Transreal Logical Space of All Propositions

Gomide, Walter; Reis, Tiago; Anderson, James A.

doi:10.1007/978-94-017-7236-5_16

Cited by 3 publications

(3 citation statements)

References 11 publications

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Section: Transreal Arithmeticmentioning

confidence: 99%

“…In [27], a transreal logical space was described which allows logical transformations to be applied simultaneously to all logical propositions. In [28], it was proved that within this space there is an infinitely dense set of hypercyclic universal worlds, each of which approximates all worlds in the space by repeated mechanical application of the backward shift operator. In [29] infinitely scaleable pipeline machines were described that employ transreal arithmetic so that no hardware or software exception handling is needed for logical exceptions, though physical faults within the machine must still be handled.…”

Section: Transreal Arithmeticmentioning

confidence: 99%

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Transreal Foundation for Floating-Point Arithmetic

Anderson¹

2023

Transmath

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Software Engineering relies, to a large extent, on formal software standards and logical means for specifying and verifying computer programs. Among these the IEEE 754 standard for floating-point arithmetic is widely used. We criticise the standard from the standpoint of transreal arithmetic. Transreal arithmetic was derived from projective geometry using a double cover to provide signed infinities on the horizon and nullity at the point of projection. These infinities and nullity have some similarities with IEEE 754 floating-point infinities and NaNs but there are important differences. We explore the differences by analysing the standard at three levels: commentary within the standard, abstract datatypes, and bit patterns. We find that all of the differences are coincident with faults in the standard. Obviously a correct standard would better support the specification, development and testing of numerical software. We discuss how the standard can be corrected, in its own terms, or by adopting transreal arithmetic as its theoretical foundation. We also discuss emulation of transreal arithmetic in IEEE 754 processing systems and address accusations that transreal arithmetic plagiarised the standard.

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Section: Transreal Arithmeticmentioning

confidence: 99%

Section: Transreal Arithmeticmentioning

confidence: 99%

Transreal Foundation for Floating-Point Arithmetic

Anderson¹

2023

Transmath

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“…The topology of the transreal [13] and transcomplex [9] numbers was developed and lead to transreal analysis [12] -all of which confirmed transreal nullity, Φ = 0/0, as the unique unordered number and the transreal numbers negative infinity, −∞ = −1/0, and positive infinity, ∞ = 1/0, as, respectively, the least and greatest of the ordered numbers. There was some development of paraconsistent logics [8] [14] and Boolean logic was generalised to a wide class of trans-Boolean logics [10]. A number of applications of transreal arithmetic were discussed in Computer Science [4] [18] [17] [5] [6] [2] and Mathematical Physics [3].…”

Section: Introductionmentioning

confidence: 99%

Foundations of Transmathematics

Anderson¹

2019

Transmath

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Transmathematics has the ambition to be a total mathematics. Many areas of the usual mathematics have been totalised in the transmathematics programme but the totalisations have all been carried out with the usual set theory, ZFC, Zermelo-Fraenkel set theory with the Axiom of Choice. This set theory is adequate but it is, itself, partial. Here we introduce a total set theory as a foundation for transmathematics. Surprisingly we adopt naive set theory. It is usually considered that the Russell Paradox demonstrates that naive set theory is incoherent because an apparently well-specified set, the Russell Set, cannot exist. We dissolve this paradox by showing that the specification of the Russell Set admits many unproblematical sets that do not contain themselves and, furthermore, unequivocally requires that the Russell Set does not contain itself because, were it to do so, that one element of the Russell Set would have contradictory membership. Having resolved the Russell Paradox, we go on to make the case that naive set theory is a paraconsistent logic. In order to demonstrate the sufficiency of naive set theory, as a basis for transmathematics, we introduce the transordinals. The von Neumann ordinals supply the usual ordinals, the simplest unordered set is identical to transreal nullity, and the Russell Set, excluding nullity, is the greatest ordinal, identical to transreal infinity. The generalisation of the transordinals to the whole of established transmathematics is already known. As naive set theory contains all other set theories, it provides a backwardly compatible foundation for the whole of mathematics.

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Transreal Numbers and Sentient Logic

Gomide

2022

Advances in Media, Entertainment, and the Arts

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This chapter introduces the basic concepts of Total Semantics based on transreal numbers, a new field of mathematics in which division of zero is allowed. In the logic created from the transreals, it is possible to evaluate non-propositional objects such as sensations or feelings as having truth value. By covering all semantical possibility, transreal semantics enables a “poetical discourse” based on images and an immediate intuition of reality, an intuition that is both sensory and mental. This describes, and defines, a new type of epistemic operation—a sentient one—and leads to the possibility of developing a formal epistemic logic that assesses how we obtain or acquire knowledge from the perspective of sentient agents.

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Transreal Logical Space of All Propositions

Cited by 3 publications

References 11 publications

Transreal Foundation for Floating-Point Arithmetic

Transreal Foundation for Floating-Point Arithmetic

Foundations of Transmathematics

Transreal Numbers and Sentient Logic

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