Aspects of Inductive Logic

Knowledge and Reflexivity

This paper is concerned with establishing broadly based system theoretic foundations and practical techniques for the problem of system identification that are rigorous, intuitively clear and conceptually powerful. A general formulation is first given in which two order relations are postulated on a class of models: a constant one of complexity and a variable one of approximation induced by an observed behaviour. An admissible model is such that any less complex model is a worse approximation. The general problem of identification is that of finding the admissible subspace of models induced by a given behaviour. It is proved under very general assumptions that, if deterministic models are required, then nearly all behaviours require models of nearly maximum complexity. A general theory of approximation between models and behaviour is then developed based on subjective probability concepts and semantic information theory. The role of structural constraints such as causality, locality, finite memory, etc., ...

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System identification, approximation and complexity

Provability in Logic. By Stig Kanger. Stockholm Studies in Philosophy I. Pp. 47. Sw. Cr. 10.00. 1957. (Almgvist and Wiksell, Stockholm)

In his pioneering Essay in Modal Logic, G. H. von Wright, after system- atising the uses of various modal words, said in a concluding note 
An important use of [the modal words] is connected with the notions of an ability and of a disposition and with the verb ‘can’. For example : Jones can speak German (=it is possible for Jones to make himself understood in German) ; Jones cannot speak German (=it is impossible for Jones to make himself understood in German). We shall call the modal concepts which refer to abilities and dispositions dynamic modalities (I am indebted for the term to Mr. Geach)… The question whether the dynamic modalities, i.e. the logic of abilities and dispositions, is subject to exactly the same formal rules as the alethic modalities will have to be investigated separately. (An Essay in Modal Logic, Amsterdam, 1951, p. 54.)

Human Abilities and Dynamic Modalities

In his Essay in Modal Logic, G. H. von Wright suggested that the logic of knowledge or epistemic logic is a branch of modal logic.1 E. J. Lemmon has recommended a system of epistemic logic based on the Feys-von Wright modal system M,2 and in Knowledge and Belief Jaakko Hintikka proposed a stronger system which corresponds to Lewis's S4.3 The distinctive axiom of S4 is 'Np^NNp',* and the epistemic counter part of this formula is the principle that knowing implies knowing that one knows, that is,

Knowing that one knows and the classical definition of knowledge

In his article ‘The Conception of Probability as a Logical Relation’ [1] A.J. Ayer has criticized the logical interpretation of inductive probability on which J. M. Keynes [22], Rudolf Carnap [6], and many other writers on the philosophy of induction have based their conceptions of inductive inference. Ayer’s criticism is concerned with the principle of total evidence and with the obvious fact that it is often reasonable to collect new evidence when we are studying the credibility of some hypothesis. According to Ayer, it is impossible to understand this simple fact and justify the principle of total evidence, if the concept of inductive probability is interpreted in the way suggested by Keynes, Jeffreys, Carnap, and other proponents of the logical interpretation.

On the Information Provided by Observations

In this paper I shall sketch a system of modal logic in which modal operators are relativised to individuals or sets of individuals. This extension of modal logic is suggested by certain expressions in ordinary language. For example, under certain circumstances we may utter the sentence (A) John can catch the trainwhich may be taken to be equivalent to (A’) It is possible for John to catch the train.

Risto Hilpinen

Papers

Knowing that one knows and the classical definition of knowledge

On the Information Provided by Observations

An Analysis of Relativised Modalities