A survey of deduction theorems for the propositional calculi

---

A survey of deduction theorems for the propositional calculi

     

Cut-free tableau calculi for some propositional normal modal logics

We give sound and complete tableau and sequent calculi for the prepositional normal modal logics S4.04, K4B and G ⁰(these logics are the smallest normal modal logics containing K and the schemata A A, A A and A ( A); A A and AA; A A and ((A A) A) A resp.) with the following properties: the calculi for S4.04 and G ⁰are cut-free and have the interpolation property, the calculus for K4B contains a restricted version of the cut-rule, the so-called analytical cut-rule.In addition we show that G ⁰is not compact (and therefore not canonical), and we proof with the tableau-method that G ⁰is characterised by the class of all finite, (transitive) trees of degenerate or simple clusters of worlds; therefore G ⁰is decidable and also characterised by the class of all frames for G ⁰.Research supported by Fonds zur Förderung der wissenschaftlichen Forschung, project number P8495-PHY.Presented by W. Rautenberg     

Non-classical propositional calculi in relation to methodological patterns of scientific investigation

Summary Modern methodology furnishes two partly competitive and partly complementary views on structure of the development of scientific investigation. According to the first view the development of science consists in enlargement of the set of empirical theorems; according to the other it consists, rather, in the narrowing of the set of possible theoretical hypotheses. A particular kind of assertion is associated with each of these views. The first is associated with the relation of assertion expressed in the statement: “the state α of scientific investigationB forces us to accept the statement Τ”. The other view is associated with the relation of assertion expressed in the statement: “the state α of scientific investigationB admits the acceptance of the statement Τ”. Under a suitable definition of forcing we obtain the conclusion that intuitionistic logic is the class of only those formulas which we are forced to accept by every state of every scientific investigation. The notion of admission under a certain definition analogically leads to intuitionistic logic; on the other hand in the case of another, more natural definition, it leads to justification of a certain system of strict implication which is stronger than Lewis’ system S4 and which overlaps system S5. It can be said that assuming quite a natural definition of admission this system contains only those formulas the acceptance of which is admitted by every state of every scientific investigation.     

Beyond the exclusively propositional era

Contemporary epistemology has assumed that knowledge is represented in sentences or propositions. However, a variety of extensions and alternatives to this view have been proposed in other areas of investigation. We review some of these proposals, focusing on (1) Ryle's notion of knowing how and Hanson's and Kuhn's accounts of theory-laden perception in science; (2) extensions of simple propositional representations in cognitive models and artificial intelligence; (3) the debate concerning imagistic versus propositional representations in cognitive psychology; (4) recent treatments of concepts and categorization which reject the notion of necessary and sufficient conditions; and (5) parallel distributed processing (connectionist) models of cognition. This last development is especially promising in providing a flexible, powerful means of representing information nonpropositionally, and carrying out at least simple forms of inference without rules. Central to several of the proposals is the notion that much of human cognition might consist in pattern recognition rather than manipulation of rules and propositions.The preparation of this article was supported in part by National Institutes of Health Grants No. NICHD-19265 and NICHD-06016. We would like to thank Rita Anderson, David Blumenfeld, Robert McCauley, and Patricia Siple for helpful discussions on points in this paper.     

On the notion of effectiveness

This paper focuses on two notions of effectiveness which are not treated in detail elsewhere. Unlike the standard computability notion, which is a property of functions themselves, both notions of effectiveness are properties of interpreted linguistic presentations of functions. It is shown that effectiveness is epistemically at least as basic as computability in the sense that decisions about computability normally involve judgments concerning effectiveness. There are many occurrences of the present notions in the writings of logicians; moreover, consideration of these notions can contribute to the clarification and, perhaps, solution of various philosophical problems, confusions and disputes.