By a "unified account of predication," they mean a general account covering all predications with the sole exception of predications leading to Russell's Paradox (so-called impredicative
Menzel gets around this difficulty by appealing to what philosophers call an impredicative
definition, which is a definition that generalizes over a totality to which the entity being defined belongs.
Poincare called such definitions impredicative
. They are especially problematic when they occur in structures that are in some sense "generative," that is, structures that, like language and formalized arithmetic, are capable of generating new units according to some rule.
Concepts such as multi-scale mosaic effects and impredicative
loop analysis stretch the imagination, yet at the same time open the door to new opportunities for integrating social and ecological components while "surfing" the complexity of time.
Potter claims that Frege adopted an impredicative
conception of the numbers in order to advance beyond demonstrating their infinity by appeal to the realm of thoughts, and argues that this impredicativity leads to two "unattractive" choices: "if ...
In particular, we are concerned with languages whose type-theoretic core combines subtyping and impredicative
polymorphism in the style of System F [Girard 1972; Reynolds 1974].
This issignificant because it makes the construction of models much easier than for impredicative
Godel notes that one of Russell's formulations of the vicious circle principle (VCP) (i.e., no totality can contain members definable only in terms of this totality) makes impredicative
definitions impossible and forces a kind of constructivity.
Plantinga eludes the classical argument by distinguishing between predicative and impredicative
singular propositions, affirming that the subject of the former, if true, has existence (e.g., Socrates was snubnosed), whereas the subject of the latter, even if true, need not exist (e.g., Socrates was not snubnosed, which is to be understood as It is false that Socrates was snubnosed).
Ordinary calculus, as taught to freshmen and sophomores, assumes certain things about existence, leading to impredicative
assumptions that are inherently non-computational in nature .
The notion of an indefinitely extensible concept and their characterization are, no doubt, crucial in appraising the consequences of the paradoxes (and the legitimacy of impredicative
definitions): Dummett's book on Frege ends by asking a pointed question of the philosophy of mathematics.
The question of the admissibility of impredicative
or circular identity criteria is investigated in the light of the view that is articulated.