This paper is concerned with the problem of giving a correct analysis of function statements as they are used in biology. Examples of such statements are (1) The function of the myelin sheath is to insulate the nerve fiber and (2) The function of chlorophyll is to enable photosynthesis to take place. After criticizing analyses of such statements developed by Braithwaite, Nagel and Hempel an analysis is presented by the author. Finally the question of whether function statements are explanations is discussed briefly.

Modifications of current theories of ordinal, interval and extensive measurement are presented, which aim to accomodate the empirical fact that perfectly exact measurement is not possible (which is inconsistent with current theories). The modification consists in dropping the assumption that equality (in measure) is observable, but continuing to assume that inequality (greater or lesser) can be observed. The modifications are formulated mathematically, and the central problems of formal measurement theory—the existence and uniqueness of numerical measures consistent with data—are re-examined. Some results also are given on a problem which does not arise in current theories: namely that of determining limits of accuracy attainable on the basis of observations.

Suppose that entities composed of two independent components are qualitatively ordered by a relation that satisfies the axioms of conjoint measurement. Suppose, in addition, that each component has a concatenation operation that, together either with the ordering induced on the component by the conjoint ordering or with its converse, satisfies the axioms of extensive measurement. Without further assumptions, nothing can be said about the relation between the numerical scales constructed from the two measurement theories except that they are strictly monotonic. An axiom is stated that relates the two types of measurement theories, seems to cover all cases of interest in physics, and is sufficient to establish that (the multiplicative form of) the conjoint measurement scales are power functions of the extensive measurement scales.

The view is advocated that to preserve a deductivist account of science against recent criticism, it is necessary to incorporate experimental error, or imprecision, in the deductive structure. The sources of imprecision in empirical variables are analyzed, and the notion of conceptual imprecision introduced and illustrated. This is then used to clarify the notion of the acceptable range of a functional law. It is further shown that imprecision may be ascribed to parameters in laws and theories without rendering the deductive structure untestable. It is claimed that this analysis explicates the relation between laws and theories in a way that invalidates certain arguments against its deductive nature.

Much of the criticism of Stevens's criterion for permissible statistics as applied to measurement data results from a lack of clarity in Stevens's position. In this paper set-theoretical notions have been used to clarify that position. We define a sig-function as a function defined on numerical assignments. If u and ℜ are empirical and numerical relational systems, respectively, then a sig-function F is constant on u with respect to ℜ if, and only if, the value of F is the same for all numerical assignments for u with respect to ℜ. Using these notions we prove rigorously certain generalizations of Stevens's results.

Some philosophers have claimed that one must be acquainted with the elements of one's ontology. Also, believing that substrata and universals are required in an adequate ontology, these philosophers have claimed acquaintance with such objects. This paper attempts to analyze what is involved in such claims and to argue that they result from a number of confusions. The paper deals largely with the claim that substrata, or bare particulars, are presented since numerical difference is a simple fact that is presented. It further attempts to argue that while neither substrata nor universals, as such, are objects of acquaintance, there are some distinctions to be drawn between talk of universals as presented objects and talk of substrata being presented.

The “problem of counterfactuals,” as proposed by Goodman and Chisholm, cannot be solved. However, a similar program, pioneered by Hiż and Mrs. Milmed, but largely neglected, can be completed and promises a satisfactory analysis of subjunctive conditionals.

Dr. Marjorie Grene has argued that criteria taken from a personalist philosophy of science have regulative force in the dispute between orthogenetic and synthetic or neo-Darwinian theories of evolution, and that these criteria commend the acceptance of the orthogenetic position. Grene's position includes two basically correct theses concerning the limitations of operationism and reductionism. However, she fails to show that personalist tenets are necessary for the validation of these two theses. Moreover, the proposed modifications of evolutionary theory depend upon additional premisses: that biology must study individuals rather than populations, and that the synthetic theory must prove that natural selection and mutation are the only possible factors for control of the direction of evolutionary change. The evidence for these premisses is called into question.

The creation-of-matter hypothesis of the Bondi-Gold-Hoyle steady-state cosmology requires that in an infinite time to which the first transfinite number may be assigned the number of atoms of matter produced would be equal to the cardinal number of the set of mathematical points in the continuum. The existence of a set of finite atoms with that cardinal number is physically unacceptable. The argument for the production of a non-denumerable set of atoms, in infinite time, is given in terms of a model which is shown to be isomorphic with the original Cantor “diagonal” proof for the existence of a non-denumerable infinity. An alternative model which meets the requirements of the steady-state theory is presented; in this model, the number of atoms is explicitly no greater than countably infinite, and remains countably infinite as long as the past time of the universe is restricted to the unlimited set of finite unit-time intervals. If the origin of the steady-state universe is taken as being within that infinite set, expressed by the negative natural numbers, the contradiction of an atom at every mathematical point does not arise. The contradiction does arise if the origin is not within the set of finite numbers, and accordingly there is a restriction as to which concept of infinite past may properly be maintained in the steady-state theory.

A consideration of some basic problems that arise in the attempt to provide an adequate characterization of statistical explanation is taken to show that an understanding of the nature of scientific explanation requires us to deal with the philosophical problems connected with the nature of prior probabilities.

The problem of the interpretation of theoretical terms is outlined, and some difficulties connected with the distinction between partial definitions and empirical postulates are discussed. A reconstruction is sketched which is intended to explicate the ‘definitional’ character of partial definitions. Finally, some implications for the methodology of theory construction are indicated.

This paper examines four arguments in support of Frege's theory of incomplete entities, the heart of his semantics and ontology. Two of these arguments are based upon Frege's contributions to the foundations of mathematics. These are shown to be question-begging. Two are based upon Frege's solution to the problem of the relation of language to thought and reality. They are metaphysical in nature and they force Frege to maintain a theory of types. The latter puts his theory of incomplete entities in the paradoxical position of maintaining that it is no theory at all. Moreover, his metaphysics rules out well-known suggestions for avoiding this difficulty.

The structure of Reichenbach's pragmatic vindication of induction is analysed in detail. The argument is seen to proceed in two stages, the first being a pragmatic justification of the frequency interpretation of probability which is taken as a license for considering the aim of induction to be the discovery of limiting relative frequencies, and the second being the pragmatic justification of induction itself. Both justifications are found to contain flaws, and the arguments used to support Reichenbach's definition of the aim of induction presuppose the availability of a type of predicate which generates paradoxes closely related to Goodman's “grue-bleen” paradox.

Next, Salmon's “Criterion of Linguistic Invariance” is evaluated as a canon of inductive logic, which singles out the “straight rule” from an infinite class of convergent inductive rules. Upon close examination, its credentials are seen to be unimpressive. In connection with Salmon's work on linguistic invariance, we take a closer look at the impact of the Goodman paradox on probability estimator rules like Reichenbach's “straight rule.” We find that its undesirable consequences for such systems of inductive logic cannot be escaped by Salmon's solution, or indeed by any solution whose motivating idea is to rule “queer” predicates out of court.

Finally, the discussion of the Goodman paradox leads us to a modest methodological proposal for systems of inductive logic which incorperate probability estimator rules of the type at issue.

The interpretation of the calculus of probability as a logic of partial belief has at least two advantages: it makes the assignment of probabilities plausible in cases where classical frequentist interpretations must find such assignments meaningless, and it gives a clear meaning to partial belief and to consistency of partial belief.

First, a comment on a pessimistic note: Salmon says we can't be sure there is any such thing as inductive inference: in demanding that some explanations have the form of correct inductive inferences, “we may be laying down a requirement which cannot be fulfilled.” To doubt that we can fulfill that requirement is to doubt that we can formalize inductive logic. It may be true, but why begin the fight by throwing in the sponge ? It is also true that there are difficulties involved in formalizing acceptance rules for inductive conclusions, but these difficulties may be overcome. It is false to claim that we ‘cannot claim to know exactly’ what these acceptance rules are. Only two years ago I did just that. It was true that my claim was shown to be erroneous a year later, but that fate can befall almost any worthwhile claim.

Four progressively ambitious systems of modal propositional logic are set forth, together with decision procedures. The simultaneous employment of parenthesis notation and parenthesis-free notation, the dual use of symbols as primitive and defined, and the introduction of a new modal operator (the truth operator) are the principal devices used to effect the development of these logics. The first two logics turn out to be “the same” as two of von Wright's systems.

This rejoinder will be divided into two parts. The first part will be devoted to what appear to be serious disagreements between Professor Kyburg and me. I shall try to show that these disagreements are far more apparent than real. The second part will contain my criticism of Kyburg's constructive suggestion on the nature of explanation.

The conclusion urged in Mr Salmon's recent article ([1], pp. 252-261) is so remarkable that it may be worth recording some difficulties. He claims to rescue Reichenbach's notorious vindication of induction. This is essentially concerned with estimating long-run frequencies. By an estimator let us mean any rule for making estimates appropriate to various bodies of information. Reichenbach thought an estimator is sensible only if it is convergent, that is, roughly speaking, only if its estimates tend to approach the truth as more and more data is collected. Mr Salmon observes that an estimator should, in addition, be independent of the language in which the information is expressed. He contends that this criterion of linguistic invariance, when added to the rest of Reichenbach's train of thought, suffices to justify the “straight rule” for estimation: if m of the n observed A's are B, estimate the long-run frequency of B among A's as m/n.

In this paper, we show that a restricted form of time travel both accords with special relativity kinematics and avoids several prima facie objections. We argue that such time travel provides a reasonable way to interpret certain phenomena which can readily be described, and the analogues of which have already been observed at the level of elementary particle reactions. We then describe how a time-traveling object could measure itself, and demonstrate how, in the appropriate circumstances, such an experiment could convince a theorist who insisted on a single criterion for length measurements that his standard had itself changed in length. The same experiment can shed light on the possibility of detecting a universal expansion; and we show that, given certain experimental results, one must conclude that every member of a class of time travelers has changed in length simultaneously, though perhaps only while going backwards in time.