The Logic of Typicality

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  1. Harry CraneSeptember 11th, 2018 at 08:19 pm

    Thanks for your comments. We will address these concerns in our next revision.

  2. Eugene PanferovSeptember 3rd, 2018 at 05:21 pm

    introduction: A huge logical Error of piling together the probabilistic statements of thermodinamics and QM with the Goldbach's Conjecture. Whereas thermodynamics statements are typical for being true in the "vast majority" of tests, The GC is true for the INFENITESSIMALLY SMALL MINORITY of tests. Apparently there are two different notions of typicality are being conflated. Indeed the section 2.2 defines infinitely many notions of typicality parametrized with {epsilon, tau} where tau is an ARBITRARY measure. Those notions of typicality are not necessarily practical, intuitive, or in any way related to each other -- they are arbitrary. The thermodinamics and QM examples are typical with the tau being set to cardinality. The GC is apparently not typical in this typicality (although the introduction creates the false impression that it is). And it remains an open question with what measure tau GC is typical? One particular typicality, the probability typicality is the origin of the notion, whereas all the rest are yet to be shown being relevant. The probabilistic approach seems to be more practical as it preserves the definition of the probability space (whithin which the typicality is questioned) and the exact value of the probability of the tested statement to be true -- the typicality language does NOT preserve this information (the probability is reduced to 1 bit of info by the predefined threshold, and the probability space definition is obscured by being pushed outside the definition of the tupicality "universe"). Moreover, the probability language (as it quantifies the "typicality" of an individual statement) allows us to quantify "typicality" of some derivative statements, taking as input the "typicality" values of its proposition statements. section 2.2: Universe is defined as a pair of sets, then there a bijection relation between these sets is introduced, thus making the universe A SET OF PAIRS (not a pair of sets) (each function is paired with its index) section 2.2: formula 4.i: the Gamma_phi is used while not being properly introduced. it becomes clear later that the definition of the Gamma_phi was borrowed/generalized from 1.ii, which is not at all obvious/rigorous. A definition of the general form of the Gamma_phi is required beforehand. Lemma 1: "for each w" WHY for each? in the loop for w the Gamma_phi sets are not restricted (by any previous reasoning) to be identical for each w. it is yet to be shown that every w will generate the same Gamma_phi. Therefore the quantor does not hold.

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Abstract

The notion of typicality appears in scientific theories, philosophical arguments, math- ematical inquiry, and everyday reasoning. Typicality is invoked in statistical mechanics to explain the behavior of gases. It is also invoked in quantum mechanics to explain the appearance of quantum probabilities. Typicality plays an implicit role in non-rigorous mathematical inquiry, as when a mathematician forms a conjecture based on personal experience of what seems typical in a given situation. Less formally, the language of typicality is a staple of the common parlance: we often claim that certain things are, or are not, typical. But despite the prominence of typicality in science, philosophy, mathematics, and everyday discourse, no formal logics for typicality have been proposed. In this paper, we propose two formal systems for reasoning about typicality. One system is based on propositional logic: it can be understood as formalizing objective facts about what is and is not typical. The other system is based on the logic of intuitionistic type theory: it can be understood as formalizing subjective judgments about typicality.

Versions

➤  Version 1 (2018-08-30)

Citation

Harry Crane and Isaac Wilhelm (2018). The Logic of Typicality. Researchers.One, https://researchers.one/articles/the-logic-of-typicality/5f52699b36a3e45f17ae7d4e/v1.

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