42,317 research outputs found

    Logical independence and quantum randomness

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    We propose a link between logical independence and quantum physics. We demonstrate that quantum systems in the eigenstates of Pauli group operators are capable of encoding mathematical axioms and show that Pauli group quantum measurements are capable of revealing whether or not a given proposition is logically dependent on the axiomatic system. Whenever a mathematical proposition is logically independent of the axioms encoded in the measured state, the measurement associated with the proposition gives random outcomes. This allows for an experimental test of logical independence. Conversely, it also allows for an explanation of the probabilities of random outcomes observed in Pauli group measurements from logical independence without invoking quantum theory. The axiomatic systems we study can be completed and are therefore not subject to Goedel's incompleteness theorem.Comment: 9 pages, 4 figures, published version plus additional experimental appendi

    The Earlier Wittgenstein on the Notion of Religious Attitude

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    I defend a new interpretation of Wittgenstein's notion of religious (or ethical) attitude in the Tractatus, one that rejects three key views from the secondary literature: firstly, the view that, for Wittgenstein, the willing subject is a transcendental condition for the religious attitude; secondly, the view that the religious attitude is an emotive response to the world or something closely modelled on this notion of emotive response; and thirdly, the view that, although the religious and ethical pseudo-propositions of the Tractatus are nonsensical, they nevertheless succeed in expressing the religious attitude endorsed by Wittgenstein. In connection to the first, I argue that the notion of willing subject as transcendental condition is abandoned by Wittgenstein in the Notebooks and is no longer a feature of his position in the Tractatus. In connection to the second, I argue that the religious attitude is dispositional rather than emotive for Wittgenstein: it is a disposition to use signs in a way that demonstrates one's conceptual clarity. Finally, in connection to the third, I argue that the religious or ethical attitude is strongly ineffable in that it cannot be described, expressed or conveyed by language at all.Peer reviewe

    G\"odel's Notre Dame Course

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    This is a companion to a paper by the authors entitled "G\"odel's natural deduction", which presented and made comments about the natural deduction system in G\"odel's unpublished notes for the elementary logic course he gave at the University of Notre Dame in 1939. In that earlier paper, which was itself a companion to a paper that examined the links between some philosophical views ascribed to G\"odel and general proof theory, one can find a brief summary of G\"odel's notes for the Notre Dame course. In order to put the earlier paper in proper perspective, a more complete summary of these interesting notes, with comments concerning them, is given here.Comment: 18 pages. minor additions, arXiv admin note: text overlap with arXiv:1604.0307

    When can statistical theories be causally closed?

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    The notion of common cause closedness of a classical, Kolmogorovian probability space with respect to a causal independence relation between the random events is defined, and propositions are presented that characterize common cause closedness for specific probability spaces. It is proved in particular that no probability space with a finite number of random events can contain common causes of all the correlations it predicts; however, it is demonstrated that probability spaces even with a finite number of random events can be common cause closed with respect to a causal independence relation that is stronger than logical independence. Furthermore it is shown that infinite, atomless probability spaces are always common cause closed in the strongest possible sense. Open problems concerning common cause closedness are formulated and the results are interpreted from the perspective of Reichenbach's Common Cause Principle (RCCP)
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