Inspired by an interesting quotation from the literature, we propose four modalities, called ‘sane belief’, ‘insane belief’, ‘reliable belief’ and ‘unreliable belief’, and introduce logics with each operator as the modal primitive. We show that the four modalities constitute a square of opposition, which indicates some interesting relationships among them. We compare the relative expressivity of these logics and other related logics, including a logic of false beliefs from the literature. The four main logics are all less expressive than the (...) standard modal logic over various model classes, and the logics of sane and insane beliefs are, respectively, equally expressive as the logics of unreliable and reliable beliefs on any class of models. The logics of reliable and unreliable beliefs are then combined into a bimodal logic, which turns out to be equally expressive as the standard modal logic. Despite this, we cannot obtain a complete axiomatization of the minimal bimodal logic, by simply translating the axioms and rules of the minimal modal logic $\textbf{K}$ into the bimodal language. We then introduce a schematic modality which unifies reliable and unreliable beliefs and axiomatize it over the class of all frames and also the class of serial frames. This line of research is finally extended to unify sane and insane beliefs and some axiomatizations are given. (shrink)

This discussion explores the possibility of distinguishing a tighter notion of contrariety evident in the Square of Opposition, especially in its modal incarnations, than as that binary relation holding statements that cannot both be true, with or without the added rider ‘though can both be false’. More than one theorist has voiced the intuition that the paradigmatic contraries of the traditional Square are related in some such tighter way—involving the specific role played by negation in contrasting them—that distinguishes them from (...) other pairs of incompatible statements constructed from the same conceptual materials. Prominent among examples, these other nonstandard pairs are the ‘new contraries’ presented by Robert Blanché’s hexagon(s) of opposition. With special, though not exclusive, attention to these cases, we investigate whether contrariety in the distinguished sense can be captured by adding to the incompatibility condition the further demand that the pair of statements concerned can be represented as the results of applying some sentence operator to the content in its scope, for one of the pair, and, for the other, the application of that same operator to the negation of that content. For one of the two cases, a Blanché case, of nonstandard contrariety singled out for attention, the question of whether such a representation is available is settled at the end of Section 4, and then in a more satisfying way in Section 5, though for the other case, noticed by Peter Simons, the question remains open, after some tentative discussion in one subsection, 6.2, of an Appendix (Section 6). (shrink)

The paper submits surprising results of systematical investigating a formal-ethical aspect of conjoining Wittgenstein’s, Moore’s, Parmenides’, Gödel’s, and Łukasiewicz’s ideas. A critique of Wittgenstein’s critique of the natural language of ethics and of metaphysics results in submitting and elaborating a new paradigm of metaphysics as formal axiology. In result, the classical metaphysics and ethics of moral rigor are represented as two-valued algebraic systems of metaphysics and formal ethics respectively. By means of this algebraic model, all the well-known scandal-making metaphysical tenets (...) of Parmenides are produced as translations of corresponding algebraic equations from the symbolic language to the natural one. At the level of submitted discrete mathematical model of formal axiology, Parmenides’ metaphysical concepts “consistency” and “inconsistency,” “completeness” and “incompleteness” are compared with Gödel’s logic ones. Formal-axiological meanings of the words “consistency,” “incompleteness,” “being,” “nonbeing,” “movement,” “knowledge,” “belief,” etc., are considered as moral-evaluation-functions determined by one moral-evaluation-variable. Binary moral-evaluation-functions are studied as well. The functions are precisely defined by tables. Precise definitions of “formal-axiological-equivalence,” “formal-axiological-law,” and “formal-axiological contradiction” are submitted. Thus, one can either generate or examine formal-axiological equations of algebra of metaphysics by “computing” relevant compositions of moral-value-functions. Using this “moral-value-table-computation-technique,” one can arrive to a surprising conclusion that both the notorious sentence of Moore and the incompleteness sentence of Gödel are formally-axiologically inconsistent ones: Hence, they are formally-axiologically equivalent. For overcoming the negative psychological effect of such a surprising result, the author has used graphic models explicating the famous Łukasiewicz’s statement “Logic is morality of thought and speech.”. (shrink)