Absolute Generality by Agustín Rayo

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By Agustín Rayo

Este libro reune ensayos de los más grandes especialistas en el tema de los angeles 'cuantificación common' abordado desde todas sus dimensiones. Ellos no sólo se limitan a hablar sobre el tema sino que después del análisis que realizan, presentan avesadas tesis de cara a los problemas que se sucitan.

Lo recomiendo, pues, creo que si todavía no lo es, en el futuro se convertirá en un clásico.

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Or again, it has been suggested that we should think of sets as being constructed in stages and that what prevents the formation of the universal or the Russell set is there being no stage at which its members are all constructed. We may grant that we should think of sets as being constructed at stages and that, under any reasonable process by which we might take them to be constructed, there will be no stage at which either the universal or the Russell set is constructed. But what is to prevent us from so understanding the quantifier over stages that it includes a stage that lies after all of the stages according to the original understanding of the quantifier (∃+ α∀β(α > β) )?

But rather than thinking of that condition as serving to define a new predicate by which the quantifier is to be restricted, we should think of it as serving to indicate how the range of the quantifier is to be extended. x∀y(y ∈ x), requiring us to introduce an object x whose members are the objects y of the given domain. x∀y(y ∈ x) is perhaps neutral as to how the required extension is to be achieved. But the intent is that there is no more fundamental understanding of what the new domain should be except as the domain that might be reached from the given domain by adding an object in conformity with the condition.

28 Kit Fine But given (6), we may drop the subscript M0 . And contraposition then yields: (12) ∀I∼UR(I)⊃∼UR(M0 ). ⁶ Of course, it should have been evident from the start that the limitavist has a difficulty in maintaining that all interpretations of the quantifier are not absolutely unrestricted, since it would follow from the truth of the claim that the interpretation of the quantifier in the claim itself was not absolutely unrestricted and hence that it could not have its intended import. What the preceding proof further demonstrates is the impossibility of maintaining a mixed position, one which grants the intelligibility of the absolutely unrestricted ‘second-order’ quantifier over all interpretations but rejects the intelligibility of the absolutely unrestricted ‘first-order’ quantifier over all objects.

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