The dialectics of infinitism and coherentism: inferential justification versus holism and coherence

This paper formally explores the common ground between mild versions of epistemological coherentism and infinitism; it proposes—and argues for—a hybrid, coherentist–infinitist account of epistemic justification. First, the epistemological regress argument and its relation to the classical taxonomy regarding epistemic justification—of foundationalism, infinitism and coherentism—is reviewed. We then recall recent results proving that an influential argument against infinite regresses of justification, which alleges their incoherence on account of probabilistic inconsistency, cannot be maintained. Furthermore, we prove that the Principle of Inferential Justification has rather unwelcome consequences—formally resembling the Sorites paradox—as soon as it is iterated and combined with a natural Bayesian perspective on probabilistic inferences. We conclude that strong versions of foundationalism and infinitism should be abandoned. Positively, we provide a rough sketch for a graded formal coherence notion, according to which infinite regresses of epistemic justification will often have more than a minimal degree of coherence.     

Mental models and probabilistic thinking

This paper outlines the theory of reasoning based on mental models, and then shows how this theory might be extended to deal with probabilistic thinking. The same explanatory framework accommodates deduction and induction: there are both deductive and inductive inferences that yield probabilistic conclusions. The framework yields a theoretical conception of strength of inference, that is, a theory of what the strength of an inference is objectively: it equals the proportion of possible states of affairs consistent with the premises in which the conclusion is true, that is, the probability that the conclusion is true given that the premises are true. Since there are infinitely many possible states of affairs consistent with any set of premises, the paper then characterizes how individuals estimate the strength of an argument. They construct mental models, which each correspond to an infinite set of possibilities (or, in some cases, a finite set of infinite sets of possibilities). The construction of models is guided by knowledge and beliefs, including lay conceptions of such matters as the “law of large numbers”. The paper illustrates how this theory can account for phenomena of probabilistic reasoning.     

The Solvability of Probabilistic Regresses. A Reply to Frederik Herzberg

We have earlier shown by construction that a proposition can have a welldefined nonzero probability, even if it is justified by an infinite probabilistic regress. We thought this to be an adequate rebuttal of foundationalist claims that probabilistic regresses must lead either to an indeterminate, or to a determinate but zero probability. In a comment, Frederik Herzberg has argued that our counterexamples are of a special kind, being what he calls ‘solvable’. In the present reaction we investigate what Herzberg means by solvability. We discuss the advantages and disadvantages of making solvability a sine qua non, and we ventilate our misgivings about Herzberg’s suggestion that the notion of solvability might help the foundationalist.     

Infinitism and probabilistic justification

According to infinitism, beliefs can be justified by an infinite chain of reasons. So far, infinitism has rarely been taken seriously and often even dismissed as inconsistent. However, Peijnenburg and Atkinson have recently argued that beliefs can indeed be justified by an infinite chain of reasons, if justification is understood probabilistically. In the following, I will discuss the formal result that has led to this conclusion. I will then introduce three probabilistic explications of justification and examine to which extent they support Peijnenburg’s and Atkinson’s claim.     

The covering law model of historical explanation

It is often argued (as by Hempel and Nagel) that genuine historical explanations — if these are to be had — must exhibit a connection between events to be explained and universal or probabilistic laws (or ‘hypotheses'). This connection may take either a ‘strong’ or ‘weak’ form. The historian may show that a statement of the event to be explained is a logical consequence of statements of reasonably well‐confirmed universal laws and occurrences linked by the laws to the event to be explained. Or the historian may show that a statement of the event to be explained has high inductive probability conferred upon it given statements of reasonably well‐confirmed probabilistic laws and occurrences so linked by the laws to the type of event to be explained that one finds the occurrence of the particular event likely. This essay focuses on ‘strong’ explanations which meet a ‘deducibility’ requirement (for reasons given in the body of the article). It is argued that explanations in history (at least where it is plausible to construe them as ‘non‐rational') may meet a ‘deducibility’ requirement and count as genuine historical explanations although they do not meet a ‘covering law’ requirement (i. e. none of the premises of these explanations state universal or probabilistic hypotheses). It is required, however, that at least one premise in such explanations assert a reasonably well‐confirmed condition (e. g., a co‐variation) which can be taken as a sign or indication of the presence of laws. Rather than appealing to laws, the historian may appeal to the well‐founded possibility of laws.     

Wittgenstein on Set Theory and the Enormously Big

Wittgenstein's conception of infinity can be seen as continuing the tradition of the potential infinite that begins with Aristotle. Transfinite cardinals in set theory might seem to render the potential infinite defunct with the actual infinite now given mathematical legitimacy. But Wittgenstein's remarks on set theory argue that the philosophical notion of the actual infinite remains philosophical and is not given a mathematical status as a result of set theory. The philosophical notion of the actual infinite is not to be found in the mathematics of set theory, only in a certain associated philosophy – what Wittgenstein calls a certain kind of “prose”.     

Challenging epistemology: Interactive proofs and zero knowledge

This essay explores what (if anything) research on interactive zero knowledge proofs has to teach philosophers about the epistemology of mathematics and theoretical computer science. Though such proof systems initially appear ‘revolutionary’ and are a nonstandard conception of ‘proof’, I will argue that they do not have much philosophical import. Possible lessons from this work for the epistemology of mathematics—our models of mathematical proof should incorporate interaction, our theories of mathematical evidence must account for probabilistic evidence, our valuation of a mathematical proof should solely focus on its persuasive power—are either misguided or old hat. And while the differences between interactive and mathematical proofs suggest the need to develop a separate epistemology of theoretical computer science (or at least complexity theory) that differs from our theory of mathematical knowledge, a casual look at the actual practice of complexity theory indicates that such a distinct epistemology may not be necessary.     

Probabilistic Justification and the Regress Problem

We discuss two objections that foundationalists have raised against infinite chains of probabilistic justification. We demonstrate that neither of the objections can be maintained. Presented by Hannes Leitgeb     

The Need for Justification

Some series can go on indefinitely, others cannot, and epistemologists want to know in which class to place epistemic chains. Is it sensible or nonsensical to speak of a proposition or belief that is justified by another proposition or belief, ad infinitum? In large part the answer depends on what we mean by “justification.” Epistemologists have failed to find a definition on which everybody agrees, and some have even advised us to stop looking altogether. In spite of this, the present essay submits a few candidate definitions. It argues that, although not giving the final word, these candidates tell us something about the possibility of infinite epistemic chains. And it shows that they can short‐circuit a debate about doxastic justification.     

The ability to look into the future (Probabilistic Prognosis)

“The Ability to Look into the Future (Probabilistic Prognosis)” is a translation of chapter five of I. M. Feigenberg’sBrain, Mind and Health [Mozg, Psikhika, Zdorov’e], published by Nauka, Moscow, in 1972. The book deals with the psychophysiology of perception, affect, and memory, as well as certain psychopathological phenomena. The main theme of the book and the author’s research is probabilistic prognosis—the prediction of future events on the basis of the probabilistic structure of the past as stored in memory. In the present essay, Feigenberg develops the concept of probabilistic prognosis in an evolutionary context, linking it with the Pavlovian conditional reflex and orienting reactions, and illustrating how the theory can be applied to both animal and human behavior.     

归纳接受与知识

归纳推理的结论进入知识集合的时候有一个“跳跃”的过程。根据什么规则接受归纳结论是归纳逻辑研究中的一个重要问题。概率接受规则与知识集合应当满足的一致性条件和演绎封闭条件之间存在着不协调,这种不协调性导致了抽彩悖论。该文介绍有代表性的归纳接受理论,指出归纳接受问题对归纳逻辑发展的影响。     

Are Mathematical Theories Reducible to Non-analytic Foundations?

In this article I intend to show that certain aspects of the axiomatical structure of mathematical theories can be, by a phenomenologically motivated approach, reduced to two distinct types of idealization, the first-level idealization associated with the concrete intuition of the objects of mathematical theories as discrete, finite sign-configurations and the second-level idealization associated with the intuition of infinite mathematical objects as extensions over constituted temporality. This is the main standpoint from which I review Cantor’s conception of infinite cardinalities and also the metatheoretical content of some later well-known theorems of mathematical foundations. These are, the Skolem-Löwenheim Theorem which, except for its importance as such, it is also chosen for an interpretation of the associated metatheoretical paradox (Skolem Paradox), and Gödel’s (first) incompleteness result which, notwithstanding its obvious influence in the mathematical foundations, is still open to philosophical inquiry. On the phenomenological level, first-level and second-level idealizations, as above, are associated respectively with intentional acts carried out in actual present and with certain modes of a temporal constitution process.     

Badiou,Priest, and the Hegelian Infinite

Abstract

Hegel’s distinction between the bad and true infinites has provoked contrasting reactions in the works of Alain Badiou and Graham Priest. Badiou claims that Hegel illegitimately attempts to impose a distinction that is only relevant to the qualitative realm onto the quantitative realm. He suggests that Cantor’s mathematical account of infinite multiplicities that are determinate and actual remains an endlessly proliferating bad infinite when placed within Hegel’s faulty schema. In contrast, Priest affirms the Hegelian true infinite, claiming that Cantor’s formal mechanisms of boundary transcendence, such as ‘diagonalization’, are implicit in Hegel’s dialectic. While arguing that a clear dividing line can be drawn here between these two interpretations of the relationship between Hegel and Cantor, this paper also mounts a defence of the Hegelian true infinite by developing Priest’s suggestion that Cantorian diagonalizing functions are prefigured by Hegel’s dialectical overcoming of limits.     

Infinity and a Critical View of Logic

Abstract

The paper explores the view that in mathematics, in particular where the infinite is involved, the application of classical logic to statements involving the infinite cannot be taken for granted. L. E. J. Brouwer’s well-known rejection of classical logic is sketched, and the views of David Hilbert and especially Hermann Weyl, both of whom used classical logic in their mathematical practice, are explored. We inquire whether arguments for a critical view can be found that are independent of constructivist premises and consider the entanglement of logic and mathematics. This offers a convincing case regarding second-order logic, but for first-order logic, it is not so clear. Still, we ask whether we understand the application of logic to the higher infinite better than we understand the higher infinite itself.     

"不可或缺性论证"与反实在论数学哲学

The indispensability argument for abstract mathematical entities has been an important issue in the philosophy of mathematics. The argument relies on several assumptions. Some objections have been made against these assumptions, but there are several serious defects in these objections. Ameliorating these defects leads to a new anti-realistic philosophy of mathematics, mainly: first, in mathematical applications, what really exist and can be used as tools are not abstract mathematical entities, but our inner representations that we create in imagining abstract mathematical entities; second, the thoughts that we create in imagining infinite mathematical entities are bounded by external conditions. __________ Translated from Zhexue Yanjiu 哲学研究 (Philosophical Researches), 2006, (8): 74–83     

Conditional Probability and Defeasible Inference

We offer a probabilistic model of rational consequence relations (Lehmann and Magidor, 1990) by appealing to the extension of the classical Ramsey–Adams test proposed by Vann McGee in (McGee, 1994). Previous and influential models of non-monotonic consequence relations have been produced in terms of the dynamics of expectations (Gärdenfors and Makinson, 1994; Gärdenfors, 1993).Expectation is a term of art in these models, which should not be confused with the notion of expected utility. The expectations of an agent are some form of belief weaker than absolute certainty. Our model offers a modified and extended version of an account of qualitative belief in terms of conditional probability, first presented in (van Fraassen, 1995). We use this model to relate probabilistic and qualitative models of non-monotonic relations in terms of expectations. In doing so we propose a probabilistic model of the notion of expectation. We provide characterization results both for logically finite languages and for logically infinite, but countable, languages. The latter case shows the relevance of the axiom of countable additivity for our probability functions. We show that a rational logic defined over a logically infinite language can only be fully characterized in terms of finitely additive conditional probability. The research of both authors was supported in part by a grant from NSF, and, for Parikh, also by support from the research foundation of CUNY.     

Grammaticality,Acceptability, and Probability: A Probabilistic View of Linguistic Knowledge

The question of whether humans represent grammatical knowledge as a binary condition on membership in a set of well‐formed sentences, or as a probabilistic property has been the subject of debate among linguists, psychologists, and cognitive scientists for many decades. Acceptability judgments present a serious problem for both classical binary and probabilistic theories of grammaticality. These judgements are gradient in nature, and so cannot be directly accommodated in a binary formal grammar. However, it is also not possible to simply reduce acceptability to probability. The acceptability of a sentence is not the same as the likelihood of its occurrence, which is, in part, determined by factors like sentence length and lexical frequency. In this paper, we present the results of a set of large‐scale experiments using crowd‐sourced acceptability judgments that demonstrate gradience to be a pervasive feature in acceptability judgments. We then show how one can predict acceptability judgments on the basis of probability by augmenting probabilistic language models with an acceptability measure. This is a function that normalizes probability values to eliminate the confounding factors of length and lexical frequency. We describe a sequence of modeling experiments with unsupervised language models drawn from state‐of‐the‐art machine learning methods in natural language processing. Several of these models achieve very encouraging levels of accuracy in the acceptability prediction task, as measured by the correlation between the acceptability measure scores and mean human acceptability values. We consider the relevance of these results to the debate on the nature of grammatical competence, and we argue that they support the view that linguistic knowledge can be intrinsically probabilistic.     

Effects of inflation on the subjective value of delayed and probabilistic rewards

In the years prior to 1994, there were very high rates of inflation in Poland, and the zloty depreciated relative to the U.S. dollar. However, the new zloty, introduced in 1995, was associated with greatly decreased rates of inflation and provided a more stable currency. We report a series of three experiments that take advantage of these changes to examine the effects of inflation on the subjective value of delayed and probabilistic rewards. Subjects were Polish citizens familiar with both zlotys and dollars. The first two experiments, conducted in 1994, used dollars and old zlotys, and the third experiment, conducted in 1996, used dollars and new zlotys. In all three experiments, the dollar and zloty rewards were of equivalent worth, according to the then current exchange rates. In Experiment 1, subjects chose between immediate and delayed rewards and, in Experiment 2, chose between certain and probabilistic rewards. The subjective value of a delayed reward was greater when its amount was specified in dollars than when it was specified in old zlotys. In contrast, the currency in which a reward was specified had no effect on the subjective value of probabilistic rewards. The results of these two experiments suggest a selective effect of inflation on decisions involving delayed rewards. This was verified in the third experiment, in which, using new zlotys, no differences in discounting were observed between the two currencies with either probabilistic or delayed rewards. Importantly, in all three experiments, the discounting of both delayed and probabilistic rewards was well described by the same simple mathematical model, suggesting that similar decision-making processes underlie both phenomena. However, the present results argue against a single-process theory in which the discounting of probabilistic rewards is derived from the discounting of delayed rewards.     

THE UNIVERSALITY OF JEWISH ETHICS: A Rejoinder to Secularist Critics

Jewish ethics like Judaism itself has often been charged with being "particularistic," and in modernity it has been unfavorably compared with the universality of secular ethics. This charge has become acute philosophically when the comparison is made with the ethics of Kant. However, at this level, much of the ethical rejection of Jewish particularism, especially its being beholden to a God who is above the universe to whom this God prescribes moral norms and judges according to them, is also a rejection of Christian (or any other monotheistic) ethics, no matter how otherwise universal. Yet this essay argues that Jewish ethics that prescribes norms for all humans, and that is knowable by all humans, actually constitutes a wider moral universe than does Kantian ethics, because it can include non-rational human objects and even non-human objects altogether. This essay also argues that a totally egalitarian moral universe, encompassing all human relations, becomes an infinite, totalizing universe, which can easily become the ideological justification ( ratio essendi ) of a totalitarian regime.     

The Kalām Cosmological Argument and the Infinite God Objection

In this article, we evaluate various responses to a noteworthy objection, namely, the infinite God objection to the kalām cosmological argument. As regards this objection, the proponents of the kalām argument face a dilemma—either an actual infinite cannot exist or God cannot be infinite. More precisely, this objection claims that God’s omniscience entails the existence of an actual infinite with God knowing an actually infinite number of future events or abstract objects, such as mathematical truths. We argue, however, that the infinite God objection is based on two questionable assumptions, namely, (1) that it is possible for an omniscient being to know an actually infinite number of things and (2) that there exist an actually infinite number of abstract objects for God to know.