact that they consistently fail the elicited-response false belief task. First, we argue that recent proposals to solve this paradox are unsatisfactory because they (i) try to give a full explanation of false belief understanding in terms of a single system, (ii) fail to provide psychological concepts that are sufficiently fine-grained to capture the cognitive requirements for the various manifestations of false belief understanding, and (iii) ignore questions about system interaction. Second, we present a dual-system solution to the developmental paradox of false belief understanding that combines a layered model of perspective taking with an inhibition-selection-representation mechanism that operates on different levels. We discuss recent experimental findings that shed light on the interaction between these two systems, and suggest a number of directions for future research.     

Clinical psychopathology, psychiatric diagnosis and therapy in psychiatry

Clinical psychopathology has a diagnostic and therapeutic purpose. On the level of phenomenology one has to deal with four paradoxes. 1. The paradox of gradual transition from normality towards pathology versus the occurrence of symptoms of a new quality, 2. the paradox of nosological nonspecificity of various symptoms versus typical constellations of special symptoms for certain diseases, 3. the paradox of pure and typical symptoms and diseases versus symptoms and diseases which are mixed and non typical. 4. The paradox of the prevalence of psychopathological phenomena versus prevalence of somatic phenomena in one and the same disease. On the level of etiology one must face the fact that there are not only typical endogenous, exogenous (organic) and psychogenic syndromes. Neurotic symptoms may have an endogenous cause. Therefore neuroses may be treated effectively with antidepressants.