Linguistically invariant inductive logic

Summary Carnap's early system of inductive logic make degrees of confirmation depend on the languages in which they are expressed. They are sensitive to which predicates are, in the language, taken as primitive. Hence they fail to be linguistically invariant. His later systems, in which prior probabilities are assigned to elements of a model rather than sentences of a language, are sensitive to which properties in the model are called primitive. Critics have often protested against these features of his work. This paper shows how to make his systems independent of any choice of primitive predicates or primitive properties.The solution is related to another criticism of inductive logic. It has been noticed that Carnap's systems are too all-embracing. Hisc(h, e) is defined for all sentencesh ande. Yet for manyh ande, the evidencee does not warrant any assessment of the probability ofh. We need an inductive logic in whichc(h, e) is defined only whene really does bear onh. This paper sketches the measure theory of such a logic, and, within this measure theory, provides relativized versions of Carnap's systems which are linguistically invariant.     

A Logic For Inductive Probabilistic Reasoning

Inductive probabilistic reasoning is understood as the application of inference patterns that use statistical background information to assign (subjective) probabilities to single events. The simplest such inference pattern is direct inference: from “70% of As are Bs” and “a is an A” infer that a is a B with probability 0.7. Direct inference is generalized by Jeffrey’s rule and the principle of cross-entropy minimization. To adequately formalize inductive probabilistic reasoning is an interesting topic for artificial intelligence, as an autonomous system acting in a complex environment may have to base its actions on a probabilistic model of its environment, and the probabilities needed to form this model can often be obtained by combining statistical background information with particular observations made, i.e., by inductive probabilistic reasoning. In this paper a formal framework for inductive probabilistic reasoning is developed: syntactically it consists of an extension of the language of first-order predicate logic that allows to express statements about both statistical and subjective probabilities. Semantics for this representation language are developed that give rise to two distinct entailment relations: a relation ⊨ that models strict, probabilistically valid, inferences, and a relation that models inductive probabilistic inferences. The inductive entailment relation is obtained by implementing cross-entropy minimization in a preferred model semantics. A main objective of our approach is to ensure that for both entailment relations complete proof systems exist. This is achieved by allowing probability distributions in our semantic models that use non-standard probability values. A number of results are presented that show that in several important aspects the resulting logic behaves just like a logic based on real-valued probabilities alone.     

An application of Carnapian inductive logic to an argument in the philosophy of statistics

I claim that an argument from the philosophy of statistics can be improved by using Carnapian inductive logic. Gelman and Shalizi [9] criticise a philosophical account of how statisticians ought to choose statistical models which they call ‘the received view of Bayesian inference’ and propose a different account inspired by falsificationist philosophy of science. I introduce another philosophical account inspired by Carnapian inductive logic and argue that it is even better than Gelman and Shalizi's falsificationist account.     

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。     

Mathematical logic in the soviet union, 1917–1980

This essay describes a variety of contributions which relate to the connection of probability with logic. Some are grand attempts at providing a logical foundation for probability and inductive inference. Others are concerned with probabilistic inference or, more generally, with the transmittance of probability through the structure (logical syntax) of language. In this latter context probability is considered as a semantic notion playing the same role as does truth value in conventional logic. At the conclusion of the essay two fully elaborated semantically based constructions of probability logic are presented.     

Non-zero probabilities for universal generalizations

This article discusses the classical problem of zero probability of universal generalizations in Rudolf Carnap’s inductive logic. A correction rule for updating the inductive method on the basis of evidence will be presented. It will be shown that this rule has the effect that infinite streams of uniform evidence assume a non-zero limit probability. Since Carnap’s inductive logic is based on finite domains of individuals, the probability of the corresponding universal quantification changes accordingly. This implies that universal generalizations can receive positive prior and posterior probabilities, even for (countably) infinite domains.     

Conceiving, Experiencing, and Conceiving Experiencing: Neo-Kantianism and the History of the Concept of Experience

It is often claimed that epistemological thought divides around the issue of the place of experience in knowledge: While empiricists argue that experience is the only legitimate source of knowledge, rationalists find other such sources. The trouble with such accounts is not that they are wrong, but that they are incomplete. On occasion, epistemological differences run deeper, raising the very notion of experience as an issue for epistemology. This paper looks at two epistemological debates which concerned not simply the place of experience in knowledge but also the appropriate account of experience itself. The first episode is the rise of Marburg Neo-Kantianism in the 1870s – in particular the seminal work of Hermann Cohen in his Kants Theorie der Erfahrung (1871). Cohen's principal point was that Kant's significance as an epistemologist was in providing a new theory of experience, one that tied experience to exact science and led to a new stress on the formal conditions of exact knowledge. The second episode is Carnap's rejection of epistemology in the 1930s in favour of a program of the logic of science. My focus in each case will be the interplay between an epistemology focused on exact science as the locus of knowledge and a concomitant call for logical methods in epistemology. This revised version was published online in June 2006 with corrections to the Cover Date.     

Hume's Inductive Logic

This paper presents a new account of Hume’s “probability of causes”. There are two main results attained in this investigation. The first, and perhaps the most significant, is that Hume developed – albeit informally – an essentially sound system of probabilistic inductive logic that turns out to be a powerful forerunner of Carnap’s systems. The Humean set of principles include, along with rules that turn out to be new for us, well known Carnapian principles, such as the axioms of semiregularity, symmetry with respect to individuals (exchangeability), predictive irrelevance and positive instantial relevance. The second result is that Hume developed an original conception of probability, which is subjective in character, although it differs from contemporary personalistic views because it includes constraints that are additional to simple consistency and do not vary between different persons. The final section is a response to Gower’s thesis, by which Hume’s probability of causes is essentially non-Bayesian in character. It is argued that, on closer examination, Gower’s reading of the relevant passages is untenable and that, on the contrary, they are in accordance with the Bayesian reconstruction presented in this paper. This revised version was published online in June 2006 with corrections to the Cover Date.     

为什么归纳逻辑需要所罗门诺夫先验?(英文)

二十世纪五十年代,卡尔纳普发展了归纳逻辑,他把概率看作一种证据对假设对"确证度";二十世纪六十年代,所罗门诺夫用通用归纳方法进行预测。为了增强归纳逻辑的归纳预测能力以及扩展所罗门诺夫通用归纳方法的表达力,本文整合二者。本文首先将所罗门诺夫先验概率的思想引入归纳逻辑中,在这个框架下,证明一阶逻辑版本的所罗门诺夫完全性定理,然后比较二者的优略。在卡尔纳普的归纳逻辑中,不管正面证据有多少,对像"所有乌鸦都是黑的"这种全称句的支持度最终都为零,而在用所罗门诺夫先验改造的归纳逻辑中,可以证明,在任何可计算的世界中,"所有乌鸦都是黑的"可以得到确证,只要在那些世界上真的所有乌鸦都是黑的。在所罗门诺夫的模型中,要证明完全性定理需要记录所有的过去信息,在修改后的归纳逻辑中,我们可以只关注某种具体的模式而忽略其它无关信息并证明类似的收敛定理。我们甚至可以不用记录所有的相关信息而采用随机抽样的方法建立合理的信念。     

Formal Requirements of Markov State Models for Paired Associate Learning

Discrete state learning models that make Markov assumptions are a powerful tool for the analysis and optimization of performance in paired associate tasks. We seek here to derive bounds on the complexity needed by such models in order to account for the critical effects of lag and retention intervals on paired associate learning. More specifically, after establishing that two different Markov chains are needed (one for describing the effects of trials where a paired associate is presented and one for describing the effects of trials where the paired associate is not presented), we determine the minimum number of states required in a Markov model with two chains. It is shown formally that, under certain psychologically plausible assumptions, more than three states are required. A model with two chains and four states is presented and it is shown empirically that it can account for the lag and retention effects in paired associate learning. Copyright 2001 Academic Press.     

Extending probabilistic dynamic epistemic logic

This paper aims to extend in two directions the probabilistic dynamic epistemic logic provided in Kooi’s paper (J Logic Lang Inform 12(4):381–408, 2003) and to relate these extensions to ones made in van Benthem et al. (Proceedings of LOFT’06. Liverpool, 2006). Kooi’s probabilistic dynamic epistemic logic adds to probabilistic epistemic logic sentences that express consequences of public announcements. The paper (van Benthem et al., Proceedings of LOFT’06. Liverpool, 2006) extends (Kooi, J Logic Lang Inform 12(4):381–408, 2003) to using action models, but in both papers, the probabilities are discrete, and are defined on trivial σ-algebras over finite sample spaces. The first extension offered in this paper is to add a previous-time operator to a probabilistic dynamic epistemic logic similar to Kooi’s in (J Logic Lang Inform 12(4):381–408, 2003). The other is to involve non-trivial σ-algebras and continuous probabilities in probabilistic dynamic epistemic logic.     

Probabilistic single function dual process theory and logic programming as approaches to non-monotonicity in human vs. artificial reasoning

In this paper, it is argued that single function dual process theory is a more credible psychological account of non-monotonicity in human conditional reasoning than recent attempts to apply logic programming (LP) approaches in artificial intelligence to these data. LP is introduced and among other critiques, it is argued that it is psychologically unrealistic in a similar way to hash coding in the classicism vs. connectionism debate. Second, it is argued that causal Bayes nets provide a framework for modelling probabilistic conditional inference in System 2 that can deal with patterns of inference LP cannot. Third, we offer some speculations on how the cognitive system may avoid problems for System 1 identified by Fodor in 1983. We conclude that while many problems remain, the probabilistic single function dual processing theory is to be preferred over LP as an account of the non-monotonicity of human reasoning.     

Laws of language use and formal logic

After a review of recent work connected with Grice's maxims of conversation, special attention is paid to the principles related to the concept of informativeness. The operation of those principles constrains logic by limiting the meaning of logical particles and, more importantly, by contradicting axioms and many theorems of the propositional calculus. It is argued that, as the development of the individual's formal operations proceeds, a conflict with laws of language use arises. The notion of a conflict between two subsystems, logical and linguistic, is used in a developmental perspective to account, in a unified manner, for the observed discrepancy between subjects' behavior in deductive and inductive reasoning tasks and predictions based on formal logic.     

Predictive probability and analogy by similarity in inductive logic

The-continuum of inductive methods was derived from an assumption, called-condition, which says that the probability of finding an individual having propertyx _j depends only on the number of observed individuals having propertyx _j and on the total number of observed individuals. So, according to that assumption, all individuals with properties which are different fromx _j have equal weight with respect to that probability and, in particular, it does not matter whether any individual was observed having some propertysimilar tox _j (the most complete proof of this result is presented in Carnap, 1980).The problem thus remained open to find some general condition, weaker than the-condition, which would allow for thederivation of probability functions which might be sensitive to similarity. Carnap himself suggested a weakening of the-condition which might allow for similarity sensitive probability functions (Carnap, 1980, p. 45) but he did not find the family of probability functions derivable from that principle. The aim of this paper is to present the family of probability functions derivable from Carnap's suggestion and to show how it is derived.In Section 1 the general problem of analogy by similarity in inductive logic is presented, Section 2 outlines the notation and the conceptual background involved in the proof, Section 3 gives the proof, Section 4 discusses Carnap's principle and the result, Section 5 is a brief review of the solutions which have previously been proposed.     

Probabilities For Two Properties

Let R(X, B) denote the class of probability functions that are defined on algebra X and that represent rationally permissible degrees of certainty for a person whose total relevant background evidence is B. This paper is concerned with characterizing R(X, B) for the case in whichX is an algebra of propositions involving two properties and B is empty. It proposes necessary conditions for a probability function to be in R(X, B), some of which involve the notion of statistical dependence. The class of probability functions that satisfy these conditions, here denoted P_I, includes a class that Carnap once proposed for the same situation. Probability functions in P_I violate Carnap's axiom of analogy but, it is argued, that axiom should be rejected. A derivation of Carnap's model by Hesse has limitations that are not present in the derivation of P_I given here. Various alternative probability models are considered and rejected.     

A RESERVED READING OF CARNAP'S AUFBAU

Abstract:  The two most popular approaches to Carnap's 1928 Aufbau are the empiricist reading of Quine and the neo-Kantian readings of Michael Friedman and Alan Richardson. This paper presents a third "reserved" interpretation that emphasizes Carnap's opposition to traditional philosophy and consequent naturalism. The main consideration presented in favor of the reserved reading is Carnap's work on a physical construction system. I argue that Carnap's construction theory was an empirical scientific discipline and that the basic relations of its construction systems need not be eliminated.     

特征归纳的关联相似性模型

作者提出特征归纳的关联相似性模型,用以描述以归纳特征的关联特征相似性为基础的归纳推理,把归纳推理中相似性和关联知识统一整合起来。该模型认为归纳强度主要取决于归纳特征与关联特征的关联强度和关联特征的相似性程度的乘积,归纳信心主要取决于关联强度,从而分离归纳信心和归纳强度。以大学生为被试的两个实验的主要结果支持关联相似性模型的主要预测。关联相似性模型能够描述解释以关联特征相似性为基础的归纳推理现象,比以往的归纳理论具有更大的解释能力和解释范围     

Lakatos's criticism of Carnapian inductive logic was mistaken

In [39], Imre Lakatos influentially argued that Carnapian inductive logic was a degenerate research programme. This paper argues that Lakatos's criticism was mistaken and that, according to Lakatos's own standards, Carnapian inductive logic was progressive rather than degenerate.     

A new formulation of the Principle of Indifference

The idea of a probabilistic logic of inductive inference based on some form of the principle of indifference has always retained a powerful appeal. However, up to now all modifications of the principle failed. In this paper, a new formulation of such a principle is provided that avoids generating paradoxes and inconsistencies. Because of these results, the thesis that probabilities cannot be logical quantities, determined in an objective way through some form of the principle of indifference, is no longer supportable. Later, the paper investigates some implications of the new principle of indifference. To conclude, a re-examination of the foundations of the so-called objective Bayesian inference is called for.