|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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2nd Order Logic/Second Order Logic/Higher Order Logic/HOL/Field: Here, the the quantifiers have no recursive method of evidence - quantification/Field: therefore it is vague and indeterminate, but even then applies: (A > logically true (A)) & (~ A > logically true (~ A)) is always true - vagueness refers to the A.
Referential indeterminacy/logical operators/2nd order Logic/Field: special case: Question: can complex logical operators - e.g., unrestricted 2nd order quantifiers ((s) via properties) have any particular truth conditions? - no: e.g.: everything that you express with them can be reformulated (reduced) with a more restricted quantification (via sets) - it does not help to say e.g. "with "for all properties" I mean for all properties" - ((s )> "Everything he said") - all/N.B./Field: the use of "all" without quotes is itself the subject of a reinterpretation. - ((s) there could be a contradictory, still undiscovered property which should not be included under "all properties." E.g. Acceleration near speed of light - here the definitive operator would again help.) - VsDeflationism: could simply say ".. all .. " is true iff all ... - Vs: in addition one needs the definitive-operator (definitive-Op), which demands conditions - but it does not specify them. - Field: dito with Higher Order Quantification (HOL).
First order Logic/2nd order/stronger/weaker/attenuation/Field: to weaken the second order logic to the 1st order, we can attenuate the second-order axioms to the axiom-schemata of first-order , namely the schema of separation. ((s) Instead of an axiom via a set, a schema for all elements?) - Problem: not many non-standard models come in. Namely, models in which quantities that are in reality infinite, satisfy the formula which usually defines straight finiteness. (> unintended model).
2nd Order Logic/Field: we have that at two places: 1. at the axiomatization of the geometry of the spacetime and at the scalar order of spacetime points we have
the "complete logic of the part-whole relation" (see Chapter 4), or the "complete logic of the Goodman sums" - 2. (in Section B, Chapter 8): the binary quantifier "less than". But we do not need this if we have Goodman's sums: - Goodman's sum: its logic is sufficient to give comparisons of powerfulness. For heuristic reasons, however, we want to keep an extra logic for powerfulness ("less than").
Realism, Mathematics and Modality Oxford New York 1989
Truth and the Absence of Fact Oxford New York 2001
Science without numbers Princeton New Jersey 1980