The Hangman Paradox has a simple solution. The amazing refutation of the judge's decree rests on the axiom of knowledge-conservation. This axiom is false under unfavourable conditions. You can have a perfect piece of knowledge in the ordinary sense, i.e. a true justified conviction, and yet be unable to conserve it. More interesting than its solution is the element of self-reference, connecting the Hangman via Moore's Paradox and Buridan's Epistemic Paradox with the Liar. This one, I think, has also a (...) natural solution, but less simple. The basic idea is given here, but the technical treatment goes far beyond this paper. It requires a strong, but conservative, extension of classical and three-valued logic to a six-valued logic with infinitely many levels of reflection. (shrink)

The set-theoretic and individual-theoretic analyses of collections make the same mistake: their ontologies are far removed from language. Our basic notion, the collector, has a clear linguistic counterpart: the definite article. Our theory furnishes a theory of definite descriptions as a special case. We use 3-valued valued logic in order to come to grips with the existential presuppositions of the collections.

Keine ausführliche Beschreibung für "Die dreiwertige Logik der Sprache" verfügbar.

SummaryIf there is truth, there is formal truth and finally truth in the set universe \V. Confiding in formal truth and reflecting this confidence we are led beyond all set and class theories to the Absolute Indefinite und formal inexpressibility of the whole concept of formal truth – a logical path to Mahlyha Buddhism.

Quine's metalogical 'quasiquotation' is formally added to classical first-Order logic; the resulting system lq is stronger and more natural than all former systems of quotational logic. Lq contains object-Variables ranging over the universe u and expression-Variables ranging over the set e of all expressions of lq; e is a subset of u. Object-Quantifiers are referential, Expression-Quantifiers are substitutional; only the latter ones bind into quasiquotations. Lq contains its own syntactic metatheory and arithmetics. Natural proofs of godel's and tarski's theorems are (...) given for lq. (shrink)

Let k be the number of all pure sets, und let K be the number of all pure classes. The Platonist, assuming that the class \V of all pure sets cannot be enlarged by any formal means, will come to the conclusion that cardinalities between k und K are conceptually impossible. So the Continuum Hypothesis is true on class level. And if this feature of \V is reflected by all infinite sets \Va, the General Continuum Hypothesis is true on set (...) level. (shrink)

Let k be the number of all pure sets, und let K be the number of all pure classes. The Platonist, assuming that the class \V of all pure sets cannot be enlarged by any formal means, will come to the conclusion that cardinalities between k und K are conceptually impossible. So the Continuum Hypothesis is true on class level. And if this feature of \V is reflected by all infinite sets \Va, the General Continuum Hypothesis is true on set (...) level. (shrink)

A direct path that has been missed for 100 years leads from the oldest paradoxes straight to mysticism, via (the concept of) logical and mathematical truth, since the purely formal truth is an absolutely univocal, absolutely timeless and absolutely unbounded reference. I present three theses in passing: (1) logicians fail to fully appreciate the basic mathematical idea of truth and consequently push the semantic paradoxes aside. Otherwise they would have come to adopt the reflexive logic LR* right after Cantor (more (...) on this briefly in Sect. 4. For a more precise elaboration see my Logik der Unbestimmtheiten und Paradoxien (Blau 2008)). (2) Mathematicians fail to fully appreciate the basic mathematical concept of number and push Zeno’s paradoxes aside. Otherwise they would have come to adopt linear arithmetic ${\mathbb{L}^{*}}$ right after Cantor (more on this briefly in Sects. 1 and 5, for further elaboration see my planned Grundparadoxien und grenzenlose Arithmetik. The technical parts are complete, yet philosophically it is as incomplete as Sect. 6). (3) Philosophers fail to fully appreciate the paradox of subject and object and push the Darwinian argument for Cartesian (mind–body)-interaction aside, otherwise they would have been en route towards mysticism since Darwin’s times (more on this in Sect. 6, for a little more precise elaboration see Blau in Die Logik der Unbestimmtheiten und Paradoxien. Synchron Publishers, Heidelberg, 2008, Ch. 1.5). (shrink)

A non-classical logic is proposed that extends classical logic and set theory as conservatively as possible with respect to three domains: the logic of natural language, the logcal foundations of mathematics, and the logical-philosophical paradoxes. A universal mechanics of consciousness connects these domains, and its best witness is the liar paradox. Its solution rests formally on a subject-object partition, mentally arising and disappearing perpetually. All deep paradoxes are paradoxes of consciousness. There are two kinds, solvable ones and unsolvable ones. The (...) solvable ones disappear by proper partitions, the unsolvable ones arise by inevitably improper partitions enforced by the natural worldview. The categorial distinctions of subject and object, certainty and truth, consciousness and being, inner freedom and outer constraint, psyche and physis, mental practice and formal theory indicate an immemorial mystical monism of consciousness without which mathematical Platonism could not survive. (shrink)