Abstract
This is a response to the papers in the special issue Logical Perspectives on Science and Cognition—The Philosophy of Gerhard Schurz.
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It is provable that Σ1≤i<nP(Wi) = 1 and for all i, P(Wi)/Σi<j<nP(Wj) = (1 − ε)/ε. Let m be the maximal number of worlds in one layer. Then τ-cumulativity is satisfied if ε < (1 − τ)/τ·(m+1).
As Votsis remarks in his fn. 10, this characterization is admittedly vague and gradual. It can be made precise in two way. (1.) By setting as threshold, e.g., by defining a concept as observational if it is learned after at most 30 training instances by at least 80% of all tp's. Since every such threshold criterion is somewhat artificial, a better alternative would be to define a quantitative degree of observability, DO, based on the shape of the learning curve (see fig. 7 in Schurz 2015b). A simple measure of this sort could be the average of the terms "(100 − nτ)+" for all thresholds τ ranging between 50 and 100%. Here nτ is the number of training instances after which τ% of all tp's have learned the concept and x+ is max(x,0); thus for all concepts that are not even learned after 100 instances by at least 50 % of tp's the DO is zero..
He remarks in fn. 7 that this fact has been proved for propositional but not yet for full first order logic. Let me add that the fact is easily provable for second order logic; thus by including certain second order consequences of T in E(T) one can always make E(T) logically equivalent with T.
There is a misspelling: instead of “va” and “ka” in should be “va” and “ka”.
Another interpretation of Kuipers' models are ∃-formulas of the form ∃xQx, where Qx is a Carnapian Q-predicate of a monadic 1st order language. ∃-formulas and their negations are the conjuncts of so-called constituents specifying which kinds of individuals (possibly) exist and which don't (cf. Niiniluoto, sec. 3, in this volume). While the full nomic truth \( {\boldsymbol{{\mathcal{T}}}} \) would correspond to a complete constituent, a two-sided Kuipers theory <M,cP> would be a partial one, M containing some positive and some negated ∃-formulas. In this interpretation, constraints are not needed, because monadic ∃- formulas cannot contradict each other.
Also the interpretation in fn. 5 is not applicable to actual truthlikeness.
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Schurz, G. Twelve great papers: comments and replies. Response to a special issue on logical perspectives on science and cognition—The philosophy of Gerhard Schurz. Synthese 197, 1661–1695 (2020). https://doi.org/10.1007/s11229-019-02329-z
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DOI: https://doi.org/10.1007/s11229-019-02329-z