Many-Valued Logic

New directions in the analysis and interactive elicitation of personal construct systems

“…In this section we show how the logical analysis extends to the multivalued logics (Rescher, 1969) with which one can analyse grids based on rating scales. Kelly (1955) presented constructs as binary categories and based his own methodology for eliciting constructs on this.…”

“…The related logical calculus derived from fuzzy set theory in the same way that the classical predicate calculus may be related to conventional set theory is of particular interest for this paper and has been presented as a system for fuzzy reasoning. This logic has been found to be one already studied by the Polish logician J~ukasiewicz (Rescher, 1969) and of particular importance since White (1979) has shown recently that it avoids paradoxes such as that of Russell's "barber" (Hughes & Brecht, 1976) which arise from the unrestricted use of the axiom of comprehension in naive set theory. Since its inception fuzzy set theory has been used to model human verbal reasoning and concept processing.…”

MacColl’s Modes of Modalities

“…Simons concludes against Nicholas Rescher [Rescher 1969] that MCL is not an essentially many-valued logic, because no one of these three criteria are respected by it. On the one hand, it fails to satisfy (MV1) insofar as the three additional values are not introduced into Five besides truth and falsehood: ε ({1.4}) belongs to τ ({1.3, 1.4, 1.6}} within the truth-class 1, since whatever is true is so plainly (without qualification), necessarily (always), or contingently (sometimes, but not always); the same goes for η and ι, with respect to the falsehood-class 0.…”

“…However, a difference between the bivalent set of classicists and MCL lies in the process of partition. Actually, such a division of the classes of truth and falsehood into a number of subclasses explains the view defended by Rescher [Rescher 1969] that MCL is both many-valued and modal: a partition augments the cardinality of the initial set beyond two elements, and the additional elements stand for different modes of truth.…”

Four-Valued Extension of Rough Sets

“…Obviously, the proposed implication → extends the usual two-valued logical implication, i.e. when we only consider the logical values t and f. It also extends the implication of the Kleene three-valued logic [5,16] in the sense that when we restrict truth values to {t, f, u} (or {t, f, i}), we obtain Kleene's implication. On the other hand, our implication differs from the material implication → proposed in [2], on the following two cases: u → i = t while u → i = i and i → u = t while i → u = i.…”