This paper takes two tasks. The one is elaborating on the relationship of inductive logic with decision theory to which later Carnap planned to apply his system (§§1-7); this is a surveying side of this article. The other is revealing the property of our prediction of the future, subjectivity (§§8-11); this is its philosophical aspect. They are both discussed under the name of belief in causation. Belief in causation is a kind of “degree of belief” born about the (...) causal effect of the action. As such, it admits of the analysis by inductive logic. (shrink)

The aim of this paper is to apply inductive logic to the field that, presumably, Carnap never expected: legal causation. Legal causation is expressible in the form of singular causal statements; but it is distinguished from the customary concept of scientific causation, because it is subjective. We try to express this subjectivity within the system of inductive logic. Further, by semantic complement, we compensate a defect found in our application, to be concrete, the impossibility of two-place (...) predicates (for causal relationship) in inductive logic. (shrink)

In section I the notions of logical and inductive probability will be discussed as well as two explicanda, viz. degree of confirmation, the base for inductive probability, and degree of evidential support, Popper's favourite explicandum. In section II it will be argued that Popper's paradox of ideal evidence is no paradox at all; however, it will also be shown that Popper's way out has its own merits.

Although written in Japanese, what the statistical inference is philosophically investigated.

Although we often see references to Carnap’s inductive logic even in modern literatures, seemingly its confusing style has long obstructed its correct understanding. So instead of Carnap, in this paper, I devote myself to its necessary and sufficient commentary. In the beginning part (Sections 2-5), I explain why Carnap began the study of inductive logic and how he related it with our thought on probability (Sections 2-4). Therein, I trace Carnap’s thought back to Wittgenstein’s Tractatus as (...) well (Section 5). In the succeeding sections, I attempt the simplest exhibition of Carnap’s earlier system, where his original thought was thoroughly provided. For this purpose, minor concepts to which researchers have not paid attention are highlighted, for example, m-function (Section 8), in-correlation (Section 10), C-correlate (Section 10), statistical distribution (Section 12), and fitting sequence (Section 17). The climax of this paper is the proof of theorem (56). Through this theorem, we will be able to overview Carnap’s whole system. (shrink)

In this thesis I defend and pursue that line about the foundations of probability theory which has come to be known as "the logicist view about probability", and, in particular, the shape which it took in Carnap's Inductive Logic. ;Most philosophers who now deal with probability theory claim that Carnap's program of Inductive Logic has failed. The main aim of my thesis is to show that this judgment is based on a fundamental misunderstanding about the nature (...) and the aim of inductive logic. To that end, I follow, in chapter 1, the events in the history of probability, logic and mathematics which led to choosing certain requirements as the requirements which any theory about the foundations of probability must fulfill in order to be acceptable. ;In chapter 2 I explain how Carnap's inductive logic fulfills those requirements and how the method it uses to give foundations to probability theory can also be used to solve another major problem about probability, namely, the problem of statistical inference. ;Chapter 3 shows how the ideas explained in chapter 2 were developed, formally, by Carnap, and how Carnap's program can be continued in a particular case, that of reasoning by analogy. ;Finally, in chapter 4, I attempt to show that subjectivism about probability theory, the rival theory to Carnap's logicism, does not succeed in meeting the most basic requirement for the acceptability of any theory about the foundations of probability. (shrink)

A basic system of inductive logic; An axiomatic foundation for the logic of inductive generalization; A survey of inductive systems; On the condition of partial exchangeability; Representation theorems of the de finetti type; De finetti's generalizations of excahngeability; The structure of probabilities defined on first-order languages; A subjectivit's guide to objective chance.

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. (shrink)

My dissertation explores the ways in which Rudolf Carnap sought to make philosophy scientific by further developing recent interpretive efforts to explain Carnap’s mature philosophical work as a form of engineering. It does this by looking in detail at his philosophical practice in his most sustained mature project, his work on pure and applied inductive logic. I, first, specify the sort of engineering Carnap is engaged in as involving an engineering design problem and then draw out the complications (...) of design problems from current work in history of engineering and technology studies. I then model Carnap’s practice based on those lessons and uncover ways in which Carnap’s technical work in inductive logic takes some of these lessons on board. This shows ways in which Carnap’s philosophical project subtly changes right through his late work on induction, providing an important corrective to interpretations that ignore the work on inductive logic. Specifically, I show that paying attention to the historical details of Carnap’s attempt to apply his work in inductive logic to decision theory and theoretical statistics in the 1950s and 1960s helps us understand how Carnap develops and rearticulates the philosophical point of the practical/theoretical distinction in his late work, offering thus a new interpretation of Carnap’s technical work within the broader context of philosophy of science and analytical philosophy in general. (shrink)

According to a popular interpretation, Carnap’s interpretation of probability had evolved from a logical towards a subjective conception. However Carnap himself insisted that his basic philosophical view of probability was always the same. I address this apparent clash between Carnap's self-identification and the subsequent interpretations of his work. Following its original intentions, I reconstruct inductive logic as an explication. The emerging picture is of a versatile linguistic framework, whose main function is not the discovery of objective logical relations (...) in the object language, but the stipulation of conceptual possibilities. Within this representation, I map out the changes that the project went through. Seen from such an explication-based perspective, inductive logic becomes quite hard to categorize using the standard labels. (shrink)

A basic system of inductive logic; An axiomatic foundation for the logic of inductive generalization; A survey of inductive systems; On the condition of partial exchangeability; Representation theorems of the de finetti type; De finetti's generalizations of excahngeability; The structure of probabilities defined on first-order languages; A subjectivit's guide to objective chance.

Carnap's Inductive Logic, like most philosophical discussions of induction, is designed for the case of independent trials. To take account of periodicities, and more generally of order, the account must be extended. From both a physical and a probabilistic point of view, the first and fundamental step is to extend Carnap's inductive logic to the case of finite Markov chains. Kuipers (1988) and Martin (1967) suggest a natural way in which this can be done. The probabilistic (...) character of Carnapian inductive logic(s) for Markov chains and their relationship to Carnap's inductive logic(s) is discussed at various levels of Bayesian analysis. (shrink)

This chapter contains sections titled: Good Old‐Fashioned Inductive Logic (GOFIL): Carnap's Program Subjectivized Inductive Logic (SIL): De Finetti Regnant New‐Fangled Inductive Logic (NFIL).

In this paper I offer a criticism of Carnap's inductive logic which also applies to other formal methods of inductive inference. Criticisms of Carnap's views have typically centered upon the justification of his particular choice of inductive method. I argue that the real problem is not that there is an agreed upon method for which no justification can be found, but that different methods are justified in different circumstances.

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 book revives inductive logic by bringing out the underlying epistemology. The resulting structural reliabilist theory propounds the view that justification supervenes on syntactic and semantic properties of sentences as justification-bearers. It is claimed to set up a genuine alternative to the prevailing theories of justification. Kawalec substantiates this claim by confronting structural reliabilism with a number of epistemological problems. While the book is addressed to both professionals and students of philosophical logic, probability, epistemology, and philosophy of (...) science, it also surveys ideas central to the development of philosophy in the 20th century. It will be a valuable companion to multifarious graduate and postgraduate courses. (shrink)

I conceive of inductive logic as a project of explication. The explicandum is one of the meanings of the word `probability' in ordinary language; I call it inductive probability and argue that it is logical, in a certain sense. The explicatum is a conditional probability function that is specified by stipulative definition. This conception of inductive logic is close to Carnap's, but common objections to Carnapian inductive logic (the probabilities don't exist, are arbitrary, (...) etc.) do not apply to this conception. (shrink)

The paper explores the handling of singular analogy in quantitative inductive logics. It concentrates on two analogical patterns coextensive with the traditional argument from analogy: perfect and imperfect analogy. Each is examined within Carnap’s λ-continuum, Carnap’s and Stegmüller’s λ-η continuum, Carnap’s Basic System, Hintikka’s α-λ continuum, and Hintikka’s and Niiniluoto’s K-dimensional system. Itis argued that these logics handle perfect analogies with ease, and that imperfect analogies, while unmanageable in some logics, are quite manageable in others. The paper concludes with (...) a modification of the K-dimensional system that synthesizes independent proposals by Kuipers and Niiniluoto. (shrink)

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. (shrink)

"Homage to Rudolph Carnap."--Hempel, C. G. Rudolf Carnap, logical empiricist.--Wedberg, A. How Carnap built the world in 1928.--Eberle, R. A construction of quality classes improved upon the Aufbau.--Carnap, R. Observation language and theoretical language.--Kaplan, D. Significance and analyticity: a comment of some recent proposals of Carnap.--Wójcicki, R. The factual content of empirical theories.--Williams, P. M. On the conservative extensions of semantical systems: a contribution to the problem of analyticity.--Winnie, J. A. Theoretical analyticity.--Wedberg, A. Decision and belief in science.--Bohnert, H. G. (...) Carnap's logicism.--Hintikka, J. Carnap's heritage in logical semantics.--Partee, B. H. The semantics of belief-sentences.--Kasher, A. Pragmatic representations and language-games.--Carnap, R. Notes on probability and induction.--Jeffrey, R. C. Carnap's inductive logic.--Hilpinen, R. Carnap's new system of inductive logic.--Kuipers, T. A. F. A generalization of Carnap's inductive logic. Essler, W. K. Hintikka versus Carnap.--Hintikka, J. Carnap and Essler versus inductive generalization.--Shimony, A. Carnap on entropy, introduction to "Two essays on entropy" by Rudolf Carnap. (shrink)

One way of explaining Rudolf Carnap’s mature philosophical view is by drawing an analogy between his technical projects—like his work on inductive logic—with a certain kind of conceptual engineering. After all, there are many mathematical similarities between Carnap’s work in inductive logic and a number of results from contemporary confirmation theory, statistics and mathematical probability theory. However, in stressing these similarities, the conceptual dependence of Carnap’s inductive logic on his work on semantics is downplayed. (...) Yet it is precisely the conceptual resources made available to Carnap from his work on semantics which allows him to understand his work on inductive logic as a kind of conceptual engineering project. The aim of this paper is to elucidate this engineering analogy in light of Carnap’s mature views through the lens of both inductive logic and semantics. (shrink)

Carnap's inductive logic (or confirmation) project is revisited from an "increase in firmness" (or probabilistic relevance) point of view. It is argued that Carnap's main desiderata can be satisfied in this setting, without the need for a theory of "logical probability." The emphasis here will be on explaining how Carnap's epistemological desiderata for inductive logic will need to be modified in this new setting. The key move is to abandon Carnap's goal of bridging confirmation and credence, (...) in favor of bridging confirmation and evidential support. (shrink)

Rudolf Carnap is principally renowned for stating with remarkable precision and rigor a rich variety of philosophical doctrines — doctrines which, thanks mainly to Carnap’s meticulous formulations, the philosophical world now holds to be clearly and fundamentally mistaken. Thus, it is Carnap who, in Meaning and Necessity (Carnap 1947), presents in detail the linguistic doctrine of logical truth and the semantic underpinnings of the analytic/synthetic distinction, providing thereby the grist for the mill of Quine’s highly influential and important attacks on (...) precisely these doctrines. Again, it was Carnap who, more than any other, precisely delineated the program of inductive logic. This program is now, thanks largely to Goodman’s (1983) New Riddle of Induction, also considered hopeless. Carnap is now firmly associated with a bewildering variety of discredited views: reductionism and the unity of science; the verification criterion of meaning; logic as an uninterpreted calculus; Russellian logicism even in the face of Gödel’s devastating incompleteness results; etc. (shrink)

We consider two formalizations of the notion of irrelevance as a rationality principle within the framework of (Carnapian) Inductive Logic: Johnson's Sufficientness Principle, JSP, which is classically important because it leads to Carnap's influential Continuum of Inductive Methods and the recently proposed Weak Irrelevance Principle, WIP. We give a complete characterization of the language invariant probability functions satisfying WIP which generalizes the Nix-Paris Continuum. We argue that the derivation of two very disparate families of inductive methods (...) from alternative perceptions of 'irrelevance' is an indication that this notion is imperfectly understood at present. (shrink)

The λ-continuum of inductive methods was derived from an assumption, called λ-condition, which says that the probability of finding an individual having property $x_{j}$ depends only on the number of observed individuals having property $x_{j}$ and on the total number of observed individuals. So, according to that assumption, all individuals with properties which are different from $x_{j}$ have equal weight with respect to that probability and, in particular, it does not matter whether any individual was observed having some property (...) similar to $x_{j}$ (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 λ-condition, which would allow for the derivation of probability functions which might be sensitive to similarity. Carnap himself suggested a weakening of the λ-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. (shrink)

This recent addition to the well-known "Studies in Logic" series is sure to be of first importance to serious students of inductive logic, confirmation theory, and related issues. The book is an anthology of fourteen papers, which are classified under five different headings: "Extensions of Inductive Logic," "Induction and Information," "Prospects of Confirmation Theory," "The Paradoxes of Confirmation," and "Probability and Foundational Problems." Needless to say, all of the papers are of uniformly high quality. Especially (...) worthy of mention are two papers by Hintikka and Hilpinen which deal with an interesting generalization of Carnap's continuum of inductive methods, and which lead to a somewhat more plausible treatment of non-logically-true closed sentences. Suppes contributes two papers dealing with the recently revived Bayesian methods and their significance for concept formation in psychology and confirmation theory in the philosophy of science. Also under the heading of confirmation theory, one might mention Max Black's paper, "Notes on the Paradoxes of Confirmation," a lucid and rather novel approach to this problem. There is also an important paper by Dana Scott and Peter Krauss entitled "Assigning Probabilities to Logical Formulas." This piece of work presents a set-theoretical method for bridging the gap between the standard treatment of probability by means of axioms defining a real measure over a Boolean algebra and the usual logical inquiries into "degree of confirmation" and the like. The reference value of the book is enhanced by a subject index of terms and concepts dealt with in the various papers.—H. P. K. (shrink)

Inductive probability is the logical concept of probability in ordinary language. It is vague but it can be explicated by defining a clear and precise concept that can serve some of the same purposes. This paper presents a general method for doing such an explication and then a particular explication due to Carnap. Common criticisms of Carnap's inductive logic are examined; it is shown that most of them are spurious and the others are not fundamental.