What Do Paraconsistent, Undecidable, Random, Computable and Incomplete mean? A Review of Godel's Way: Exploits into an undecidable world by Gregory Chaitin, Francisco A Doria, Newton C.A. da Costa 160p (2012) (review revised 2019)

In ‘Godel’s Way’ three eminent scientists discuss issues such as undecidability, incompleteness, randomness, computability and paraconsistency. I approach these issues from the Wittgensteinian viewpoint that there are two basic issues which have completely different solutions. There are the scientific or empirical issues, which are facts about the world that need to be investigated observationally and philosophical issues as to how language can be used intelligibly (which include certain questions in mathematics and logic), which need to be decided by looking at how we actually use words in particular contexts. When we get clear about which language game we are playing, these topics are seen to be ordinary scientific and mathematical questions like any others. Wittgenstein’s insights have seldom been equaled and never surpassed and are as pertinent today as they were 80 years ago when he dictated the Blue and Brown Books. In spite of its failings—really a series of notes rather than a finished book—this is a unique source of the work of these three famous scholars who have been working at the bleeding edges of physics, math and philosophy for over half a century. Da Costa and Doria are cited by Wolpert (see below or my articles on Wolpert and my review of Yanofsky’s ‘The Outer Limits of Reason’) since they wrote on universal computation, and among his many accomplishments, Da Costa is a pioneer in paraconsistency. Those wishing a comprehensive up to date framework for human behavior from the modern two systems view may consult my book ‘The Logical Structure of Philosophy, Psychology, Mind and Language in Ludwig Wittgenstein and John Searle’ 2nd ed (2019). Those interested in more of my writings may see ‘Talking Monkeys--Philosophy, Psychology, Science, Religion and Politics on a Doomed Planet--Articles and Reviews 2006-2019 3rd ed (2019), The Logical Structure of Human Behavior (2019), and Suicidal Utopian Delusions in the 21st Century 4th ed (2019

Combining paraconsistent and dynamic logic for Qiskit

Dissertação de mestrado integrado em Engenharia FísicaThis dissertation introduces a logic aimed at combining dynamic logic and paraconsistent logic for application to the quantum domain, to reason about quantum phase properties: Paraconsistent Phased Logic Of Quantum Programs (PhLQP◦ ). In the design PhLQP◦ , firstly the dynamic was built first, Phased Logic Of Quantum Programs (PhLQP). PhLQP is itself a dynamic logic capable of dealing with quantum phase properties, quantum measurements, unitary evolutions, and entanglements in compound systems , since it is a redesign of the already existing Logic Of Quantum Programs (LQP), [14], over a representation of quantum states restricted to a space B equipped with only two computational basis, standard and Hadamard. As instances of applications of the logic PhLQP, there is a formal proof of the correctness of the Quantum Teleportation Protocol, of the 2-party and 4-party of the Quantum Leader Election (QLE) protocol, and of the Quantum Fourier Transform (QFT) operator for 1, 2 and 3 qubits . On a second stage, PhLQP was extended with the connective ◦ known as the consistency operator, a typical connective of the paraconsistent logics Logics of Formal Inconsistency (LFIs), [8, 21, 22]. The definition of consistent quantum state and a set of proper para consistent axioms for the quantum domain, Fundamental Paraconsistent Quantum Axioms (FParQAxs), were provided. An example of application of PhLQP◦ is the possibility of express and prove correctness of the universal quantum gate, the Deustch gate.Esta dissertação introduz uma lógica que tem como objectivo combinar lógica dinâmica e lógica paraconsistente com aplicação no domínio quântico, assim como expressar propriedades relacionadas com fases quânticas: PhLQP◦. No projetar da PhLQP◦, primeiramente concebeu-se a sua componente dinâmica, PhLQP. PhLQP por si só é uma lógica capaz de lidar com propriedades de fases quânticas, evoluções unitárias, e entrelaçamento em sistemas compostos, uma vez que é um redesenhar da já existente LQP, [14], sobre uma representação de estados quânticos restrita a um espaço B munido de apenas duas bases computacionais, standard e Hadamard. Como instâncias de aplicação da lógica PhLQP, há uma prova formal para a correção do protocolo de Teletransporte Quântico, para o protocolo QLE para uma party quer de 2 quer de 4 agentes, e para o operador de QFT de 1, 2, e 3 qubits. Numa segunda fase, PhLQP é extendida com a conectiva ◦, conhecida como operador de consistência, uma conectiva característica das LFIs, [8, 21, 22]. E a partir desta conectiva a definição de estado quântico consistente é um conjunto de axiomas paraconsistentes próprios para o domínio quântico, FParQAxs. Um exemplo de aplicação da PhLQP◦ é a possibilidade de expressar e permitir correção para o comportamento da gate quântica universal, a Deutsch-gate

Paraconsistent transition systems

Often in Software Engineering, a modeling formalism has to support scenarios of inconsistency in which several requirements either reinforce or contradict each other. Paraconsistent transition systems are proposed in this paper as one such formalism: states evolve through two accessibility relations capturing weighted evidence of a transition or its absence, respectively. Their weights come from a specific residuated lattice. A category of these systems, and the corresponding algebra, is defined as providing a formal setting to model different application scenarios. One of them, dealing with the effect of quantum decoherence in quantum programs, is used for illustration purposes.publishe