A Wittgensteinian Solution to the Sorites

I develop a solution to the Sorites Paradox, according to which a concatenation of valid arguments need not itself be valid. I specify which chains of valid arguments are those that do not preserve validity: those that pass the vague boundary between cases where the relevant concept applies and cases where that concept does not apply. I also develop various criticisms of this solution and show why they fail; basically, they all involve a petitio at some stage. I criticise the conviction that if every short argument in a long concatenated argument is valid, so is the long argument: it is, I argue, the result of an unjustified generalisation from the case of arguments that do not employ vague concepts (as in mathematics) to arguments that do employ them. My approach is Wittgensteinian in its “leaving everything as it is,” in its claiming that the “beginning” has been searched too far back (see paper's epigraph) and in its claim that the paradox was generated by a misapplication of a partial picture of the behaviour of arguments. I conclude my paper by comparing and contrasting my approach to the few precedents found in the vagueness literature and by answering a few additional objections that were raised there.     

Laws,causality and the intentional explanation of action

Whether or not an intentional explanation of action necessarily involves law-like statements is related to another question, namely, is it a causal explanation? The Popper-Hempel Thesis, which answers both questions affirmatively, inevitably faces a dilemma between realistic and universalistic requirements. However, in terms of W.C. Salmon’s concept of causal explanation, intentional explanation can be a causal one even if it does not rely on any laws. Based on this, we are able to refute three characteristic arguments for the claim “reason is not a cause of action,” namely, the “proper logical” argument, the “logical relation” argument, and the “rule-following” argument. This rebuttal suggests that the causal relationship between reason and action can provide a justification for intentional explanations.     

Moore’s Paradox is not just another pragmatic paradox

One version of Moore’s Paradox is the challenge to account for the absurdity of beliefs purportedly expressed by someone who asserts sentences of the form ‘p & I do not believe that p’ (‘Moorean sentences’). The absurdity of these beliefs is philosophically puzzling, given that Moorean sentences (i) are contingent and often true; and (ii) express contents that are unproblematic when presented in the third-person. In this paper I critically examine the most popular proposed solution to these two puzzles, according to which Moorean beliefs are absurd because Moorean sentences are instances of pragmatic paradox; that is to say, the propositions they express are necessarily false-when-believed. My conclusion is that while a Moorean belief is a pragmatic paradox, it is not just another pragmatic paradox, because this diagnosis does not explain all the puzzling features of Moorean beliefs. In particularly, while this analysis is plausible in relation to the puzzle posed by characteristic (i) of Moorean sentences, I argue that it fails to account for (ii). I do so in the course of an attempt to formulate the definition of a pragmatic paradox in more precise formal terms, in order to see whether the definition is satisfied by Moorean sentences, but not by their third-person transpositions. For only an account which can do so could address (ii) adequately. After rejecting a number of attempted formalizations, I arrive at a definition which delivers the right results. The problem with this definition, however, is that it has to be couched in first-person terms, making an essential use of ‘I’. Thus the problem of accounting for first-/third-person asymmetry recurs at a higher order, which shows that the Pragmatic Paradox Resolution fails to identify the source of such asymmetry highlighted by Moore’s Paradox.     

Decision-making under uncertainty: biases and Bayesians

Animals (including humans) often face circumstances in which the best choice of action is not certain. Environmental cues may be ambiguous, and choices may be risky. This paper reviews the theoretical side of decision-making under uncertainty, particularly with regard to unknown risk (ambiguity). We use simple models to show that, irrespective of pay-offs, whether it is optimal to bias probability estimates depends upon how those estimates have been generated. In particular, if estimates have been calculated in a Bayesian framework with a sensible prior, it is best to use unbiased estimates. We review the extent of evidence for and against viewing animals (including humans) as Bayesian decision-makers. We pay particular attention to the Ellsberg Paradox, a classic result from experimental economics, in which human subjects appear to deviate from optimal decision-making by demonstrating an apparent aversion to ambiguity in a choice between two options with equal expected rewards. The paradox initially seems to be an example where decision-making estimates are biased relative to the Bayesian optimum. We discuss the extent to which the Bayesian paradigm might be applied to the evolution of decision-makers and how the Ellsberg Paradox may, with a deeper understanding, be resolved.     

Accounting for occurrences: An explanation for some novel tendencies in causal judgment from contingency information

Contingency information is information about empirical associations between possible causes and outcomes. In the present research, it is shown that, under some circumstances, there is a tendency for negative contingencies to lead to positive causal judgments and for positive contingencies to lead to negative causal judgments. If there is a high proportion of instances in which a candidate cause (CC) being judged is present, these tendencies are predicted by weighted averaging models of causal judgment. If the proportion of such instances is low, the predictions of weighted averaging models break down. It is argued that one of the main aims of causal judgment is to account for occurrences of the outcome. Thus, a CC is not given a high causal judgment if there are few or no occurrences of it, regardless of the objective contingency. This argument predicts that, if there is a low proportion of instances in which a CC is present, causal judgments are determined mainly by the number of Cell A instances (i.e., CC present, outcome occurs), and that this explains why weighted averaging models fail to predict judgmental tendencies under these circumstances. Experimental results support this argument.     

The oldest solution to the circularity problem for Humeanism about the laws of nature

According to Humeanism about the laws, the laws of nature are nothing over and above certain kinds of regularities about particular facts (the “Humean mosaic”). Humeanism has often been accused of circularity: according to scientific practice laws often explain their instances, but on the Humean view they also reduce to the mosaic, which includes those instances. In this paper I formulate the circularity problem in a way that avoids a number of controversial assumptions routinely taken for granted in the literature, and against which many extant responses are therefore ineffective. I then propose a solution that denies the alleged Humean commitment that laws are explained by their instances. The solution satisfies three desiderata that other solutions don’t: it provides independent motivation against the idea that Humean laws are explained by their instances; it specifies the sense in which Humean laws are nonetheless “nothing over and above” their instances; and it gives an alternative account of what does explain the laws, if not their instances. This solution, I will argue, is not only the simplest but also the oldest one: it appeals only to tools and theses whose first appearance predates the earliest statements of the circularity problem itself.

     

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.     

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.     

An unwelcome consequence of the Multiverse Thesis

The Multiverse Thesis is a proposed solution to the Grandfather Paradox. It is popular and well promulgated, found in fiction, philosophy and (most importantly) physics. I first offer a short explanation on behalf of its advocates as to why it qualifies as a theory of time travel (as opposed to mere ‘universe hopping’). Then I argue that the thesis nevertheless has an unwelcome consequence: that extended objects cannot travel in time. Whilst this does not demonstrate that the Multiverse Thesis is false, the consequence should give pause for concern. Even if it does not lead one to reject the thesis, I briefly detail some reasons to think it is interesting nonetheless.     

Causation and Induction1

The connection between views about causation and attempts to justify inductive reasoning is sufficiently close that some philosophers² have taken success at the latter as a litmus test for the truth of the former. I do not agree with this approach. Like Hume, I believe that the nature of causal connections must be understood prior to, and independently of, solutions to the problem of induction. Like Hume, I also hold that the problem of induction cannot be solved if Hume's analysis of causal connections is correct. But unlike Hume, I believe that that analysis is incorrect. However, I shall not attempt to establish this crucial thesis here. I mention it because this paper presupposes it. Hume's difficulty about causation must—and can—be faced head-on. There are phenomenological grounds for affirming that we sometimes directly experience nonlogical, necessary connections between events. I shall only briefly summarize these grounds, which will be argued for in detail elsewhere. The purpose of this paper is to explore the extent to which a necessitarian theory of causation can bring the problem of induction closer to solution.     

Source incompatibilism and the foreknowledge dilemma

The problem that divine foreknowledge poses for free will is one that is notoriously difficult to solve. If God believes in advance how an agent will act, this fact about the past eradicates all alternatives for the actor, given the infallibility of God’s beliefs. And if we assume, with many theists, that free will requires alternatives possibilities, then it looks as if God’s omniscience is incompatible with our free will. One solution to this problem, introduced and defended by David Hunt, draws on the source incompatibilist position in the secular free debate. According to source incompatibilists, free will does not require alternative possibilities but is also not compatible with causal determinism. Hunt argues that because God’s foreknowledge does not eliminate future alternatives through causal means, it is compatible with free will. In this paper, I challenge Hunt’s position using Kevin Timpe’s distinction between “wide” and “narrow” source incompatibilists. I argue that if one wishes to be an incompatibilist concerning free will and causal determinism, one must accept that alternatives are required for free will. And if one must accept that alternatives are a necessary condition for free will, then Hunt’s solution to the foreknowledge dilemma will not succeed.     

Intentions and Causes, Actions and Right Actions

I argue in this essay that belief/desire explanations are not logically true and not causal, and further that the antecedent of a true belief/desire conditional cannot be strengthened in such a way as to transform it into a true causal statement. I also argue that belief/desire explanations are not dispensable: they are presupposed in our justifications of scientific claims. The proposal is not that psychological determinism is false, but that some at least of our activities are not describable in causal terms. These arguments prepare the ground for a puzzle. If all human intentional behaviour is caused, then all actual linkages between psychological states and behaviour should be expressed in causal statements. But neither the action of asserting a causal statement nor the action of justifying the assertion can be described as the result of a cause. Therefore if one accepts that scientific claims can be justified, not all linkages between psychological states and subsequent action are expressible in causal statements. I do not offer a solution to this puzzle.     

Modelling mechanisms with causal cycles

Mechanistic philosophy of science views a large part of scientific activity as engaged in modelling mechanisms. While science textbooks tend to offer qualitative models of mechanisms, there is increasing demand for models from which one can draw quantitative predictions and explanations. Casini et al. (Theoria 26(1):5–33, 2011) put forward the Recursive Bayesian Networks (RBN) formalism as well suited to this end. The RBN formalism is an extension of the standard Bayesian net formalism, an extension that allows for modelling the hierarchical nature of mechanisms. Like the standard Bayesian net formalism, it models causal relationships using directed acyclic graphs. Given this appeal to acyclicity, causal cycles pose a prima facie problem for the RBN approach. This paper argues that the problem is a significant one given the ubiquity of causal cycles in mechanisms, but that the problem can be solved by combining two sorts of solution strategy in a judicious way.     

Buridan's Solution to the Liar Paradox

Jean Buridan has offered a solution to the Liar Paradox, i.e. to the problem of assigning a truth-value to the sentence ‘What I am saying is false’. It has been argued that either (1) this solution is ad hoc since it would only apply to self-referencing sentences [Read, S. 2002. ‘The Liar Paradox from John Buridan back to Thomas Bradwardine’, Vivarium, 40 (2), 189–218] or else (2) it weakens his theory of truth, making his ‘a logic without truth’ [Klima, G. 2008. ‘Logic without truth: Buridan on the Liar’, in S. Rahman, T. Tulenheimo and E. Genot, Unity, Truth and the Liar: The Modern Relevance of Medieval Solutions to the Liar Paradox, Berlin: Springer, 87–112 (Chapter 5); Dutilh Novaes, C. 2011. ‘Lessons on truth from mediaeval solutions to the Liar Paradox’, The Philosophical Quarterly, 61 (242), 58–78]. Against (1), I will argue that Buridan's solution by means of truth by supposition does not involve new principles. Self-referential sentences force us to handle supposition more carefully, which does not warrant the accusation of adhoccery. I will also argue, against (2), that it is exaggerated to assert that this solution leads to a ‘weakened’ theory of truth, since it is consistent with other passages of the Sophismata, which only gives necessary conditions for the truth of affirmative propositions, but sufficient conditions for falsity.     

Mental causation and explanatory exclusion

Kim argues that we can never have more than one complete and independent explanation for a single event. The existence of both mental and physical explanations for behavior would seem to violate this principle. We can avoid violating it only if we suppose that mental causal relationships supervene on physical causal relationships. I argue that although his solution is attractive in many respects, it will not do as it stands. I propose an alternate understanding of supervenient causation which preserves the advantages of Kim's account while avoiding the problems. My analysis involves appeal to counterfactuals. Any counterfactual analysis must confront the problem that mental states appear to be screened off from causal relevance by physical states. I argue that screening off is not a problem, because cases in which mental states appear to be screened off are cases in which background conditions are not held constant.     

SUBSET REALIZATION,PARTHOOD, AND CAUSAL OVERDETERMINATION

Defenders of the subset view of realization have claimed that we can resolve well‐known worries about mental‐physical causal overdetermination by holding that mental properties are subset realized by physical properties, that instances of subset realized properties are parts of physical realizers, and that part‐whole overdetermination is unproblematic. I challenge the claim that the overdetermination generated by the subset view can be legitimated by appealing to more mundane part‐whole overdetermination. I conclude that the subset view does not provide a unique solution to overdetermination worries.     

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.     

Causality, Propensity, and Bayesian Networks

This paper investigates the relations between causality and propensity. Aparticular version of the propensity theory of probability is introduced, and it is argued that propensities in this sense are not causes. Some conclusions regarding propensities can, however, be inferred from causal statements, but these hold only under restrictive conditions which prevent cause being defined in terms of propensity. The notion of a Bayesian propensity network is introduced, and the relations between such networks and causal networks is investigated. It is argued that causal networks cannot be identified with Bayesian propensity networks, but that causal networks can be a valuable heuristic guide for the construction of Bayesian propensity networks.     

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.