Vincenzo Ciccarelli

This paper examines the implications of Bedau’s notion of weak emergence when combined with both a hierarchical ontological model and Wolfram’s principle of computational equivalence. I argue that if a system exhibits weak emergence, the very fact of weak emergence of some of its states must itself be a weakly emergent state. Interpreted through Wolfram’s principle of computational equivalence, this implies that a system with weakly emergent states would need to ‘decide’ whether its own dynamical trajectories are incompressible—which contradicts with the formal undecidability of the incompressibility predicate. I term this aporetic conclusion the paradox of weak emergence. After evaluating possible responses, I contend that Bedau’s approach exemplifies what I call the derivational account of metaphysical relations, a framework that I will show to be highly problematic insofar as it implies a mismatch between the intrinsic limitations of formal derivations and the explanatory ambitions of metaphysical notions.

A famous passage in Section 64 of Frege’s Grundlagen may be seen as a justification for the truth of abstraction principles. The justification is grounded in the procedureofcontent recarvingwhich Frege describes in the passage. In this paper I argue that Frege’sprocedure of content recarving while possibly correct in the case of first-order equivalencerelations is insufficient to grant the truth of second-order abstractions. Moreover, I propose apossible way of justifying second-order abstractions by referring to the operation of contentrecarving and I show that the proposal relies to a certain extent on the Basic Law V. Therefore,if we are to justify the truth of second-order abstractions by invoking the content recarvingprocedure we are committed to a special status of some instances of the Basic Law V and thusto a special status of extensions of concepts as abstract objects.

According to the Bad Company objection, the fact that Frege’s infamous Basic Law V instantiates the general definitional pattern of higher-order abstraction principles is a good reason to doubt the soundness of this sort of definitions. In this paper I argue against this objection by showing that the definitional pattern of abstraction principles – as extrapolated from §64 of Frege’s Grundlagen– includes an additional requirement (which I call the specificity condition) that is not satisfied by the Basic Law V while is satisfied by other higher-order abstractions such as Hume’s Principle. I also show that the failure of this additional requirement in the case of Basic Law V is engendered by an essential feature of Frege’s conception of logic and thus that Frege himself should not have regarded the Basic Law V as a definition by abstraction.