|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._____________Annotation: The above characterizations of concepts are neither definitions nor exhausting presentations of problems related to them. Instead, they are intended to give a short introduction to the contributions below. – Lexicon of Arguments. |
|Logic 2nd Level/Bigelow/Pargetter: has many fascinating but also astonishing properties: we mention only one: for example, we can define the identity predicate with it without introducing another basic concept with axioms, simply by an abbreviation, for this we need Leibniz' law:
(x)(y) ((x = y) ⇔ (ψ)(ψx equi ψy)).
Two things are identical, iff everything that is one thing applies also to the other.
Logic of the 2nd level: the problem is of semantic, not of syntactic nature.
Question: Which language should we choose for our semantic theory?
Standard solution: frames a semantic theory for 2nd level logic within a 1st level logic. I.e. the meta language contains quantifiers of the 1st level and a predicate of being contained in a set (with axioms of the Zermelo-Fraenkelian set theory).
N.B.: then the 2nd level logic becomes equivalent to a fragment of set theory. For example, by a sentence like
Everyday language translation: "There is something that is a" - "It is somehow like a is" (There is somehow that a is).
That actually says that
(Ey)(a ε y)
Everyday language translation: "There is a set to which a belongs".
That is also logic of the 2nd level.
The semantics for this is as follows: for each predicate we attribute a set to it. So is a simple sentence
True, if the referent of a is an element of the set corresponding to F.
If F is replaced by a variable , this variable will go over all sets that a predicate such as F could refer to. Therefore,
becomes true iff there is a set that a belongs to.
Leibniz' Law/Identity/Bigelow/Pargetter: can be rephrased like this:
Two things are identical iff they belong to exactly the same set.
Bigelow/Pargetter: Problem: this should not be taken as a definition of identity, because when defining sets the concept of identity was already used (circularly).
Solution: one could dispel the concerns of circularity by pointing out that identity has been defined in a language while set theory is used in a meta-language.
Circle/Bigelow/Pargetter: nevertheless the circularity is not harmless.
Logic of the 2nd level: Level/Bigelow/Pargetter: should not be treated as a notational variant of set theory. For example, the assertion that x is a set:
Then we can use logic of the 2nd Level to claim that all sets have something in common, that something exists, something that all sets are:
This is also true insofar as "being a set" is actually "something" that they all are.
Problem: this cannot be interpreted set theoretically, because otherwise we get a paradox:
There is a set that contains all the sets.
Cantor: showed that this is wrong. There is no set of all sets, no universal set.
Logic 2nd level/Bigelow/Pargetter: this is the reason why it must not be combined with set theory. ((s) Although all sets have something in common, this must not be a set, although otherwise all properties can be considered sets).
Solution/Bigelow/Pargetter: our semantics for a 2nd level logic must be a semantics in a 2nd level language, not a semantics in a 1st level language.
Universals/Bigelow/Pargetter: we are only concerned here, however, with the connection between universals and their instances.
Universal/Bigelow/Pargetter: can be denoted with a name (refers) and quantified via it with quantifiers of the 1st level.
Open sentence/predicate/Bigelow/Pargetter: not every open sentence corresponds to a universal, e.g. one cannot conclude from
There's something that these things are
(Eψ)(ψ x1 uψx2 u...
without additional premises on the fact that...
...there is something that these things have in common
(Ez)(x1 instantiates z u x2 instantiates z u.....)
Universals/Bigelow/Pargetter: this identifies universals as "recurrences" (see above recurrence). ((s) as "recurring" in various things).
Existence of 2nd level/Bigelow/Pargetter: of "something" that things "are" (that things are/(s) like things are) does not secure the existence of the 1st level of something that somehow stands by all these things.
Semantics/Bigelow/Pargetter: this shows that the primary arguments for universals are not semantic. ((s) Because they are not figurative truthmakers)._____________Explanation of symbols: Roman numerals indicate the source, arabic numerals indicate the page number. The corresponding books are indicated on the right hand side. ((s)…): Comment by the sender of the contribution.
J. Bigelow, R. Pargetter
Science and Necessity Cambridge 1990