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4059 results in Logic

Page 30 of 203

  • 10 - Maximal Conservative Theories

  • from Part II - Conservativity
  • Cezary Cieśliński
  • Book:
    The Epistemic Lightness of Truth
    Published online:
    30 November 2017
    Print publication:
    07 December 2017, pp 174-182

  • Summary

    How useful to the deflationist are conservative theories of truth? In this chapter, I will discuss one particular strategy of employing the conservativity condition in order to foster the deflationist's philosophical goals. The main idea is to combine two attractive properties of truth theories: syntactic conservativeness and maximality.

    As emphasised in the last paragraph of the previous chapter, conservative truth theories are very attractive to the deflationist in certain important respects. Now, maximality can be viewed as another desirable property. To begin with, imagine the disquotationalist who wants to extend his base theory of syntax with some chosen instantiations of the T-schema. Obviously, the set of instantiations has to be limited in some way in order to avoid the liar-type paradoxes. Unfortunately, this is easier said than done. Which instantiations should be adopted? Which of them should be rejected? What sort of choices can be made without inviting the charge of arbitrariness? Paul Horwich gives the following answer:

    The principles governing our selection of excluded instances are, in order of priority: (a) that the minimal theory not engender ‘liar-type’ contradictions; (b) that the set of excluded instances be as small as possible; and – perhaps just as important as (b) – (c) that there be a constructive specification of the excluded instances that is as simple as possible. (Horwich 1999, p. 42)

    In view of this, one option which suggests itself consists in trying to include as many instances of the T-schema as possible in an attempt to build a maximal consistent extension of our base theory.

    This possibility has been investigated by McGee (1992). At the start, McGee observes that it is indeed possible to extend a chosen base theory (say, Peano arithmetic) in such a way. However, in the end he notices two problems.

    Firstly, there are continuum many maximal consistent extensions of Peano arithmetic. In effect, some additional principles will be required to select a truth theory of our choice. In short, maximality is not enough. We still have to “make comparative judgments about which instances of (T) we regard as essential and which we are willing to relinquish”.

  • 9 - Conservativity: Philosophical Motivations

  • from Part II - Conservativity
  • Cezary Cieśliński
  • Book:
    The Epistemic Lightness of Truth
    Published online:
    30 November 2017
    Print publication:
    07 December 2017, pp 145-173

  • Summary

    Semantic Conservativity

    The presentation of the formal framework has been completed, and I turn now to a discussion of philosophical motivation for conservativity demands. In this section the conservativity requirement in its semantic version will be considered.

    Model-theoretic conservativity is a relatively recent candidate for the role of the property which would be able to give meaning to the slogans of thinness of truth. To my knowledge, very few authors have tried to provide careful philosophical argumentation supporting semantically conservative truth theories as somehow ‘proper’ or ‘adequate’ for the adherents of the innocence claim. It is my aim now to consider such arguments. Why should semantic conservativity be a virtue?

    Indeed, some authors have tried to portray semantic conservativity as a consequence of the traditional deflationist claims. Thus, in a recent paper, Andrea Strollo states quite explicitly that semantic non-conservativity contradicts the tenets of deflationism. If our theory of truth were to exclude some models, then “in open contrast with the deflationist claim, the property of truth would enter reality as a robust ingredient” (Strollo 2013, p. 530).

    In itself, this amounts to little more than ascribing one's intuitions to the deflationist without stating any clear reason for doing so. When no citations are given and no attempt is made to relate the semantic conservativity doctrine to explicitly formulated deflationary tenets, one is fully entitled to conclude that no link whatsoever has been established with what the deflationists have actually been saying. We could, of course, decide to define ‘robust’ in such a way that semantically non-conservative truth will emerge as robust. However, the question should be asked whether the ‘open contrast with the deflationist claim’ is real or illusory. Why should it matter to the deflationist whether his theory excludes some models?

    Strollo himself does not answer this question in his paper; he remarks only that “good reasons had been put forward” for the conservativeness demand, referring the reader to (Shapiro 1998). Shapiro's argumentation will be discussed in the next section. However, let us just note that in the specific context of discussing the semantic conservativity demand, referring the reader to Shapiro's paper is hardly satisfactory.

  • Part II - Conservativity

  • Cezary Cieśliński
  • Book:
    The Epistemic Lightness of Truth
    Published online:
    30 November 2017
    Print publication:
    07 December 2017, pp 83-89

  • Summary

    The central notion to be discussed in the chapters to follow is that of a conservative extension. In the contemporary literature, conservativity is sometimes presented either as an explication of the lightness of truth or at least as a demand which deflationary truth theories must satisfy. This introductory section describes the philosophical intuitions behind such claims. It contains also the plan of Part II.

    To my knowledge, in the context of a discussion about the light notion of truth, conservativity was introduced for the first time by Leon Horsten, who made the following comment about Horwich's ‘minimal’ (disquotational) theory of truth:

    The minimalist theory entails that a truth predicate should be conservative over a given theory that is stated without the truth predicate (or any other semantical notions). (Horsten 1995, p. 183)

    Horsten's stated motivation for ascribing to the deflationists (and to Horwich in particular) the commitment to conservativity was his incapacity otherwise to see “what the neutrality of the notion of truth according to deflationism amounts to”. Since then, many authors have treated conservativeness as at least a promising explication of the ‘lightness’ of the notion of truth, typically repeating Horsten's question: how otherwise could the deflationist claim that the notion of truth is contentless or ‘metaphysically thin’?

    Definition 1.3.1 characterised two notions of conservativity, which could be proposed as tools for the deflationist to explicate his position. As we saw, one notion is syntactic; the intended meaning is that a conservative extension does not prove new theorems of the base language. The second is semantic and concerns the possibility of expanding models: every model of a base theory can be expanded to a model of its conservative extension in the semantic sense.

    Both notions were invoked by Shapiro (1998), who also attributed conservativity constraint to deflationary truth theories. On the face of it, Shapiro's motivation for accepting the conservativeness demand was quite similar to that of Horsten.

  • 12 - The Strength of Reflection Principles

  • from Part III - Reflection Principles
  • Cezary Cieśliński
  • Book:
    The Epistemic Lightness of Truth
    Published online:
    30 November 2017
    Print publication:
    07 December 2017, pp 207-231

  • Summary

    In this chapter, I am going to discuss various reflection principles, taking into account both their arithmetical and truth-theoretical strength. As for the arithmetical component, it would be good to know at least whether adding a given reflection principle leads us beyond the base theory (Peano arithmetic). On the other hand, when assessing truth-theoretic strength, one needs to know how powerful the reflection principles are in proving truth-theoretic generalisations. All in all, one needs to be clear about how helpful reflection can be in solving the problem of deductive weakness of some truth theories (conservative or disquotational ones).Without such a knowledge, there seems to be little point of entering a philosophical debate about reflection.

    Definition 1.5.1 has introduced three types of reflection principles for an arithmetical theory S: the global, the uniform, and the local one. Here I will consider reflection in a more general setting. Intuitively, a reflection principle for a class Γ (there is no assumption that Γ contains arithmetical sentences only) is a sentence or a set of sentences stating the soundness of Γ. The principle in question can have a form of a global principle, or it can be a schematic expression of soundness, in the style of (UR) or (LR), that is, it can be (cf. again Definition 1.5.1). In general, the strength of the principle will depend on the choice of Γ.

    In the context of discussing axiomatic characterisations of light notions of truth, an obvious choice of Γ is the set of disquotational axioms. Indeed, it turns out that, when applied to such a set, some forms of reflection produce theories which are truth-theoretically strong. I will start, however, with a discussion of reflection for some classes Γ containing arithmetical sentences only. This will permit to get a clearer overall picture of the situation. At this initial stage, before venturing further, I want (at the very least) to indicate some classes Γ for which reflection seems intuitively natural and obvious. A more ambitious objective would be to approach the question of what really explains the arithmetical strength of reflection principles.

  • 4 - Why Do We Need Disquotational Truth?

  • from Part I - Disquotation
  • Cezary Cieśliński
  • Book:
    The Epistemic Lightness of Truth
    Published online:
    30 November 2017
    Print publication:
    07 December 2017, pp 58-67

  • Summary

    What is the truth predicate for? If the concept of truth is so light as to be fully characterised by the disquotational schema, why do we need it? Perhaps we could just as well eliminate it from our language? Contemporary disquotationalists usually answer the last question with an emphatic ‘no’. They typically say that the truth predicate has an important role to play; namely, it permits us to express general observations which otherwise could not be expressed. In this context, the disquotationalists often invoke Willard V. Quine as one of the precursors of their philosophical views on truth.

    According to Quine (1986), speaking about the truth of a single sentence is plainly redundant: “So long as we are speaking only of the truth of singly given sentences, the perfect theory of truth is what Wilfrid Sellars has called the disappearance theory of truth” (p. 11). Truth is disquotation, and when applied to a concrete sentence, it can just as well be dispensed with. Nevertheless:

    Where the truth predicate has its utility is in just those places where […] we are impelled by certain technical complications to mention sentences. (Quine 1986, p. 11)

    I find the choice of words in the quoted passage very telling. Not only is truth dispensable in many applications, but where it is not, the reasons are not deep and mysterious at all. It is mere technical complications that explain our need for the truth predicate. But what are they? To quote Quine again:

    We can generalize on ‘Tom is mortal’, ‘Dick is mortal’, and so on, without talking of truth or of sentences; we can say ‘All men are mortal’ […] When on the other hand we want to generalize on ‘Tom is mortal or Tom is not mortal’, ‘Snow is white or snow is not white’, and so on, we ascend to talk of truth and of sentences, saying ‘Every sentence of the form “p or not p” is true’, or ‘Every alternation of a sentence with its negation is true’.

    According to Quine, the ‘technical complication’ in question is that we are not able to make sense of quantifying into sentence position.

  • 13 - Deflationism and Truth-Theoretical Strength

  • from Part III - Reflection Principles
  • Cezary Cieśliński
  • Book:
    The Epistemic Lightness of Truth
    Published online:
    30 November 2017
    Print publication:
    07 December 2017, pp 232-278

  • Summary

    In Chapter 5, I discussed the generalisation problem as the basic difficulty facing disquotational truth theories. It is worth stressing that criticisms based on the conservativeness argument go in a somewhat similar direction. After all, the argument in question points to the deductive weaknesses of truth theories (not necessarily disquotational ones) and to their unsuitability for proving certain truth-theoretic generalisations. I have already presented – and deemed inadequate – some solutions proposed in the literature (cf. Chapter 5 and Section 11.2.1 of Chapter 11.2). The main objective here will be a presentation and a defence of a strategy employing reflection principles, which will permit me to present a uniform answer to the critics of deflationism.

    Both the generalisation problem and the conservativeness argument will be treated here as variants of one and the same broader question, which takes the following form:

    (Q) Given that we accept a theory K, why should we accept various additional statements, which are unprovable in K?

    This perspective could be adopted by the critic whose starting point is the observation that K is a theory of truth accepted by the deflationist. If K is disquotational, the critic can argue that the acceptance of T-biconditionals commits the deflationist to the acceptance of compositional principles not derivable in K. Thus, with this option the generalisation problem is introduced as the challenge of explaining such commitments. If K is conservative (not necessarily disquotational), then the critic proceeds in exactly the same manner, indicating that the various statements unprovable in K (Godel sentences, consistency statements, reflection principles) should be accepted once we accept K. Thus, the conservativeness argument appears on the scene.

    Of course the critic might choose a different presentation of his reasoning, one which does not require accepting K as a premise. Even so, the deflationist is fully entitled to employ the acceptance of K as a key premise in his own defence against the challenge. Answering (Q) permits him to do just that. If he is able to show that the acceptance of, say, compositional principles becomes fully rational once K is accepted, then the critic's challenge is met. Indeed, this is exactly the strategy which will be pursued here.

  • 5 - The Generalisation Problem

  • from Part I - Disquotation
  • Cezary Cieśliński
  • Book:
    The Epistemic Lightness of Truth
    Published online:
    30 November 2017
    Print publication:
    07 December 2017, pp 68-82

  • Summary

    Even if we agree that truth permits us to express generalisations which would otherwise remain inexpressible, does it really show the usefulness of the notion of truth? Assume that, as users of a given base theory, we accept (separately) each sentence from the set A. Assume also that we have at our disposal the sentence τ, stating (in intuitive terms) that all elements of A are true, together with a convincing explanation of what it means for τ to express our simultaneous acceptance of all elements of A. Are we finished? Is there really nothing else required?

    The worry is that even with all of this at our disposal, we still have not answered the question of when – or under what conditions – we are entitled to accept τ. What are the assertability conditions for this sentence? And if none are produced – or, alternatively, if the ones which are produced are unattainable – what is to be gained in recognising that τ expresses our simultaneous acceptance of all elements of A? What is the point of merely expressing it if we are never able to establish τ as acceptable?

    Employing the previous example again, assume that we are able to recognise that the sentence ‘All substitutions of ϕ→ϕ are true’ expresses our simultaneous acceptance of each concrete instance of ‘ϕ → ϕ’. The question is: so what? What is the point, if we are not able to assert this sentence? This concern can be formulated as the following challenge: ‘So you claim that the role of truth is to express generalisations? Fine. But generalisations which cannot be asserted are useless. And if the only role of truth is to permit us to express something useless, then the notion of truth is useless as well’.

    One rejoinder consists in observing that the theory of truth is not the only admissible source of the assertability conditions. We may also have inductive empirical grounds for accepting a given truth-involving generalisation. When analysing a large sample of the pope's past theological statements, I discover to my surprise that all of them – without even a single exception – were true.

  • Afterword

  • Cezary Cieśliński
  • Book:
    The Epistemic Lightness of Truth
    Published online:
    30 November 2017
    Print publication:
    07 December 2017, pp 279-280

  • Summary

    This book focuses on disquotational and conservative axiomatic truth theories. As a matter of fact, disquotation and conservativity have been advocated in the literature (usually separately) as either commitments of the deflationary standpoint or at least as ideas that give meaning to slogans of the lightness of truth. Since I view deflationism as an array of diverse conceptions only loosely connected by the theme of the insubstantiality of truth, the idea of ‘commitments’ is not appealing. Nonetheless, it is also my view that the approach to truth which takes disquotation and conservativity as central features of truth theories is indeed natural and fundamentally correct.

    Any defence of disquotation and conservativity has to overcome two serious objections which have been raised in the literature: the generalisation problem and the conservativeness argument. As I see it, this is also the point at which disquotation and conservativity meet and where they can be seen to fit together. Namely, the most serious reservations against both standpoints turn out to be very closely related – so closely that they can be overcome by a uniform epistemic strategy advocated in the final chapter of this book.

    Indeed, I do think that the proposed epistemic strategy permits the deflationist to vindicate the thesis of the lightness of truth. With such a weapon at hand, the deductive weakness of disquotational conservative theories of truth is no longer a concern. “No-one will drive us out of Tarski's truth-theoretic paradise”, one critic has objected. By taking the approach presented here, I believe that the road to Tarski's paradise is now open.

  • Part III - Reflection Principles

  • Cezary Cieśliński
  • Book:
    The Epistemic Lightness of Truth
    Published online:
    30 November 2017
    Print publication:
    07 December 2017, pp 203-206

  • Summary

    Parts I and II left us with two serious challenges, which undermine the deflationist's philosophical project: the generalisation problem and the conservativeness argument. A solution has been promised in a couple of places which, in spite of these objections, permits the deflationist to vindicate the idea of disquotational and conservative theories as satisfactory characterisations of the content of the truth predicate. This is the task I am going to undertake in Part III of this book.

    When discussing disquotationalism, I emphasised that Horwich's second solution to the generalisation problem seems to involve an appeal to some form of reflection (see the final remarks of Section 5.2). It has also been observed that the reflection principles come to the foreground both in some formulations of the conservativeness argument and in some of the proposed defences against the charges of the critics, notably, in the defence offered by Tennant. In each of these cases, I have found the defences insufficient. In my opinion, appeals to reflection principles require a more thorough and deeper philosophical analysis.

    Nevertheless, I view Horwich's and Tennant's attempts as steps in the right direction. Moreover, it seems to me that both challenges (the generalisation problem and the conservativeness argument) are similar enough to motivate the search for a unified, single answer – that is, for a proposal which would deal with both of them at the same time. In this context, the investigation of reflection principles is a promising direction of research. After all, both challenges point to the truth-theoretic or arithmetical weakness of the relevant disquotational/conservative theories. Reflection principles are not only intuitively correct, but they also increase the deductive power of theories, meaning that they can be viewed indeed as attractive remedies for the weaknesses of deflationary truth theories. Still, the following key question remains unanswered: exactly in what way can they be used by the deflationist without compromising his philosophical position? How can I accept, for example, a global or a local reflection principle for Peano arithmetic if my disquotational/conservative truth theory does not prove it?

  • Introduction

  • Cezary Cieśliński
  • Book:
    The Epistemic Lightness of Truth
    Published online:
    30 November 2017
    Print publication:
    07 December 2017, pp ix-xiv

  • Summary

    Is there anything more familiar and obvious than the opposition of truth and falsity? It is true that the earth is round. It is false that dragons eat virgins. (As everyone knows, dragons eat only pistachio marzipan with vanilla truffle.) Elementary, is it not? However, if this is so familiar, what then is truth? When confronted with such a direct question, many of us are tempted to repeat the famous words of Saint Augustine: “If no one asks me, I know what it is. If I wish to explain it to him who asks, I do not know”.

    Being that no decent philosopher can rest satisfied with ignorabimus, some answers have naturally been proposed. Indeed, answers have proliferated, with various philosophical schools promoting their own worldviews and agendas. Unfortunately, no lasting consensus has emerged, with the only exception perhaps being the following. Philosophers seemed to agree that the task of explaining the nature of truth is a daunting one; it is hard, complicated, deep and far-reaching. However, in recent times, serious doubts have emerged even here. Some modern philosophers have reacted to the ancient puzzles with a bold claim; they have said that, in fact, truth has no nature, and the very concept of truth is, in some sense, innocent or trivial. This book is devoted to the analysis and assessment of this claim.

    So, what is truth? Here is a selection of quotes giving answers to this question.

  • • ‘To say of what is that it is not, or of what is not that it is, is false, while to say of what is that it is, and of what is not that it is not, is true.’ (Aristotle, Metaphysics, IV 7, 1011b27)

  • • ‘Veritas est adaequatio intellectus et rei.’ (‘Truth is the conformity of the intellect to the things.’ Thomas Aquinas, Summa Theologica I, Q 16)

  • • ‘The nominal definition of truth, namely that it is the agreement of cognition with its object, is here granted and presupposed.’ (I. Kant, Critique of Pure Reason, A 57-8/B 82)

    • In one crucial respect, the first of these classical formulations is rather different from the other two.

  • 11 - The Conservativeness Argument

  • from Part II - Conservativity
  • Cezary Cieśliński
  • Book:
    The Epistemic Lightness of Truth
    Published online:
    30 November 2017
    Print publication:
    07 December 2017, pp 183-202

  • Summary

    As mentioned earlier, the conservativeness demand was proposed not by the deflationists themselves but by their critics, who subsequently used the demand in their argumentation against deflationary conceptions of truth. According to the critics, the deflationary truth theory should but cannot be conservative – that is the general upshot of the (so-called) ‘conservativeness argument’.

    I have already discussed the reasons for ascribing conservativity doctrine to the deflationists. My conclusions have been sceptical; that is, the justifications for it are very weak. Nevertheless, I have also concluded that conservativity would be a useful property; in addition, it has been emphasised that syntactic conservativity could still well function as a new explication of the lightness doctrine. In view of this, I find the conservativeness argument to be worth considering. If nothing else, its validity would convince us that this road is closed; that the attractive property in question is inaccessible to the adherent of the ‘lightness of truth’ doctrine.

    In Chapter 5 the basic problem has been described concerning the strength of truth theories, which – if real – seriously restricts our choice of such theories. The problem is that even if we take for granted that truth permits us to express generalisations, the question still remains about the point of our being able to express them. Why should we want the generalisations to be expressible in our language if we do not have the slightest idea of how to arrive at them – if we are not in a position to reasonably assert them – or, in formal contexts, to prove them?

    As I take it, that is exactly the starting point of the proponents of the conservativeness argument. First, without proving basic generalisations involving truth, our theory makes the predicate pretty useless. Second, weak theories of truth do not fit with the facts of the actual usage of the truth predicate, both in mathematics and in colloquial discourse. The issue is that we do actually formulate and accept some fairly general truth principles. The perspective on truth which does not take this into account would make a lot of our practice incomprehensible.

The Epistemic Lightness of Truth
  • Deflationism and its Logic
  • Cezary Cieśliński
  • Published online:
    30 November 2017
    Print publication:
    07 December 2017
  • This book analyses and defends the deflationist claim that there is nothing deep about our notion of truth. According to this view, truth is a 'light' and innocent concept, devoid of any essence which could be revealed by scientific inquiry. Cezary Cieśliński considers this claim in light of recent formal results on axiomatic truth theories, which are crucial for understanding and evaluating the philosophical thesis of the innocence of truth. Providing an up-to-date discussion and original perspectives on this central and controversial issue, his book will be important for those with a background in logic who are interested in formal truth theories and in current philosophical debates about the deflationary conception of truth.

  • 6 - Reflection

  • Simon M. Huttegger, University of California, Irvine
  • Book:
    The Probabilistic Foundations of Rational Learning
    Published online:
    25 October 2017
    Print publication:
    19 October 2017, pp 126-148

  • Summary

    I have beliefs. I may not be able to describe “the precise reasons” why I hold these beliefs. Even so, the rules implied by coherence provide logical guidance formy expression of these beliefs.My beliefs will change. I may not now be able to describe the precise reasons how or why these changes will occur. Even so, just as withmy beliefs, the rules implied by coherence provide logical guidance for my expression of belief as to how these beliefs will change.

    Michael Goldstein The Prevision of a Prevision

    In the preceding chapter, we saw how central reflection and martingale principles are to a generalized theory of learning. The purpose of this chapter is to examine those principles in more detail. Reflection requires one's future beliefs to cohere with one's prior beliefs. There are many examples of cases where this would not seem advisable.What these examples have in common is that factors other than learning new information have an influence on how beliefs change. If we restrict belief revisions to learning events, the reflection principle can be derived as a requirement of rational belief change in three different ways. The first one, based on dynamic Dutch book arguments, is well known. The other two approaches are new. According to one, reflection principles can be derived within an accuracy framework, in which degrees of belief are regarded as best estimates of truth values. The third approach starts with the value of knowledge theorem of decision theory. The value of knowledge theorem says that you never expect to make worse decisions after having conditioned your prior probabilities on some new information: typically, new information is valuable. The same should be true for learning in general, not just for Bayesian conditioning. Reflection principles are an immediate consequence of requiring that the value of knowledge property holds for generalized learning.

    Probabilities of Future Probabilities

    Recall the black box learning situation introduced in the previous chapter. You are about to change your degree of belief for some proposition A in response to an unspecified learning experience.

The Probabilistic Foundations of Rational Learning
  • Simon M. Huttegger
  • Published online:
    25 October 2017
    Print publication:
    19 October 2017
  • According to Bayesian epistemology, rational learning from experience is consistent learning, that is learning should incorporate new information consistently into one's old system of beliefs. Simon M. Huttegger argues that this core idea can be transferred to situations where the learner's informational inputs are much more limited than Bayesianism assumes, thereby significantly expanding the reach of a Bayesian type of epistemology. What results from this is a unified account of probabilistic learning in the tradition of Richard Jeffrey's 'radical probabilism'. Along the way, Huttegger addresses a number of debates in epistemology and the philosophy of science, including the status of prior probabilities, whether Bayes' rule is the only legitimate form of learning from experience, and whether rational agents can have sustained disagreements. His book will be of interest to students and scholars of epistemology, of game and decision theory, and of cognitive, economic, and computer sciences.

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