Lexicon of Arguments

 

I XII
Def vicious circle principle/Russell/Gödel: no totality can contain elements that can be defined only in terms of this totality or elements which include or imply this totality. - Vicious circle principle, VCP.
I XII
Circle fault principle/GödelVsRussell: the Principia themselves do not fulfil the principle in their first edition if "definable" means "definable within the system" and no definition methods outside are known, except for those that include even more extensive totalities than those that occur in the system - Gödel: I would rather see this as proof that the circle fault principle is wrong than that classical mathematics is wrong - because one can argue that the reference to a totality necessarily implies a reference to all of its individual elements or, in other words, that "all" means the same as an infinite logical conjunction.
I XIV
"All"/solution/Carnap: "All" alludes to analyticity or necessity or provability. - Circle Fault Principle/Gödel: seems to apply only to entities constructed by ourselves - otherwise totality is nothing absurd.
>Self-reference, >Wholes.
I 55f
Circle fault principle/Russell: Propositions: only form multiplicities, no entities. - (s) Entities are formed by terms, i.e. that you cannot set up a sentence about "all of its elements". (> "Everything he said"/(s): "say" does not form a category like "next to", "similar" "son of"; "nothing" does not either nor does it form an entity, only a multiplicity but "father of" (unambiguous) (Russell: function, not only relation).
>Relation/Russell, >Function/Russell.
I 57
Circle/Principia Mathematica⁽¹⁾/Russell: arises when one allows values as possible arguments of a propositional function ​​that require the function.
I 61
Circle fault principle/circle/entitiy/totality/Principia Mathematica⁽¹⁾/Russell: there must be no propositions about all propositions. - E.g. All propositions are false - therefore two kinds of truth/falsehood: 1st kind: "φ a is true" (special value) - 2nd kind: "Every value of φx^ has truth of the 1st kind".


1. Whitehead, A.N. and Russel, B. (1910). Principia Mathematica. Cambridge: Cambridge University Press.