Philosophy Lexicon of Arguments

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2nd order Logic: Predicate logic of the 2nd order goes beyond predicate logic of the 1st level allowing quantification over properties and relations, and not just objects. Thus comparisons of the powerfulness of sets become possible. Problems which are expressed in everyday terms with terms such as "greater", "between", etc., and e.g. the specification of all the properties of an object require predicate logic of the 2nd order. Since the 2nd level logic is not complete (because there are, for example, an infinite number of properties of properties), one often tries to get on with the logic of the 1st order.


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Re III 62f
"All the characteristics of a great commander"/compact/1st Level/2nd level/Read: a categorical set of axioms for arithmetic must be a 2nd level logic - logical form: "for every quality f, if for every person x, if x was a great general, then x had f, then Napoleon had f - but: purely syntactically you cannot decide whether this is 1st or 2nd level - what distinguishes the two is their semantics!
The definition area can be arbitrary, provided it is not empty - Russell: Addition: "... and these are all..." - ReadVs: either superfluous in explicitly specified conjunction or wrong - Omega rule: needs the addition, however it cannot be expressed in the 1st level logic - to exclude non-standard models, but it should be formulated in the 1st level (i.e. in logical terms).
Re III 152f
Logic 1st level: individuals, 2nd level variables for predicates, distribution of predicates by quantifiers - 1st level allows restricted vocabulary of the 2nd level: existence and universal quantifier - other properties 2nd level are not definable in the logic of the 1st stage: e.g. to be finite, or to be true of most things.

Logic Texts
Me I Albert Menne Folgerichtig Denken Darmstadt 1988
HH II Hoyningen-Huene Formale Logik, Stuttgart 1998
Re III Stephen Read Philosophie der Logik Hamburg 1997
Sal IV Wesley C. Salmon Logik Stuttgart 1983
Sai V R.M.Sainsbury Paradoxien Stuttgart 2001

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Ed. Martin Schulz, access date 2018-04-23