We will study several weak axiom systems that use the Subtraction and Division primitives (rather than Addition and Multiplication) to formally encode the theorems of Arithmetic. Provided such axiom systems do not recognize Multiplication as a total function, we will show that it is feasible for them to verify their Semantic Tableaux, Herbrand, and Cut-Free consistencies. If our axiom systems additionally do not recognize Addition as a total function, they will be capable of recognizing the consistency of their Hilbert-style deductive (...) proofs. Our axiom systems will not be strong enough to recognize their Canonical Reflection principle, but they will be capable of recognizing an approximation of it, called the "Tangibility Reflection Principle". We will also prove some new versions of the Second Incompleteness Theorem stating essentially that it is not possible to extend our exceptions to the Incompleteness Theorem much further. (shrink)
Let us recall that Raphael Robinson's Arithmetic Q is an axiom system that differs from Peano Arithmetic essentially by containing no Induction axioms , . We will generalize the semantic-tableaux version of the Second Incompleteness Theorem almost to the level of System Q. We will prove that there exists a single rather long Π 1 sentence, valid in the standard model of the Natural Numbers and denoted as V, such that if α is any finite consistent extension of Q + (...) V then α will be unable to prove its Semantic Tableaux consistency. The same result will also apply to axiom systems α with infinite cardinality when these infinite-sized axiom systems satisfy a minor additional constraint, called the Conventional Encoding Property. Our formalism will also imply that the semantic-tableaux version of the Second Incompleteness Theorem generalizes for the axiom system IΣ 0 , as well as for all its natural extensions. (This answers an open question raised twenty years ago by Paris and Wilkie .). (shrink)
This article will study a class of deduction systems that allow for a limited use of the modus ponens method of deduction. We will show that it is possible to devise axiom systems α that can recognize their consistency under a deduction system D provided that: (1) α treats multiplication as a 3-way relation (rather than as a total function), and that (2) D does not allow for the use of a modus ponens methodology above essentially the levels of Π1 (...) and Σ1 formulae. Part of what will make this boundary-case exception to the Second Incompleteness Theorem interesting is that we will also characterize generalizations of the Second Incompleteness Theorem that take force when we only slightly weaken the assumptions of our boundary-case exceptions in any of several further directions. (shrink)
This paper will introduce the notion of a naming convention and use this paradigm to both develop a new version of the Second Incompleteness Theorem and to describe when an axiom system can partially evade the Second Incompleteness Theorem.
In 1981, Paris and Wilkie raised the open question about whether and to what extent the axiom system did satisfy the Second Incompleteness Theorem under Semantic Tableaux deduction. Our prior work showed that the semantic tableaux version of the Second Incompleteness Theorem did generalize for the most common definition of appearing in the standard textbooks.However, there was an alternate interesting definition of this axiom system in the Wilkie–Paris article in the Annals of Pure and Applied Logic 35 , pp. 261–302 (...) which we did not examine in our year-2002 article in the Journal of Symbolic Logic. Our first goal is to show that the incompleteness results of our prior paper can generalize in this alternate context. We will also develop a formal analysis, using a new technique called Passive Induction, that is simpler than the formalism we had used before.A further reason our results are of interest is that we have shown in a companion paper published in Electronic Notes in Theoretical Computer Science 165 , pp. 213–226 that some very unorthodox axiomizations for are anti-thresholds for the Herbrandized version of the Second Incompleteness Theorem. Thus, different axiomizations for have nearly fully opposite incompleteness properties.This paper is self-contained. It will not require a knowledge of our earlier results. (shrink)
Gödel’s Second Incompleteness Theorem states axiom systems of sufficient strength are unable to verify their own consistency. We will show that axiomatizations for a computer’s floating point arithmetic can recognize their cut-free consistency in a stronger respect than is feasible under integer arithmetics. This paper will include both new generalizations of the Second Incompleteness Theorem and techniques for evading it.
O presente artigo disserta sobre duas características centrais ao holismo epistemológico de Willard Quine: a sustentação de critérios empíricos de avaliação por meio das sentenças observacionais e a defesa de que a ontologia fisicalista é decisiva para a avaliação de teorias.
This paper deals with Quine's several attempts To define the concept of underdetermination of scientifics theories in some of his articles and with the dependence of this definition on other concepts of Quine's semantic holism. To define "underdetermination”, Quine needs to explain the relationship between theory and observation. His position concerning this subject can be criticized, on the one hand, by saying that it gives an insufficient criterion for "underdetermination", and, on the other hand, by asserting that it is still (...) too close to the reductionist's conception of truth. (shrink)
Resumo : O objetivo deste artigo será o de tentar verificar qual a relação de Willard Van Orman Quine com a teoria deflacionária da verdade. Para tanto, começaremos com o exame dos portadores de verdade, com a intenção de saber a opinião de Quine sobre qual, dentre outros, seria o verdadeiro portador ou veículo da verdade, ou seja, saber, segundo Quine, de que dizemos ser verdadeiro ou falso. Reduziremos nosso exame às sentenças e proposições, e uma vez identificado o (...) portador de verdade escolhido por nosso filósofo aqui em questão, a saber, a sentença , faremos uma breve apresentação das terias “deflacionárias” e “da correspondência” no que diz respeito à verdade. Por fim, com base em tal apresentação, trataremos de identificar a posição quiniana no que se refere à verdade e, consequentemente, sua relação com o deflacionismo. (shrink)
The contemporary popularity of the prefix post has found its expression also in the realm of analytic philosophy - there arises something which has come to be called postanalytic philosophy. We put forward that this branch of the analytic movement, germinating in the writings of the late Ludwig Wittgenstein, of Willard Van Orman Quine and Willfrid Sellars, and coming to full blossom with Nelson Goodman, Donald Davidson, Hilary Putnam and Richard Rorty, springs first and foremost from the repudiation of (...) the doctrine of logical atomism as entertained by Bertrand Russell and the early Wittgenstein. The dismantling of this doctrine means especially the denial of the following four points: (1) the atomistic character of language; (2) the sharp boundary between analytic and contingent statements; (3) the sharp boundary between 'given' and 'inferred' knowledge; and also (4) the essential cummulativity of knowledge. Postanalytic philosophy is thus essentially holistic; and we put forward that in fact postanalytic philosophy equals analytic philosophy minus logical atomism. (shrink)
--The energy of the new world, By E. E. Slosson.--The new energies and the new man, by W. D. Scott.--The future of our economic system, by F S. Deibler.--Business in the new era, by W. B. Hotchkiss.--Consumers in the modern world, by Stuart Chase.
The strong weak truth table (sw) reducibility was suggested by Downey, Hirschfeldt, and LaForte as a measure of relative randomness, alternative to the Solovay reducibility. It also occurs naturally in proofs in classical computability theory as well as in the recent work of Soare, Nabutovsky, and Weinberger on applications of computability to differential geometry. We study the sw-degrees of c.e. reals and construct a c.e. real which has no random c.e. real (i.e., Ω number) sw-above it.