Encoding many-valued logic in {\lambda}-calculus

We extend the well-known Church encoding of two-valued Boolean Logic in $\lambda$-calculus to encodings of $n$-valued propositional logic (for $3\leq n\leq 5$) in well-chosen infinitary extensions in $\lambda$-calculus. In case of three-valued logic we use the infinitary extension of the finite $\lambda$-calculus in which all terms have their B\"ohm tree as their unique normal form. We refine this construction for $n\in\{4,5\}$. These $n$-valued logics are all variants of McCarthy's left-sequential, three-valued propositional calculus. The four- and five-valued logic have been given complete axiomatisations by Bergstra and Van de Pol. The encodings of these $n$-valued logics are of interest because they can be used to calculate the truth values of infinitary propositions. With a novel application of McCarthy's three-valued logic we can now resolve Russell's paradox. Since B\"ohm trees are always finite in Church's original $\lambda{\mathbf I}$-calculus, we believe their construction to be within the technical means of Church. Arguably he could have found this encoding of three-valued logic and used it to resolve Russell's paradox.Comment: 15 page

Комплексная теория истины самореферентных предложений для (¬,↔)-языка

Комплексная теория истины самореферентных предложений для (¬,↔)-язык

Formal foundations for semantic theories of nominalisation

This paper develops the formal foundations of semantic theories dealing with various kinds of nominalisations. It introduces a combination of an event-calculus with a type-free theory which allows a compositional description to be given of such phenomena like Vendler's distinction between perfect and imperfect nominals, iteration of gerunds and Cresswell's notorious non-urrival of'the train examples. Moreover, the approach argued for in this paper allows a semantic explanation to be given for a wide range of grammatical observations such as the behaviour of certain tpes of nominals with respect to their verbal contexts or the distribution of negation in nominals

An Epistemicist Solution to Curry's Paradox

This paper targets a series of potential issues for the discussion of, and modal resolution to, the alethic paradoxes advanced by Scharp (2013). I aim, then, to provide a novel, epistemicist treatment to Curry's Paradox. The epistemicist solution that I advance enables the retention of both classical logic and the traditional rules for the alethic predicate: truth-elimination and truth-introduction

Some remarks on formal description of God's omnipotence

There are proposed two simple formal descriptions of the notion of God’s omnipotence which are inspired by formalizations of C. Christian and E. Nieznański. Our first proposal is expressed in a modal sentential language with quantifires. The second one is formulated in first order predicate language. In frame of the second aproach we admit using self-referential expressions. In effect we link our considerations with so called paradox of God’s omnipotence and reconstruct some argumentation against the possibility of reference God’s omnipotence to a lack of itself

Gestalt Shifts in the Liar Or Why KT4M Is the Logic of Semantic Modalities

ABSTRACT: This chapter offers a revenge-free solution to the liar paradox (at the centre of which is the notion of Gestalt shift) and presents a formal representation of truth in, or for, a natural language like English, which proposes to show both why -- and how -- truth is coherent and how it appears to be incoherent, while preserving classical logic and most principles that some philosophers have taken to be central to the concept of truth and our use of that notion. The chapter argues that, by using a truth operator rather than truth predicate, it is possible to provide a coherent, model-theoretic representation of truth with various desirable features. After investigating what features of liar sentences are responsible for their paradoxicality, the chapter identifies the logic as the normal modal logic KT4M (= S4M). Drawing on the structure of KT4M (=S4M), the author proposes that, pace deflationism, truth has content, that the content of truth is bivalence, and that the notions of both truth and bivalence are semideterminable

Field's Logic of Truth

Saving Truth from Paradox is a re-exciting development. The 70s and 80s were a time of excitement among people working on the semantic paradoxes. There were continual formal developments, with the constant hope that these results would yield deep insights. The enthusiasm wore off, however, as people became more cognizant of the disparity between what they had accomplished, impressive as it was, and what they had hoped to accomplish. They moved onto other problems that they hoped would prove more yielding. That, at least, was how it seemed to me, so I was delighted to see a dramatically new formal development that is likely to rekindle our enthusiasm