Propositional Part Research Articles

I. A comment on performative, subject, and proposition in habermas's theory of communication

Habermas claims that the concept of ‘communicative action’ can be explained by illocutionary acts alone. It appears to me that his explanation collapses into a sort of intentional theory (2[i]). Habermas maintains further that a speech act consists of three components which are ‘correlated’ to three worlds and to three validity claims. However, he also seems to mean that all worlds and validity claims are correlated to just one; the so‐called propositional component. One consequence is that the propositional content, not the illocutionary act, determines the main mode of at least some speech acts. Another is that the T as used in an expressive speech act will occur in the propositional part of the act and not, as claimed by Habermas, in the performative sentence (2[iii]). In 2(ii) two other problems concerning Habermas's view on the concept of ‘I’ ('the subject') are discussed.

Nondeterministic three-valued logic: Isotonic and guarded truth-functions

Nondeterministic programs occurring in recently developed programming languages define nondeterminate partial functions. Formulas (Boolean expressions) of such nondeterministic languages are interpreted by a nonempty subset of {T (“true”), F (“false”), U (“undefined)}. As a semantic basis for the propositional part of a corresponding nondeterministic three-valued logic we study the notion of a truth-function over {T, F, U} which is computable by a nondeterministic evaluation procedure. The main result is that these truth-functions are precisely the functions satisfying four basic properties, called\( \subseteq \)-isotonic,\( \subseteq \)−-isotonic, hereditarily guarded, and hereditarily guard-using, and that a function satisfies these properties iff it is explicitly definable (in a certain normal form) from “if..then..else..fi”, binary choice, and constants.

An investigation of the propositional calculus used in a particular system of logic

This paper contains a summary of some of the results obtained by the author during a study of the propositional part (denoted by A) of a system of free-from-contradiction, type-free logic set up in 1950 by Ackermann (1). It is shown that in its original form the calculus does not possess the desired properties with respect to equivalence of formulae. A calculus A′, which it is shown may be considered as a ‘minimal’ satisfactory extension of A, is constructed. A′ is compared and contrasted with an alternative form A″ of A given by Ackermann in a paper published in 1952 (2). It is proved that A″ is a proper extension of A′. Among the properties of A′ and A″ which are obtained is the resuit that neither calculus possesses a finite complete model. Reference is made to the solution of the decision problem for A, and it is indicated that it is thought that the corresponding problems can probably be solved for A′ and A″. The proofs of many of the results mentioned in the paper are, if given in detail, rather long. In such cases, from space considerations, only outline proofs are given. Complete proofs are contained in (5), which reference also contains several additional properties of the calculi considered.