Artículos con la etiqueta ‘paradojas’

Arrow’s Theorem by Arrow Theory

Por • 21 ene, 2014 • Category: Filosofía

We give a categorical account of Arrow’s theorem, a seminal result in social choice theory.



Paradoxes of rational agency and formal systems that verify their own soundness

Por • 13 dic, 2013 • Category: Filosofía

We consider extensions of Peano arithmetic which include an assertibility predicate. Any such system which is arithmetically sound effectively verifies its own soundness. This leads to the resolution of a range of paradoxes involving rational agents who are licensed to act under precisely defined conditions.



Paradoxes in Social Networks with Multiple Products

Por • 4 feb, 2013 • Category: Opinion

Recently, we introduced in arXiv:1105.2434 a model for product adoption in social networks with multiple products, where the agents, influenced by their neighbours, can adopt one out of several alternatives. We identify and analyze here four types of paradoxes that can arise in these networks. To this end, we use social network games that we recently introduced in arXiv:1211.5938. These paradoxes shed light on possible inefficiencies arising when one modifies the sets of products available to the agents forming a social network. One of the paradoxes corresponds to the well-known Braess paradox in congestion games and shows that by adding more choices to a node, the network may end up in a situation that is (weakly) worse for everybody.



UNDECIDABLE PROPOSITIONS BY ODE’S

Por • 23 ene, 2013 • Category: Crítica

Starting with elementary functions, we generate new functions by multiplication, integration and by solving ODE’s so as to obtain a family M of real holomorphic functions such that: (*) if E \subseteq N is recursively enumerable then there is f \in M such that n \in E iff \int^{+\pi}_{-\pi} f(x)e-inx dx \neq 0. Constructive aspects and relations to hypercomputation are discussed.



The Quantum Zeno Effect — Watched Pots in the Quantum World

Por • 19 nov, 2012 • Category: Filosofía

In the 5th century B.C.,the philosopher and logician Zeno of Elea posed several paradoxes which remained unresolved for over two thousand five hundred years. The $20^{th}$ century saw some resolutions to Zeno’s mind boggling problems. This long journey saw many significant milestones in the form of discoveries like the tools of converging series and theories on infinite sets in mathematics. In recent times, the Zeno effect made an intriguing appearance in a rather unlikely place – a situation involving the time evolution of a quantum system, which is subject to «observations» over a period of time. Leonid Khalfin working in the former USSR in the 1960s and ECG Sudarshan and B. Misra at the University of Texas, Austin, first drew attention to this problem. In 1977, ECG Sudarshan and B. Misra published a paper on the quantum Zeno effect, called «The Zeno’s paradox in quantum theory».



The Rule of Global Necessitation

Por • 5 oct, 2012 • Category: Crítica

For half a century, authors have weakened the rule of necessitation in various more or less ad hoc ways in order to make inconsistent systems consistent. More recently, necessitation was weakened in a systematic way, not for the purpose of resolving paradoxes but rather to salvage the deduction theorem for modal logic. We show how this systematic weakening can be applied to the older problem of paradox resolution. Four examples are given: a predicate symbol S4 consistent with arithmetic; a resolution of the surprise examination paradox; a resolution of Fitch’s paradox; and finally, the construction of a knowing machine which knows its own code. We discuss a technique for possibly finding answers to a question of P. \’Egr\’e and J. van Benthem.



Truth and the liar paradox

Por • 8 ene, 2012 • Category: Filosofía

We analyze the informal notion of truth and conclude that it can be formalized in essentially two distinct ways: constructively, in terms of provability, or classically, as a hierarchy of concepts which satisfy Tarski’s biconditional in limited settings. This leads to a complete resolution of the liar paradox.



Complete Totalities

Por • 22 jul, 2011 • Category: Filosofía

The cumulative hierarchy conception of set, which is based on a metaphor of elements of sets being prior to their collection, is generally considered to be a good way to create a set conception that seems safe from contradictions. This imposes two restrictions on sets. One is a «limitation of size,» and the other is the rejection of non well-founded sets. Quine’s NF system of axioms, does not have any of the two restrictions, but it has a formal restriction on allowed formulas in its comprehension axiom schema, which reflects a similar notion of elements being prior to sets. The suggestion made here is that a possible reason for set antinomies is the tension between our perception of sets and their elements as lying on a different conceptual planes, and our wish to be able to refer to mathematical objects without contemplating their relation with other objects. A new approach to sets as totalities is presented which is based on a notion of «concurrent aggregation,» which instead of avoiding «viscous circles,» acknowledges the inherent circularities of some predicates, and provides a way to learn from their natural emergence.