arXiv subjects

X. Y. Newberry

Publications and source records attributed to X. Y. Newberry.

6 recordsLinked to original sources

arXiv2023

Semantics for a Logic of Presuppositions

In 1952 P. F. Strawson proposed a logic of presuppositions. It is an interpretation of Aristotelian logic, i.e. of the logic of the traditional syllogism. In 1981 Richard Diaz published a monograph in which he presented truth-relevant logic. This paper shows that truth-relevant logic is but a propositional version of the logic of presuppositions. A semantics of the logic of presuppositions is developed using truth-relevant logic. The semantics is then further extended to polyadic logic and some consequences discussed.

cs.LO↗

arXiv2023

Truth-relevant Logic, Propositional Calculus

The thesis of this paper is that truth-relevant logic is a better foundation for mathematics than classical logic. It is a system proposed by Richard Diaz in 1981. In a certain sense t-relevant logic is based on Kleene strong tables. These define a system with three values: true, false, unknown. It turns out that there exist tautologies with the following property: there exists a proper subset of propositional variables (t-relevant variables) such that for all combinations true/false the tautology will be true, that is, the rest of the variables (redundant variables) occurring in the tautology can be unknown. We consider such compound sentences as neither true nor false. Philosophical justification is provided. Proof system based on tableaux is proposed. The following theorem is proved: tautology L = (R1 v ~R1) & (R2 v ~R2) & ... & (Rm v ~Rm), where {R1, R2, ... Rm} is a subset of t-relevant propositional variables, and = is equivalence according to Kleene strong tables.

math.LO↗

arXiv2023

A Proof System for a Logic of Presuppositions

The paper proposes a derivation system for a logic of presuppositions as introduced by P. F. Strawson. It is based on truth-relevant logic described by M. Richard Diaz in 1981. In another paper I outlined a derivation system for t-relevant logic based on truth trees. The conclusion was that a tautology is truth-relevant iff all the variables in a tree are self-contradicted. It is possible that a tree terminates without all the variables self-contradicting themselves. In this case the pertaining formula is still a tautology, but not a truth-relevant tautology. This concept is extended to the predicate calculus, i.e. a logic of presuppositions or a variant thereof.

cs.LO↗

arXiv2022

Getting around the Halting Problem

There are numbers k and s and a URM program A(n,m) satisfying the following conditions. 1. If A(n,m) halts, then Cn(m) diverges. 2. For all n, C_k(n) = A(n,n) and C_s(n) = C_k(s). 3. A(k,s) halts and for all n, A(s,n) diverges. Here C_n(_) is a program with index n in some exhaustive enumeration of all possible programs. This has implications for solving the liar paradox and for generalization of Gödel incompleteness theorem to formal systems other than PA.

cs.LO↗

arXiv2020

The Diagonal Lemma Fails in Aristotelian Logic

In the 1950-ies P. F. Strawson proposed the Logic of Presuppositions. In this system sentences with empty subject are considered neither true nor false. Strawson considered only simple sentences with two predicate letters. But the concept can be extended to arbitrary monadic, even polyadic sentences utilizing truth-relevant logic developed by Richard Diaz. It can be shown that self-referential Gödel's sentence in fact has an empty subject, and can thus be classified by the Logic of Presuppositions as neither true nor false.

cs.LO↗

arXiv2018

The Recursion Theorem from a Different Angle

This paper is about computability. I claim the likely existence of a program DoesHalt(Program, Input) such that DoesHalt( HaltsOnItself, AntiSelf ) halts with resounding 'NO'. HaltsOnItself( Program ) is simply DoesHalt( Program, Program ). AntiSelf() is a self-referential self-contradictory program that loops when HaltsOnItself() returns 'YES' and halts when HaltsOnItself() returns 'NO'.

cs.LO↗