The so‐called problem of irrelevant conjunction has been seen as a serious challenge for theories of confirmation. It involves the consequences of conjoining irrelevant statements to a hypothesis that is confirmed by some piece of evidence. Following Hawthorne and Fitelson, we reconstruct the problem with reference to Bayesian confirmation theory. Then we extend it to the case of conjoining irrelevant statements to a hypothesis that is dis confirmed by some piece of evidence. As a consequence, we obtain and formally present (...) a novel and more troublesome problem of irrelevant conjunction. We conclude by indicating a possible solution based on a measure‐sensitive approach and by critically discussing a major alternative way to address the problem. *Received December 2008; revised August 2009. †To contact the authors, please write to: Department of Philosophy, University of Turin, via Sant'Ottavio 20, 10124 Turin, Italy; e‐mail: [email protected] ; [email protected] or [email protected]. (shrink)

I outline four competing probabilistic accounts of contrastive evidential support and consider various considerations that might help arbitrate between these. The upshot of the discussion is that the so-called ‘Law of Likelihood’ is to be preferred to any of the alternatives considered.

I outline four competing probabilistic accounts of contrastive evidential support and consider various considerations that might help arbitrate between these. The upshot of the discussion is that the so-called 'Law of Likelihood' is to be preferred to any of the alternatives considered.

APA PsycNET abstract: This is the first volume of a two-volume work on Probability and Induction. Because the writer holds that probability logic is identical with inductive logic, this work is devoted to philosophical problems concerning the nature of probability and inductive reasoning. The author rejects a statistical frequency basis for probability in favor of a logical relation between two statements or propositions. Probability "is the degree of confirmation of a hypothesis (or conclusion) on the basis of some given evidence (...) (or premises)." Furthermore, all principles and theorems of inductive logic are analytic, and the entire system is to be constructed by means of symbolic logic and semantic methods. This means that the author confines himself to the formalistic procedures of word and symbol systems. The resulting sentence or language structures are presumed to separate off logic from all subjectivist or psychological elements. Despite the abstractionism, the claim is made that if an inductive probability system of logic can be constructed it will have its practical application in mathematical statistics, and in various sciences. 16-page bibliography. (PsycINFO Database Record (c) 2016 APA, all rights reserved). (shrink)

Fitelson (1999) demonstrates that the validity of various arguments within Bayesian confirmation theory depends on which confirmation measure is adopted. The present paper adds to the results set out in Fitelson (1999), expanding on them in two principal respects. First, it considers more confirmation measures. Second, it shows that there are important arguments within Bayesian confirmation theory and that there is no confirmation measure that renders them all valid. Finally, the paper reviews the ramifications that this "strengthened problem of measure (...) sensitivity" has for Bayesian confirmation theory and discusses whether it points at pluralism about notions of confirmation. (shrink)

1 Knowledge and Justification This book is an investigation of one central problem which arises in the attempt to give a philosophical account of empirical ...

In the recent literature on confirmation there are two leading approaches to the provision of a probabilistic measure of the degree to which a hypothesis is confirmed by evidence. The first is to construe the degree to which evidence E confirms hypothesis H as a function that is directly proportional to p and inversely proportional to p . I shall refer to this as the probability approach. The second approach construes the notion as a function that is directly proportional to (...) the true-positive rate – the probability of the evidence if the hypothesis is true, p – and inversely proportional to the false-positive rate – the probability of the evidence if the hypothesis is false, p . These reverse conditional probabilities – of the evidence on the truth or falsehood of the hypothesis – are sometimes known as likelihoods. I shall refer to the approach to confirmation that uses them as the likelihood approach.For each of these approaches, there are two obvious options as to how to define the confirmation function. The first is to define it as the difference between the magnitude that is treated as increasing confirmation or p ) and the magnitude that is treated as decreasing confirmation or p ). The second is to define confirmation as the quotient of these two magnitudes.This yields four different measures of confirmation, represented in the following table: Differences RatiosProbabilities PD = p – p1 PR = Graphic2Likelihoods LD = p – p3 LR = Graphic4All these measures agree on whether or not E provides some support for H. Intuitively, we want to say that E provides some support for H just in case learning that E is true would make us assign a higher probability to H, i.e. when p(H …. (shrink)

It is shown that the probabilistic theories of coherence proposed up to now produce a number of counter-intuitive results. The last section provides some reasons for believing that no probabilistic measure will ever be able to adequately capture coherence. First, there can be no function whose arguments are nothing but tuples of probabilities, and which assigns different values to pairs of propositions {A, B} and {A, C} if A implies both B and C, or their negations, and if P(B)=P(C). But (...) such sets may indeed differ in their degree of coherence. Second, coherence is sensitive to explanatory relations between the propositions in question. Explanation, however, can hardly be captured solely in terms of probability. (shrink)

Zwart and Franssen’s impossibility theorem reveals a conflict between the possible-world-based content-definition and the possible-world-based likeness-definition of verisimilitude. In Sect. 2 we show that the possible-world-based content-definition violates four basic intuitions of Popper’s consequence-based content-account to verisimilitude, and therefore cannot be said to be in the spirit of Popper’s account, although this is the opinion of some prominent authors. In Sect. 3 we argue that in consequence-accounts , content-aspects and likeness-aspects of verisimilitude are not in conflict with each other, but (...) in agreement . We explain this fact by pointing towards the deep difference between possible-world- and the consequence-accounts, which does not lie in the difference between syntactic (object-language) versus semantic (meta-language) formulations, but in the difference between ‘disjunction-of-possible-worlds’ versus ‘conjunction-of-parts’ representations of theories. Drawing on earlier work, we explain in Sect. 4 how the shortcomings of Popper’s original definition can be repaired by what we call the relevant element approach. We propose a quantitative likeness-definition of verisimilitude based on relevant elements which provably agrees with the qualitative relevant content-definition of verisimilitude on all pairs of comparable theories. We conclude the paper with a plea for consequence-accounts and a brief analysis of the problem of language-dependence (Sect. 6). (shrink)

Zwart and Franssen’s impossibility theorem reveals a conflict between the possible-world-based content-definition and the possible-world-based likeness-definition of verisimilitude. In Sect. 2 we show that the possible-world-based content-definition violates four basic intuitions of Popper’s consequence-based content-account to verisimilitude, and therefore cannot be said to be in the spirit of Popper’s account, although this is the opinion of some prominent authors. In Sect. 3 we argue that in consequence-accounts, content-aspects and likeness-aspects of verisimilitude are not in conflict with each other, but in (...) agreement. We explain this fact by pointing towards the deep difference between possible-world- and the consequence-accounts, which does not lie in the difference between syntactic versus semantic formulations, but in the difference between ‘disjunction-of-possible-worlds’ versus ‘conjunction-of-parts’ representations of theories. Drawing on earlier work, we explain in Sect. 4 how the shortcomings of Popper’s original definition can be repaired by what we call the relevant element approach. We propose a quantitative likeness-definition of verisimilitude based on relevant elements which provably agrees with the qualitative relevant content-definition of verisimilitude on all pairs of comparable theories. We conclude the paper with a plea for consequence-accounts and a brief analysis of the problem of language-dependence. (shrink)

One of the integral parts of Bayesian coherentism is the view that the relation of ‘being no less coherent than’ is fully determined by the probabilistic features of the sets of propositions to be ordered. In the last one and a half decades, a variety of probabilistic measures of coherence have been put forward. However, there is large disagreement as to which of these measures best captures the pre-theoretic notion of coherence. This paper contributes to the debate on coherence measures (...) by considering three classes of adequacy constraints. Various independence and dependence relations between the members of each class will be taken into account in order to reveal the ‘grammar’ of probabilistic coherence measures. Afterwards, existing proposals are examined with respect to this list of desiderata. Given that for purely mathematical reasons there can be no measure that satisfies all constraints, the grammar allows the coherentist to articulate an informed pluralist stance as regards probabilistic measures of coherence. (shrink)

This paper presents an outline of a new theory of relevant deduction which arose from the purpose of solving paradoxes in various fields of analytic philosophy. In distinction to relevance logics, this approach does not replace classical logic by a new one, but distinguishes between relevance and validity. It is argued that irrelevant arguments are, although formally valid, nonsensical and even harmful in practical applications. The basic idea is this: a valid deduction is relevant iff no subformula of the conclusion (...) is replaceable on some of its occurrences by any other formula salva validitate of the deduction. The paper first motivates the approach by showing that four paradoxes seemingly very distant from each other have a common source. Then the exact definition of relevant deduction is given and its logical properties are investigated. An extension to relevance of premises is discussed. Finally the paper presents an overview of its applications in philosophy of science, ethics, cognitive psychology and artificial intelligence. (shrink)

The debate on probabilistic measures of coherence flourishes for about 15 years now. Initiated by papers that have been published around the turn of the millennium, many different proposals have since then been put forward. This contribution is partly devoted to a reassessment of extant coherence measures. Focusing on a small number of reasonable adequacy constraints I show that (i) there can be no coherence measure that satisfies all constraints, and that (ii) subsets of these adequacy constraints motivate two different (...) classes of coherence measures. These classes do not coincide with the common distinction between coherence as mutual support and coherence as relative set-theoretic overlap. Finally, I put forward arguments to the effect that for each such class of coherence measures there is an outstanding measure that outperforms all other extant proposals. One of these measures has recently been put forward in the literature, while the other one is based on a novel probabilistic measure of confirmation. (shrink)

The basic theory of scientific understanding presented in Sections 1–2 exploits three main ideas.First, that to understand a phenomenonP (for a given agent) is to be able to fitP into the cognitive background corpusC (of the agent).Second, that to fitP intoC is to connectP with parts ofC (via arguments in a very broad sense) such that the unification ofC increases.Third, that the cognitive changes involved in unification can be treated as sequences of shifts of phenomena inC. How the theory fits (...) typical examples of understanding and how it excludes spurious unifications is explained in detail. Section 3 gives a formal description of the structure of cognitive corpuses which contain descriptive as well as inferential components. The theory of unification is then refined in the light of so called puzzling phenomena, to enable important distinctions, such as that between consonant and dissonant understanding. In Section 4, the refined theory is applied to several examples, among them a case study of the development of the atomic model. The final part contains a classification of kinds of understanding and a discussion of the relation between understanding and explanation. (shrink)

In this paper a theory of finitistic and frequentistic approximations — in short: f-approximations — of probability measures P over a countably infinite outcome space N is developed. The family of subsets of N for which f-approximations converge to a frequency limit forms a pre-Dynkin system $${{D\subseteq\wp(N)}}$$. The limiting probability measure over D can always be extended to a probability measure over $${{\wp(N)}}$$, but this measure is not always σ-additive. We conclude that probability measures can be regarded as idealizations of (...) limiting frequencies if and only if σ-additivity is not assumed as a necessary axiom for probabilities. We prove that σ-additive probability measures can be characterized in terms of so-called canonical and in terms of so-called full f-approximations. We also show that every non-σ-additive probability measure is f-approximable, though neither canonically nor fully f-approximable. Finally, we transfer our results to probability measures on open or closed formulas of first-order languages. (shrink)

According to the comparative Bayesian concept of confirmation, rationalized versions of creationism come out as empirically confirmed. From a scientific viewpoint, however, they are pseudo-explanations because with their help all kinds of experiences are explainable in an ex-post fashion, by way of ad-hoc fitting of an empirically empty theoretical framework to the given evidence. An alternative concept of confirmation that attempts to capture this intuition is the use novelty criterion of confirmation. Serious objections have been raised against this criterion. In (...) this paper I suggest solutions to these objections. Based on them, I develop an account of genuine confirmation that unifies the UN-criterion with a refined probabilistic confirmation concept that is explicated in terms of the confirmation of evidence-transcending content parts of the hypothesis. (shrink)

There is a plethora of confirmation measures in the literature. Zalabardo considers four such measures: PD, PR, LD, and LR. He argues for LR and against each of PD, PR, and LD. First, he argues that PR is the better of the two probability measures. Next, he argues that LR is the better of the two likelihood measures. Finally, he argues that LR is superior to PR. I set aside LD and focus on the trio of PD, PR, and LR. (...) The question I address is whether Zalabardo succeeds in showing that LR is superior to each of PD and PR. I argue that the answer is negative. I also argue, though, that measures such as PD and PR, on one hand, and measures such as LR, on the other hand, are naturally understood as explications of distinct senses of confirmation. (shrink)

I show that the two most devastating objections to Shogenji's formal account of coherence necessarily involve information sets of cardinality . Given this, I surmise that the problem with Shogenji's measure has more to do with his means of generalizing the measure than with the measure itself. I defend this claim by offering an alternative generalization of Shogenji's measure. This alternative retains the intuitive merits of the original measure while avoiding both of the relevant problems that befall it. In the (...) light of all of this, I suggest that there is new hope for Shogenji's analysis: Shogenji's early and influential attempt at measuring coherence, when generalized in a subset-sensitive way, is able to clear its most troubling objections. (shrink)

Coherentism in epistemology has long suffered from lack of formal and quantitative explication of the notion of coherence. One might hope that probabilistic accounts of coherence such as those proposed by Lewis, Shogenji, Olsson, Fitelson, and Bovens and Hartmann will finally help solve this problem. This paper shows, however, that those accounts have a serious common problem: the problem of belief individuation. The coherence degree that each of the accounts assigns to an information set (or the verdict it gives as (...) to whether the set is coherent tout court) depends on how beliefs (or propositions) that represent the set are individuated. Indeed, logically equivalent belief sets that represent the same information set can be given drastically different degrees of coherence. This feature clashes with our natural and reasonable expectation that the coherence degree of a belief set does not change unless the believer adds essentially new information to the set or drops old information from it; or, to put it simply, that the believer cannot raise or lower the degree of coherence by purely logical reasoning. None of the accounts in question can adequately deal with coherence once logical inferences get into the picture. Toward the end of the paper, another notion of coherence that takes into account not only the contents but also the origins (or sources) of the relevant beliefs is considered. It is argued that this notion of coherence is of dubious significance, and that it does not help solve the problem of belief individuation. (shrink)

Plausibly, when we adopt a probabilistic standpoint any measure Cb of the degree to which evidence e confirms hypothesis h relative to background knowledge b should meet these five desiderata: Cb > 0 when P > P < 0 when P < P; Cb = 0 when P = P. Cb is some function of the values P and P assume on the at most sixteen truth-functional combinations of e and h. If P < P and P = P then (...) Cb ≤ Cb; if P = P and P < P then Cb ≥ Cb. Cb – Cb is fully determined by Cb and Cbe – Cbe; if Cb = 0 then Cb + Cbe = 0. If P = P then Cb = Cb. (shrink)

In this paper I consider whether there is a measure of coherence that could be rightly claimed to generalize the notion of logical equivalence. I show that Fitelson’s (2003) proposal to that effect encounters some serious difficulties. Furthermore, there is reason to believe that no mutual-support measure could ever be suitable for the formalization of coherence as generalized logical equivalence. Instead, it appears that the only plausible candidate for such a measure is one of relative overlap. The measure I propose (...) in this paper is quite similar to Olsson’s (2002) proposal but differs from it by not being susceptible to the type of counterexample that Bovens and Hartmann (2003) have devised against it. (shrink)

Surprisingly, modified versions of the confirmation theory (Carnap and Hempel) and truth approximation theory (Popper) turn out to be smoothly sythesizable. The glue between the two appears to be the instrumentalist methodology, rather than that of the falsificationalist. The instrumentalist methodology, used in the separate, comparative evaluation of theories in terms of their successes and problems (hence, even if already falsified), provides in theory and practice the straight road to short-term empirical progress in science ( à la Laudan). It is (...) also argued that such progress is also functional for all kinds of truth approximation: observational, referential, and theoretical. This sheds new light on the long-term dynamics of science and hence on the relation between the main epistemological positions, viz., instrumentalism (Toulmin, Laudan), constructive empiricism (Van Fraassen), referential realism (Hacking, Cartwright), and theory realism of a non-essentialist nature (constructive realism à la Popper). Readership: Open minded philosophers and scientists. The book explains and justifies the scientist's intuition that the debate among philosophers about instrumentalism and realism has almost no practical consequences. (shrink)

Coherence is the property of propositions hanging or fitting together. Intuitively, adding a proposition to a set of propositions should be compatible with either increasing or decreasing the set’s degree of coherence. In this paper we show that probabilistic coherence measures based on relative overlap are in conflict with this intuitive verdict. More precisely, we prove that according to the naive overlap measure it is impossible to increase a set’s degree of coherence by adding propositions and that according to the (...) refined overlap measure no set’s degree of coherence exceeds the degree of coherence of its maximally coherent subset. We also show that this result carries over to all other subset-sensitive refinements of the naive overlap measure. As both results stand in sharp contrast to elementary coherence intuitions, we conclude that extant relative overlap measures of coherence are inadequate. (shrink)

Naive deductivist accounts of confirmation have the undesirable consequence that if E confirms H, then E also confirms the conjunction H·X, for any X—even if X is completely irrelevant to E and H. Bayesian accounts of confirmation may appear to have the same problem. In a recent article in this journal Fitelson (2002) argued that existing Bayesian attempts to resolve of this problem are inadequate in several important respects. Fitelson then proposes a new‐and‐improved Bayesian account that overcomes the problem of (...) irrelevant conjunction, and does so in a more general setting than past attempts. We will show how to simplify and improve upon Fitelson's solution. (shrink)

In Gemes (1990) I examined certain formal versions of hypothetico-deductivism (H-D) showing that they have the unacceptable consequence that "Abe is a white raven" confirms "All ravens are black"! In Gemes (1992) I developed a new notion of content that could save H-D from this bizarre consequence. In this paper, I argue that more traditional formulations of H-D also need recourse to this new notion of content. I present a new account of the vexing notion of the natural axiomatization of (...) a theory. The notion is used to construct a form of H-D that allows for the type of selective confirmation without which Glymour (1980a,b) claims H-D is hopeless. (shrink)

Highly idealized models, such as the Hawk-Dove game, are pervasive in biological theorizing. We argue that the process and motivation that leads to the introduction of various idealizations into these models is not adequately captured by Michael Weisberg’s taxonomy of three kinds of idealization. Consequently, a fourth kind of idealization is required, which we call hypothetical pattern idealization. This kind of idealization is used to construct models that aim to be explanatory but do not aim to be explanations.

Contemporary Bayesian confirmation theorists measure degree of (incremental) confirmation using a variety of non-equivalent relevance measures. As a result, a great many of the arguments surrounding quantitative Bayesian confirmation theory are implicitly sensitive to choice of measure of confirmation. Such arguments are enthymematic, since they tacitly presuppose that certain relevance measures should be used (for various purposes) rather than other relevance measures that have been proposed and defended in the philosophical literature. I present a survey of this pervasive class of (...) Bayesian confirmation-theoretic enthymemes, and a brief analysis of some recent attempts to resolve the problem of measure sensitivity. (shrink)

Naive deductive accounts of confirmation have the undesirable consequence that if E confirms H, then E also confirms the conjunction H & X, for any X—even if X is utterly irrelevant to H (and E). Bayesian accounts of confirmation also have this property (in the case of deductive evidence). Several Bayesians have attempted to soften the impact of this fact by arguing that—according to Bayesian accounts of confirmation— E will confirm the conjunction H & X less strongly than E confirms (...) H (again, in the case of deductive evidence). I argue that existing Bayesian “resolutions” of this problem are inadequate in several important respects. In the end, I suggest a new‐and‐improved Bayesian account (and understanding) of the problem of irrelevant conjunction. (shrink)

Likelihoodists and Bayesians seem to have a fundamental disagreement about the proper probabilistic explication of relational (or contrastive) conceptions of evidential support (or confirmation). In this paper, I will survey some recent arguments and results in this area, with an eye toward pinpointing the nexus of the dispute. This will lead, first, to an important shift in the way the debate has been couched, and, second, to an alternative explication of relational support, which is in some sense a "middle way" (...) between Likelihoodism and Bayesianism. In the process, I will propose some new work for an old probability puzzle: the "Monty Hall" problem. (shrink)

Let E be a set of n propositions E1, ..., En. We seek a probabilistic measure C(E) of the ‘degree of coherence’ of E. Intuitively, we want C to be a quantitative, probabilistic generalization of the (deductive) logical coherence of E. So, in particular, we require C to satisfy the following..

Confirmation of a hypothesis by evidence can be measured by one of the so far known incremental measures of confirmation. As we show, incremental measures can be formally defined as the measures of confirmation satisfying a certain small set of basic conditions. Moreover, several kinds of incremental measure may be characterized on the basis of appropriate structural properties. In particular, we focus on the so-called Matthew properties: we introduce a family of six Matthew properties including the reverse Matthew effect; we (...) further prove that incremental measures endowed with reverse Matthew effect are possible; finally, we shortly consider the problem of the plausibility of Matthew properties. (shrink)

There is currently no viable alternative to the Bayesian analysis of scientific inference, yet the available versions of Bayesianism fail to do justice to several aspects of the testing and confirmation of scientific hypotheses. Bayes or Bust? provides the first balanced treatment of the complex set of issues involved in this nagging conundrum in the philosophy of science. Both Bayesians and anti-Bayesians will find a wealth of new insights on topics ranging from Bayes’s original paper to contemporary formal learning theory.In (...) a paper published posthumously in 1763, the Reverend Thomas Bayes made a seminal contribution to the understanding of "analogical or inductive reasoning." Building on his insights, modem Bayesians have developed an account of scientific inference that has attracted numerous champions as well as numerous detractors. Earman argues that Bayesianism provides the best hope for a comprehensive and unified account of scientific inference, yet the presently available versions of Bayesianisin fail to do justice to several aspects of the testing and confirming of scientific theories and hypotheses. By focusing on the need for a resolution to this impasse, Earman sharpens the issues on which a resolution turns. John Earman is Professor of History and Philosophy of Science at the University of Pittsburgh. (shrink)

This paper aims to contribute to our understanding of the notion of coherence by explicating in probabilistic terms, step by step, what seem to be our most basic intuitions about that notion, to wit, that coherence is a matter of hanging or fitting together, and that coherence is a matter of degree. A qualitative theory of coherence will serve as a stepping stone to formulate a set of quantitative measures of coherence, each of which seems to capture well the aforementioned (...) intuitions. Subsequently it will be argued that one of those measures does better than the others in light of some more specific intuitions about coherence. This measure will be defended against two seemingly obvious objections. (shrink)