Samuel R. Buss and Aleksandar Ignjatović. Unprovability of Consistency Statements in Fragments of Bounded Arithmetic.

The author gentzenizes the positive fragmentsT + andR + of relevantT andR using formulas with, prefixes (subscripts). There are three main Gentzen formulations ofS +{T+,R +} calledW 1 S +,W 2 S + andG 2 S +. The first two have the rule of modus ponens. All of them have a weak rule DL for disjunction introduction on the left. DL is not admissible inS + but it is needed in the proof of a cut elimination theorem forG 2 S (...) +.W 1 S + has a weak rule of weakeningW 1 and it is not closed under a general transitivity rule. This allows the proof that A inS + iff A inW 1 S +. From the cut elimination theorem forG 2 S + it follows that if A inS +, then A inG 2 S +. In order to prove the converse,W 2 S + is needed. It contains modus ponens, transitivity, and a restricted weakening rule.G 2 S + is contained inW 2 S + and there is a proof that A inW 2 S + iff A inW 1 S +. (shrink)

This paper is a study of four subscripted Gentzen systems G u R +, G u T +, G u RW + and G u TW +. [16] shows that the first three are equivalent to the semilattice relevant logics u R +, u T + and u RW + and conjectures that G u TW + is, equivalent to u TW +. Here we prove Cut Theorems for these systems, and then show that modus ponens is admissible — which (...) is not so trivial as one normally expects. Finally, we give decision procedures for the contractionless systems, G u TW + and G u RW +. (shrink)

In this article we present a p-adic valued probabilistic logic equation image which is a complete and decidable extension of classical propositional logic. The key feature of equation image lies in ability to formally express boundaries of probability values of classical formulas in the field equation image of p-adic numbers via classical connectives and modal-like operators of the form Kr, ρ. Namely, equation image is designed in such a way that the elementary probability sentences Kr, ρα actually do have their (...) intended meaning—the probability of propositional formula α is in the equation image-ball with the center r and the radius ρ. Due to modal nature of the operators Kr, ρ, it was natural to use the probability Kripke like models as equation image-structures, provided that probability functions range over equation image instead of equation image or equation image. (shrink)

History of Mongolian philosophy, intellectual life, and politics; covers until 14th century.

A Brief Description of Neoplatonism R. Baine Harris Old Dominion University There are essentially three ways in which Neoplatonism may be considered to be ...

The time taken to react voluntarily to a stimulus is far longer than can be accounted for by ordinary processes of nerve conduction and synaptic delay, and varies unpredictably from trial to trial. Though random, the distribution of reaction times usually follows a relatively simple law, which in turn can be explained by the LATER model, in which a decision signal, representing belief in the existence of the target, rises in response to incoming sensory evidence from an initial value to (...) a criterion level at which action is initiated. But the rate of rise fluctuates randomly from trial, to an extent that cannot be explained by sensory noise at the input. These conclusions, confirmed by recording from neurons in the frontal eye fields, suggest that the randomness of reaction time is due to a deliberate, gratuitous, neural process which confers certain biological advantages as well as having implications for general ideas about the nature of ‘free will'. (shrink)

This paper examines the life and work of T. C. Chamberlin, a prominent glacial geologist who developed an interest in interdisciplinary earth science. His work on the geological agency of the atmosphere informed his understanding of climate change and other terrestrial phenomena and led him to propose a new theory of the formation of the Earth and the solar system.Chamberlin's graduate seminar at the University of Chicago in 1896 contained all the themes that informed his research programme over the next (...) three decades. These included the carbon dioxide theory of climate change in its relationship to diastrophism and oceanic circulation, the role of water vapour feedbacks in the climate system, and the relationship between multiple glaciations, the climate system, and the formation of the planet. (shrink)

In recent years, one of the key philosophical problems of modern science - the problem of determinism - has been attracting increasing attention from researchers. As a consequence, the need has arisen to draw conclusions from the investigations conducted and to outline the range of questions requiring further treatment.