Popper on Definitions

Summary  In the present paper I shall first summarize Popper’s criticism of the traditional method of definition, and then go on to comment critically on his own views on the form and function of so-called nominalist definitions.     

Relevant deduction

This paper presents an outline of a new theory of relevant deduction which arose from the purpose of solving paradoxes in various fields of analytic philosophy. In distinction to relevance logics, this approach does not replace classical logic by a new one, but distinguishes between relevance and validity. It is argued that irrelevant arguments are, although formally valid, nonsensical and even harmful in practical applications. The basic idea is this: a valid deduction is relevant iff no subformula of the conclusion is replaceable on some of its occurrences by any other formula salva validitate of the deduction. The paper first motivates the approach by showing that four paradoxes seemingly very distant from each other have a common source. Then the exact definition of relevant deduction is given and its logical properties are investigated. An extension to relevance of premises is discussed. Finally the paper presents an overview of its applications in philosophy of science, ethics, cognitive psychology and artificial intelligence.This paper is based on the first part of my habilitation Schurz (1989). For various helps and comments I am indebted to Paul Weingartner. Andrzej Wrónski, Georg Kreisel, David Miller, Kit Fine, Terence Parsons and Peter Woodruff.     

Harmonising natural deduction

Prawitz proved a theorem, formalising ‘harmony’ in Natural Deduction systems, which showed that, corresponding to any deduction there is one to the same effect but in which no formula occurrence is both the consequence of an application of an introduction rule and major premise of an application of the related elimination rule. As Gentzen ordered the rules, certain rules in Classical Logic had to be excepted, but if we see the appropriate rules instead as rules for Contradiction, then we can extend the theorem to the classical case. Properly arranged there is a thoroughgoing ‘harmony’, in the classical rules. Indeed, as we shall see, they are, all together, far more ‘harmonious’ in the general sense than has been commonly observed. As this paper will show, the appearance of disharmony has only arisen because of the illogical way in which natural deduction rules for Classical Logic have been presented.     

Aspects of analytic deduction

Let ? be the ordinary deduction relation of classical first-order logic. We provide an “analytic” subrelation ?³ of ? which for propositional logic is defined by the usual “containment” criterion $$\Gamma \vdash ^a \varphi iff \Gamma \vdash \varphi and Atom(\varphi ) \subseteq Atom(\Gamma ),$$ whereas for predicate logic, ?^a is defined by the extended criterion $$\Gamma \vdash ^a \varphi iff \Gamma \vdash \varphi and Atom(\varphi ) \subseteq ' Atom(\Gamma ),$$ where Atom(?) $ \subseteq '$ Atom(Γ) means that every atomic formula occurring in ? “essentially occurs” also in Γ. If Γ, ? are quantifier-free, then the notions “occurs” and “essentially occurs” for atoms between Γ and ? coincide. If ? is formalized by Gentzen's calculus of sequents, then we show that ?^a is axiomatizable by a proper fragment of analytic inference rules. This is mainly due to cut elimination. By “analytic inference rule” we understand here a rule r such that, if the sequent over the line is analytic, then so is the sequent under the line. We also discuss the notion of semantic relevance as contrasted to the previous syntactic one. We show that when introducing semantic sequents as axioms, i.e. when extending the pure logical axioms and rules by mathematical ones, the property of syntactic relevance is lost, since cut elimination no longer holds. We conclude that no purely syntactic notion of analytic deduction can ever replace successfully the complex semantico-syntactic deduction we already possess.     

Mental models and deduction

According to the mental-model theory of deductive reasoning, reasoners use the meanings of assertions together with general knowledge to construct mental models of the possibilities compatible with the premises. Each model represents what is true in a possibility. A conclusion is held to be valid if it holds in all the models of the premises. Recent evidence described here shows that the fewer models an inference calls for, the easier the inference is. Errors arise because reasoners fail to consider all possible models, and because models do not normally represent what is false, even though reasoners can construct counterexamples to refute invalid conclusions.