Logical Probability

This is part II in a series of papers outlining Abstraction Theory, a theory that I propose provides a solution to the characterisation or epistemological problem of induction. Logic is built from first principles severed from language such that there is one universal logic independent of specific logical languages. A theory of (non-linguistic) meaning is developed which provides the basis for the dissolution of the `grue' problem and problems of the non-uniqueness of probabilities in inductive logics. The problem of counterfactual (...) conditionals is generalised to a problem of truth conditions of hypotheses and this general problem is then solved by the notion of abstractions. The probability calculus is developed with examples given. In future parts of the series the full decision theory is developed and its properties explored. (shrink)

Epistemic modals have peculiar logical features that are challenging to account for in a broadly classical framework. For instance, while a sentence of the form ‘p, but it might be that not p’ appears to be a contradiction, 'might not p' does not entail 'not p', which would follow in classical logic. Likewise, the classical laws of distributivity and disjunctive syllogism fail for epistemic modals. Existing attempts to account for these facts generally either under- or over-correct. Some theories predict that (...) 'p and might not p', a so-called epistemic contradiction, is a contradiction only in an etiolated sense, under a notion of entailment that does not allow substitution of logical equivalents; these theories underpredict the infelicity of embedded epistemic contradictions. Other theories savage classical logic, eliminating not just rules that intuitively fail, like distributivity and disjunctive syllogism, but also rules like non-contradiction, excluded middle, De Morgan’s laws, and disjunction introduction, which intuitively remain valid for epistemic modals. In this paper, we aim for a middle ground, developing a semantics and logic for epistemic modals that makes epistemic contradictions genuine contradictions and that invalidates distributivity and disjunctive syllogism but that otherwise preserves classical laws that intuitively remain valid. We start with an algebraic semantics, based on ortholattices instead of Boolean algebras, and then propose a more concrete possibility semantics, based on partial possibilities related by compatibility. Both semantics yield the same consequence relation, which we axiomatize. Then we show how to extend our semantics to explain parallel phenomena involving probabilities and conditionals. The goal throughout is to retain what is desirable about classical logic while accounting for the non-classicality of epistemic vocabulary. (shrink)

We prove that quantum mechanics is the unique theory compatible with normalization- induced curvature on complex Hilbert space. Two rigidity theorems establish: (1) Lin- earity: continuous, reversible, scale-invariant flows necessarily have linear generators, and (2) Born rule: the assignment P (ϕ|ψ) = |⟨ϕ, ψ⟩|2 is the unique probability mea- sure compatible with cone geometry, unitary invariance, and orthogonal additivity. These are not interpretations but uniqueness results—no other mathematical struc- tures are possible given the geometric assumptions. We derive the Schr¨odinger equa- (...) tion, interference, Berry phase, and uncertainty relations as consequences of the master equation CurvC(X, Y ) = [X, Y ] − ⟨∇N,[X,Y ]⟩ ⟨∇N,x⟩ x applied to the quantum cone. The re- construction inverts standard quantum foundations: rather than postulating Hilbert space structure, we show it emerges from reversible dynamics on normalized states. (shrink)

Choices rarely deal with certainties; and, where assertoric logic and modal logic are insufficient, those seeking to be reasonable turn to one or more things called “probability.” These things typically have a shared mathematical form, which is an arithmetic construct. The construct is often felt to be unsatisfactory for various reasons. A more general construct is that of a preordering, which may even be incomplete, allowing for cases in which there is no known probability relation between two propositions or between (...) two events. Previous discussion of incomplete preorderings has been as if incidental, with researchers focusing upon preorderings for which quantifications are possible. This article presents formal axioms for the more general case. Challenges peculiar to some specific interpretations of the nature of probability are brought to light in the context of these propositions. A qualitative interpretation is offered for probability differences that are often taken to be quantified. A generalization of Bayesian updating is defended without dependence upon coherence. Qualitative hypothesis testing is offered as a possible alternative in cases for which quantitative hypothesis testing is shown to be unsuitable. [NB: There is a transcription error in (A6), the quantification “∀(Y,c₂)” should be applied within the large parentheses to the proposition on the third row. You have my apologies!]. (shrink)

This paper argues that Maximum Likelihood Estimation (MLE) is the wrong approach for parameter estimation, both conceptually and in its results. We propose a new estimation method, called PLE, that offers four benefits over MLE: 1) PLE produces less biased estimates of the true parameters 2) PLE reduces overfitting 3) PLE more fairly represents minority (low-frequency) data and 4) PLE is more resilient to MAD collapse. We show how these benefits result from PLE's incorporation of counterfactual samples.

John Maynard Keynes’s concept of the weight of evidence is one of the most enduring—and mysterious—features of Keynes’s epistemology. One popular interpretation of Keynesian weight comes courtesy of Jochen Runde, who argues that weight is best construed as the ratio of the amount of known relevant evidence to the amount of total relevant information. Runde’s interpretation aims to establish a strong link between Keynesian weight and rational confidence, which in turn is meant to bolster the connection between Keynesian weight of (...) evidence and Keynes’s later economic work. Despite this advantage, Runde’s interpretation also implies the existence of significant inconsistencies in Keynes’s account of weight of evidence. As a consequence, Runde’s interpretation paints Keynes as particularly confused about Keynesian weight of evidence. In this work, I provide a distinction between estimated Keynesian weight and actual Keynesian weight in order to preserve the desirable features of Runde’s interpretation while simultaneously resolving the textual inconsistencies that arise from that interpretation. The upshot is a novel interpretation that makes Keynesian weight less ambiguous, more consistent with the rest of Keynes’s epistemology, and better-situated for future research. (shrink)

As the geometers of old reduced the heavens to conic sections and epicycles, so do we now reduce probability to the projection of curved geometries upon the manifold of observation.” — The Author The ancients believed that chance was the domain of the gods, beyond human comprehension. The moderns declared probability an axiom, a primitive concept admitting no further reduction. We propose a third path: that probability is de- rived geometry, the inevitable consequence of dimensional collapse under the coarea formula. (...) When a system of higher dimension projects onto a space of observation, the induced density obeys an iron law—not of randomness, but of geometric neces- sity. This density is the exponential of an effective Hamiltonian, which unifies two contributions of equal status: The intrinsic curvature of the hidden dimensions, and The logarithmic cost of projection itself. That geometry should act as energy, and projection as thermodynamic partition, is no mere analogy but mathematical identity. The Jacobian of projection enters the exponent on precisely the same footing as potential energy or field curvature. Herein we demonstrate three grades of truth: First, that probability densities are partition functions over fibers, governed by an effective Hamiltonian combining curvature and geometric distortion. Second, that curvature—whether of connections or of spaces—remains invisible to first derivatives and reveals itself exclusively through second-order comparison, yielding the universal acceleration law that unifies all forces. Third, that for constant curvature, the functional form of probability is rigid: bounded normalized observables must evolve as the square of the cosine, this being the unique solution compatible with geometric constraints. These are not hypotheses to be tested, but theorems to be proven. We do not conjecture that the Born rule arises from geometry; we derive it from the Hopf vii viii Principia Probabilitatis fibration and the coarea formula. We do not speculate that forces are projection artifacts; we demonstrate it through geodesic deviation in fiber bundles. The philosophy that animates this work is simple: What appears random is often necessary, viewed from insufficient dimension. Probability is the shadow cast by deterministic geometry when hidden dimensions are integrated out. Uncertainty is dimensional inaccessibility, not ontological indeterminism. We write in the manner of the ancients—with axioms, propositions, and scholia— not from affectation but from conviction that geometric truth deserves geometric presentation. Let the reader judge whether we have succeeded in revealing the necessity beneath probability’s veil. (shrink)

This book investigates the interplay between two new and influential subdisciplines in the philosophy of science and philosophy: contemporary scientific metaphysics and the philosophy of information. Scientific metaphysics embodies various scientific realisms and has a partial intellectual heritage in some forms of neo-positivism, but is far more attuned than the latter to statistical science, theory defeasibility, scale variability, and pluralist ontological and explanatory commitments, and is averse to a-priori conceptual analysis. The philosophy of information is the combination of what has (...) been called the informational turn in philosophy and ontology, with the informational turn in logic. The book explores the intersecting theoretical commitments and metaphysical basis of both. It applies scientific metaphysics to the philosophy of information, and also correspondingly proposes a new informational interpretation of scientific metaphysics. These are applied to numerous outstanding ontological, philosophical, and theoretical problems in the philosophy of information, scientific realism, cognitive science, theories of semantic information, and informational logic. The book investigates known and new problems, and advances debates and insights in the philosophy of information. It will be of interest to philosophers of information and science, metaphysicians, and cognitive scientists. (shrink)

本稿は応答可能性を振幅構造として定式化する理論枠組みを提示する。従来の評価理論では、評価対象は単一値あるいは確率分布として扱われ、評価は写像 Eval : M(V) → D として定式化されてきた。しかし確率分布は可能性の統計的強度を与えるが、位相関係や干渉構造を表現することはできない。本稿はこの点に着目し、確率分布をより基礎的な可能性構造から誘導される統計的表現として再 解釈する。 そのため応答空間 V 上の複素値関数 ψ : V → ℂ を応答振幅として導入する。振幅 ψ からはボルン写像 μ(v) = |ψ(v)|² により応答分布 μ ∈ M(V) が誘導される。振幅空間はヒルベルト空間 H = L²(V) を形成し、振幅の線形合成には干渉項 2Re(ψ₁\overline{ψ₂}) が現れることにより、確率分布では表現できない可能性間の干渉構造が記述される。さらに生成経路集合 Σ 上の振幅 α : Σ → ℂ を導入し、応答振幅が生成経路振幅の線形合成として構成されることを示す。 この枠組みにおいて応答分布は振幅構造から誘導される統計的表現となり、従来の分布評価 Eval : M(V) → D はボルン写像 B : H → M(V) を通じて振幅空間上の写像 Eval ∘ B : H → D として位置づけられる。これにより本理論は 振幅 → 分布 → 評価 という階層構造を与え、確率分布に基づく従来の評価理論を拡張する。 本稿は、前作（2026）で提示した生成構造における評価段階に先立つ応答可能性の構造を振幅として定式化する。.

The unique and continuous nature of subjective experience—the persistent "I" that unifies a lifetime of perceptions—represents the most profound unsolved problem in science. While modern neuroscience has made substantial progress in identifying the neural correlates of conscious states, leading theories, including the Global Neuronal Workspace Theory (GNWT), Integrated Information Theory (IIT), and the Orchestrated Objective Reduction (Orch-OR) model, provide mechanistic accounts of awareness but fail to explain the unicity and temporal persistence of the self. These theories cannot logically resolve scenarios (...) of physical disruption or perfect replication. Here, we introduce a fundamental theoretical framework positing that the unique, streaming character of consciousness necessitates a physical principle beyond emergent neural computation. Through a series of rigorous thought experiments—brain revival, molecular disassembly/reassembly, and synthetic replication—we demonstrate the categorical failure of existing models to account for the self. We propose that a unique, non-local identity factor, potentially operating on principles extending current quantum physics, binds to the specific magnetic field topologies of a complex neural system. This "Consciousness Identity Factor" (CIF) hypothesis provides a parsimonious resolution to the duplication paradox and the problem of personal identity. Our framework is fully compatible with established neuroscience while generating specific, falsifiable predictions for studies of anesthesia, brain revival, meditation, and quantum biology. This theory bridges the explanatory gap between the objective brain and subjective identity, proposing a new physical paradigm for the most intimate fact of our existence. (shrink)

Reasoning from inconclusive evidence, or 'induction', is central to science and any applications we make of it. For that reason alone it demands the attention of philosophers of science. This element explores the prospects of using probability theory to provide an inductive logic: a framework for representing evidential support. Constraints on the ideal evaluation of hypotheses suggest that the overall standing of a hypothesis is represented by its probability in light of the total evidence, and incremental support, or confirmation, indicated (...) by the hypothesis having a higher probability conditional on some evidence than it does unconditionally. This proposal is shown to have the capacity to reconstruct many canons of the scientific method and inductive inference. Along the way, significant objections are discussed, such as the challenge of inductive scepticism, and the objection that the probabilistic approach makes evidential support arbitrary. (shrink)

These interlinking essays are connected by a core theme: the necessary structures in reality that allow certain knowledge of absolute truths. Franklin’s Aristotelian realist philosophy of mathematics shows how mathematical truths are directly about physical reality, and at the same time certainly and provably true. Ranging from mathematics to evidence evaluation to ethics, his philosophy of probability sees the relation of evidence to hypothesis, such as in science and law, as purely logical, hence necessary. Across ethics and the philosophy of (...) religion, the theme of necessity is repeated: basic ethical truths (such as the worth of persons and the wrongness of murder) are shown to have the same certainty as mathematics. Focus on the history of ideas connects the philosophical work in the present with the medieval scholastic tradition, which defended similar necessities but is now neglected. Here is an up-to-date introduction to Franklin’s overall perspective. Recalling Western philosophy to its roots, it reveals the way absolute necessities are discoverable across the abstract fields of mathematics, logical evidence and ethics. (shrink)

Probabilidad Imposible Vol 2 reúne los fundamentos de una investigación desarrollada entre 2001 y 2018, cuyo propósito es replantear las bases del conocimiento racional y su relación con la probabilidad empírica y teórica. La obra introduce conceptos como el Impacto del Defecto y el Segundo Método, proponiendo un marco alternativo frente a la epistemología de Karl Popper y orientado hacia la búsqueda del conocimiento puro. Este proyecto, originado en el cruce entre matemáticas, inteligencia artificial y psicología, constituye una semilla teórica (...) para el desarrollo de la Inteligencia Artificial Global y para la construcción de una sociedad tecnológica basada en un nuevo paradigma del saber. -/- Files. (shrink)

The work presented here, Introduction to Impossible Probability, is essential for understanding the original meaning of the Global Artificial Intelligence. Its origins can be traced back to my writings in 2002 on Astro-ecology, the science that understands the cosmos as a single environment that must be studied from a unified, single science. In essence, this is what the Global Artificial Intelligence seeks to achieve. At the same time, in 2002 I developed the formulations of Impossible Probability, beginning in the early (...) hours of September 11, 2001 with the Impact of the Defect, which was later expanded into the Second Method on October 16, 2002. (shrink)

I argue that when we use ‘probability’ language in epistemic contexts—e.g., when we ask how probable some hypothesis is, given the evidence available to us—we are talking about degrees of support, rather than degrees of belief. The epistemic probability of A given B is the mind-independent degree to which B supports A, not the degree to which someone with B as their evidence believes A, or the degree to which someone would or should believe A if they had B as (...) their evidence. My central argument is that the degree-of-support interpretation lets us better model good reasoning in certain cases involving old evidence. Degree-of-belief interpretations make the wrong predictions not only about whether old evidence confirms new hypotheses, but about the values of the probabilities that enter into Bayes’ Theorem when we calculate the probability of hypotheses conditional on old evidence and new background information. (shrink)

Recently, Yudkowsky and Soares (2018) and Levinstein and Soares (2020) have developed a novel decision theory, Functional Decision Theory (FDT). They claim FDT outperforms both Evidential Decision Theory (EDT) and Causal Decision Theory (CDT). Yet FDT faces several challenges. First, it yields some very counterintuitive results (Schwarz 2018; MacAskill 2019). Second, it requires a theory of counterpossibles, for which even Yudkowsky and Soares (2018) and Levinstein and Soares (2020) admit we lack a “full” or “satisfactory” account. Here, I focus on (...) the latter problem of counterpossibles. My aim is to establish two claims. First, the problem of counterpossibles does not even arise without a fairly strong assumptionone that rarely applies to human agents, but may apply to artificial agents. And second, even given this assumption, the problem is solvable, though how best to solve it remains an open question. (shrink)

As part of a book symposium on Erwin Dekker's Jan Tinbergen (1903–1994) and the Rise of Economic Expertise (2021), William Peden reflects on shared views on the objectivity and nature of statistics between Tinbergen and Keynes underlying the Tinbergen-Keynes debates.

Pondering the question of free will in the context of probability allows us to take a fresh look at a number of old problems. We are able to avoid deterministic entrapments and attempt to look at free will as an outcome of the entire decision-making system. In my paper, I will argue that free will should be considered in the context of a complex system of decisions, not individual cases. The proposed system will be probabilistic in character, so it will (...) be embedded in the calculus of probability. To achieve the stated goal, I will refer to two areas of Carnap’s interest: the relationship between free will and determinism, and the probability-based decision-making system. First, I will present Carnap’s compatibilist position. On this basis, I will show how free will can be examined on deterministic grounds. Then I will present Carnap’s probabilistic project—the so-called logical interpretation of probability. In addition to presenting its characteristics and functionality, I will argue for its usefulness in the context of decision analysis and its immunity to problems associated with determinism. Finally, I will show how the two mentioned elements can be combined, as a result of which I will present a concept for a probabilistic analysis of free will. In this context, I will identify free will with the individual characteristics of the system. My main aim is to present the theme of free will in the light of a formal analysis based on probability rather than metaphysical assumptions. (shrink)

David Stove was a philosopher strong on argument and polemic. His work on the logical intepretation of probability led to a defence of induction in The Rationality of Induction (1986). It resulted too in his denunciation of Popper, Kuhn, Lakatos and Feyeraband as irrationalists because of their "deductivism" (the thesis that the only logic is deductive logic). Stove also defended controversial views on the intelligence of women and on Darwinism. The article surveys his life and work.

John Maynard Keynes’s A Treatise on Probability is the seminal text for the logical interpretation of probability. According to his analysis, probabilities are evidential relations between a hypothesis and some evidence, just like the relations of deductive logic. While some philosophers had suggested similar ideas prior to Keynes, it was not until his Treatise that the logical interpretation of probability was advocated in a clear, systematic and rigorous way. I trace Keynes’s influence in the philosophy of probability through a heterogeneous (...) sample of thinkers who adopted his interpretation. This sample consists of Frederick C. Benenson, Roy Harrod, Donald C. Williams, Henry E. Kyburg and David Stove. The ideas of Keynes prove to be adaptable to their diverse theories of probability. My discussion indicates both the robustness of Keynes’s probability theory and the importance of its influence on the philosophers whom I describe. I also discuss the Problem of the Priors. I argue that none of those I discuss have obviously improved on Keynes’s theory with respect to this issue. (shrink)

Epistemic theories of objective chance hold that chances are idealised epistemic probabilities of some sort. After giving a brief history of this approach to objective chance, I argue for a particular version of this view, that the chance of an event E is its epistemic probability, given maximal knowledge of the possible causes of E. The main argument for this view is the demonstration that it entails all of the commonly-accepted properties of chance. For example, this analysis entails that chances (...) supervene on the physical facts, that the chance function is an expert probability, that the existence of stable frequencies results from invariant chances in repeated experiments, and that chances are probably close to long-run relative frequencies in repeated experiments. Despite these virtues, epistemic approaches to chance have been neglected in recent decades, on account of their conflict with accepted views about closely related topics such as causation, laws of nature, and epistemic probability. However, existing views on these topics are also very problematic, and I believe that the epistemic view of chance is a key piece of this puzzle that, once in place, allows all the other pieces to fit together into a new and coherent way. I also respond to some criticisms of the epistemic theory that I favour. (shrink)

In this contribution we will present a generalization of de Finetti's betting game in which a gambler is allowed to buy and sell unknown events' betting odds from more than one bookmaker. In such a framework, the sole coherence of the books the gambler can play with is not sucient, as in the original de Finetti's frame, to bar the gambler from a sure-win opportunity. The notion of joint coherence which we will introduce in this paper characterizes those coherent books (...) on which sure- win is impossible. Our main results provide geometric characterizations of the space of all books which are jointly coherent with a xed one. As a consequence we will also show that joint coherence is decidable. (shrink)

Many philosophers argue that Keynes’s concept of the “weight of arguments” is an important aspect of argument appraisal. The weight of an argument is the quantity of relevant evidence cited in the premises. However, this dimension of argumentation does not have a received method for formalisation. Kyburg has suggested a measure of weight that uses the degree of imprecision in his system of “Evidential Probability” to quantify weight. I develop and defend this approach to measuring weight. I illustrate the usefulness (...) of this measure by employing it to develop an answer to Popper’s Paradox of Ideal Evidence. (shrink)

I argue that none of the usual interpretations of probability provide an adequate interpretation of probabilistic theories in science. Assuming that the aim of such theories is to capture noisy relationships in the world, I suggest that we do not have to give them classical truth-conditional content at all: their probabilities can remain uninterpreted. Indirectly, this account turns out to explain what is right about the frequency interpretation, the best-systems interpretation, and the epistemic interpretation.

Tomoji Shogenji is generally assumed to be the first author to have presented a probabilistic measure of coherence. Interestingly, Rudolf Carnap in his Logical Foundations of Probability discussed a function that is based on the very same idea, namely his well-known relevance measure. This function is largely neglected in the coherence literature because it has been proposed as a measure of evidential support and still is widely conceived as such. The aim of this paper is therefore to investigate Carnap’s measure (...) regarding its plausibility as a candidate for a probabilistic measure of coherence by comparing it to Shogenji’s. It turns out that both measures satisfy and violate the same adequacy constraints, despite not being ordinally equivalent exhibit a strong correlation with each other in a Monte Carlo simulation and perform similarly in a series of test cases for probabilistic coherence measures. (shrink)

B.K.Matilal, and earlier J.F.Staal, have suggested a reading of the `Nyaya five limb schema' (also sometimes referred to as the Indian Schema or Hindu Syllogism) from Gotama's Nyaya-Sutra in terms of a binary occurrence relation. In this paper we provide a rational justification of a version of this reading as Analogical Reasoning within the framework of Polyadic Pure Inductive Logic.

In a recent paper in this journal, James Hawthorne, Jürgen Landes, Christian Wallmann, and Jon Williamson argue that the principal principle entails the principle of indifference. In this paper, I argue that it does not. Lewis’s version of the principal principle notoriously depends on a notion of admissibility, which Lewis uses to restrict its application. HLWW base their argument on certain intuitions concerning when one proposition is admissible for another: Conditions 1 and 2. There are two ways of reading their (...) argument, depending on how you understand the status of these conditions. Reading 1: The correct account of admissibility is determined independently of these two principles, and yet these two principles follow from that correct account. Reading 2: The correct account of admissibility is determined in part by these two principles, so that the principles follow from that account but only because the correct account is constrained so that it must satisfy them. HLWWshow that, given an account of admissibility on which Conditions 1 and 2 hold, the principal principle entails the principle of indifference. I argue that, on either reading of the argument, it fails. First, I argue that there is a plausible account of admissibility on which Conditions 1 and 2 are false. That defeats reading 1. Next, I argue that the intuitions that lead us to assent to Condition 2 also lead us to assent to other very closely related principles that are inconsistent with Condition 2. This, I claim, casts doubt on the reliability of those intuitions, and thus removes our justification for Condition 2. This defeats the second reading of the HLWW argument. Thus, the argument fails. (shrink)

В книгата се изследва идеята за случайността през нейното историческо и концептуално развитие и са отделени пет основни и типични понятия. Анализът тръгва от класическите примери – Платон, Аристотел, Имануел Кант и Георг Хегел (противопоставени и усложнени през размишленията на Жак Дерида, Жак Лакан и Карл Густав Юнг) и стига до съвременния контекст на случайността, който е представен през а) теорията на вероятностите и б) теорията на сложността. Съпоставени са най-важните интерпретации върху вероятността, както и ключови автори като Рихард фон (...) Мизес, Пер Мартин-Льоф и Андрей Колмогоров, редом с по-периферни и радикални изследователи (Антоан Курно, Емил Бутру и т.н.). Някои от изведените понятия са формализирани и имат по-логически, математически или пък информационен характер, а други са по-скоро физически или субектни, но всички те са представени в една обща историческа типология. В нея се включва и парадигмата срещу случайността, която има теоретична, културна и дори органична природа. В рамките на книгата е представено като предложение и едно шесто и максимално абстрактно понятие за случайност. (shrink)

We investigate the relative probabilistic support afforded by the combination of two analogies based on possibly different, structural similarity (as opposed to e.g. shared predicates) within the context of Pure Inductive Logic and under the assumption of Language Invariance. We show that whilst repeated analogies grounded on the same structural similarity only strengthen the probabilistic support this need not be the case when combining analogies based on different structural similarities. That is, two analogies may provide less support than each would (...) individually. (shrink)

Spectrum Exchangeability, Sx, is an irrelevance principle of Pure Inductive Logic, and arguably the most natural extension of Atom Exchangeability to polyadic languages. It has been shown1that all probability functions which satisfy Sx are comprised of a mixture of two essential types of probability functions; heterogeneous and homogeneous functions. We determine the theory of Spectrum Exchangeability, which for a fixed languageLis the set of sentences ofLwhich must be assigned probability 1 by every probability function satisfying Sx, by examining separately the (...) theories of heterogeneity and homogeneity. We find that the theory of Sx is equal to the theory of finite structures, i.e., those sentences true in all finite structures forL, and it emerges that Sx is inconsistent with the principle of Super-Regularity. As a further consequence we are able to characterize those probability functions which satisfy Sx and the Finite Values Property. (shrink)

We propose two principles of inductive reasoning related to how observed information is handled by conditioning, and justify why they may be said to represent aspects of rational reasoning. A partial classification is given of the probability functions which satisfy these principles.

In Pure Inductive Logic, the rational principle of Predicate Exchangeability states that permuting the predicates in a given language L and replacing each occurrence of a predicate in an L-sentence phi according to this permutation should not change our belief in the truth of phi. In this paper we study when a prior probability function w on a purely unary language L satisfying Predicate Exchangeability also satisfies the principle of Unary Language Invariance.

We extend the framework of Inductive Logic to Second Order languages and introduce Wilmers' Principle, a rational principle for probability functions on Second Order languages. We derive a representation theorem for functions satisfying this principle and investigate its relationship to the first order principles of Regularity and Super Regularity.

Sometimes we receive evidence in a form that standard conditioning (or Jeffrey conditioning) cannot accommodate. The principle of maximum entropy (MAXENT) provides a unique solution for the posterior probability distribution based on the intuition that the information gain consistent with assumptions and evidence should be minimal. Opponents of objective methods to determine these probabilities prominently cite van Fraassen’s Judy Benjamin case to undermine the generality of maxent. This article shows that an intuitive approach to Judy Benjamin’s case supports maxent. This (...) is surprising because based on independence assumptions the anticipated result is that it would support the opponents. It also demonstrates that opponents improperly apply independence assumptions to the problem. (shrink)

In 1977 van Fraassen showed convincingly, and in detail, how one can give a dissentive answer to the question `[a]re there necessities in nature?'. In this paper I follow his lead and show in a similar fashion and detail, how it is possible to give a dissentive answer to: Are there probabilities in nature? This is achieved by giving a partial analysis—with the aid of Kaplanian pragmatics—of objective chance in terms of that credence that is reasonable where prevailing laws and (...) conditions exhaust one's evidence. This template belongs firmly within the established Bayesian program of analysing objective chance as ultimate belief. Its contribution to that program is the same as van Fraassen's contribution to the empiricist program of analysing physical necessity; namely, it demonstrates the logical possibility of such an analysis. (shrink)

The goal of this paper is to correct a widespread misconception about the work of Robert Leslie Ellis and John Venn, namely that it can be considered as the ‘British empiricist’ reaction against the traditional theory of probability. It is argued, instead, that there was no unified ‘British school’ of frequentism during the nineteenth century. Where Ellis arrived at frequentism from a metaphysical idealist transformation of probability theory’s mathematical calculations, Venn did so on the basis of an empiricist critique of (...) its ‘inverse application’. (shrink)

This volume is a collection of cutting-edge research papers in scientifically informed metaphysics, tackling a range of philosophical puzzles which have emerged from recent work on chance and temporal asymmetry. How do the probabilities found in fundamental physics and the probabilities of the special sciences relate to one another? How can we account for the normative significance of chance? Can constraints on the initial conditions of the universe underwrite the second law of thermodynamics, and potentially also all other lawlike regularities? (...) How does contemporary quantum theory bear on debates over the nature of chance and the arrow of time? What grounds do we have for believing in a fundamental temporal direction, or flow? And how do all these questions connect up with one another? The aim of the volume is both to survey and summarize recent debates about chance and temporal asymmetry and to push them forward. The authors bring perspectives from metaphysics, from philosophy of physics and from philosophy of probability. Mainstream approaches are subjected to searching new critiques, and bold new proposals are made concerning (inter alia) the semantics of chance-attributions, the justification of the Principal Principle connecting chance and degree of belief, the limits on inter-theoretic reduction and the source of the temporal asymmetry of human experience. (shrink)

Mathematicians often speak of conjectures, yet unproved, as probable or well-confirmed by evidence. The Riemann Hypothesis, for example, is widely believed to be almost certainly true. There seems no initial reason to distinguish such probability from the same notion in empirical science. Yet it is hard to see how there could be probabilistic relations between the necessary truths of pure mathematics. The existence of such logical relations, short of certainty, is defended using the theory of logical probability (or objective Bayesianism (...) or non-deductive logic), and some detailed examples of its use in mathematics surveyed. Examples of inductive reasoning in experimental mathematics are given and it is argued that the problem of induction is best appreciated in the mathematical case. (shrink)

We propose an Analogy Principle in the context of Unary Inductive Logic and characterize the probability functions which satisfy it. In particular in the case of a language with just two predicates the probability functions satisfying this principle correspond to solutions of Skyrmsʼ ‘Wheel of Fortune’.

This note has three goals. First, we discuss a presentation of Bertrand's paradox in a recent issue of Philosophia Mathematica, which we believe to be a subtle but important misinterpretation of the problem. We compare claims made about Bertrand with his 1889 Calcul des Probabilités. Second, we use this source to understand Bertrand's true intention in describing what we now call his paradox, comparing it both to another problem he describes in the same section and to a modern treatment. Finally, (...) we briefly consider the importance of knowing when a specific example represents a general case. (shrink)

We characterize those identities and independencies which hold for all probability functions on a unary language satisfying the Principle of Atom Exchangeability. We then show that if this is strengthen to the requirement that Johnson's Sufficientness Principle holds, thus giving Carnap's Continuum of inductive methods for languages with at least two predicates, then new and somewhat inexplicable identities and independencies emerge, the latter even in the case of Carnap's Continuum for the language with just a single predicate.

This article shows that Popper’s measure of corroboration is inapplicable if, as Popper argued, the logical probability of synthetic universal statements is zero relative to any evidence that we might possess. It goes on to show that Popper’s definition of degree of testability, in terms of degree of logical content, suffers from a similar problem. 1 The Corroboration Function and P(h|b) 2 Degrees of Testability and P(h|b).

This paper shows that Bertrand's proposed 'solutions' to his own question, which generates his chord paradox, are inapplicable. It uses a simple analogy with cake cutting. The problem is that none of Bertrand's solutions considers all possible cuts. This is no solace for the defenders of the principle of indifference, however, because it emerges that the paradox is harder to solve than previously anticipated.

A family of symmetries of polyadic inductive logic are described which in turn give rise to the purportedly rational Permutation Invariance Principle stating that a rational assignment of probabilities should respect these symmetries. An equivalent, and more practical, version of this principle is then derived.

I consider the notions of logical probability, degree of belief, and objective chance, and argue that a different formalism for conditional probability is appropriate for each.

The objective Bayesian view of proof (or logical probability, or evidential support) is explained and defended: that the relation of evidence to hypothesis (in legal trials, science etc) is a strictly logical one, comparable to deductive logic. This view is distinguished from the thesis, which had some popularity in law in the 1980s, that legal evidence ought to be evaluated using numerical probabilities and formulas. While numbers are not always useful, a central role is played in uncertain reasoning by the (...) ‘proportional syllogism’, or argument from frequencies, such as ‘nearly all aeroplane flights arrive safely, so my flight is very likely to arrive safely’. Such arguments raise the ‘problem of the reference class’, arising from the fact that an individual case may be a member of many different classes in which frequencies differ. For example, if 15 per cent of swans are black and 60 per cent of fauna in the zoo is black, what should I think about the likelihood of a swan in the zoo being black? The nature of the problem is explained, and legal cases where it arises are given. It is explained how recent work in data mining on the relevance of features for prediction provides a solution to the reference class problem. (shrink)