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Full Idea
The dogmatic Frege is more right than wrong in denying that numerical terms can stand for numerical quantifiers, for there cannot be a language in which object-quantifiers and objects are simultaneously viewed as level zero.
Clarification
'Level zero' is for objects, and 'level-one' is for quantifying over objects
Gist of Idea
Numerical terms can't really stand for quantifiers, because that would make them first-level
Source
Harold Hodes (Logicism and Ontological Commits. of Arithmetic [1984], p.142)
Book Ref
-: 'Journal of Philosophy' [-], p.142
A Reaction
Subtle. We see why Frege goes on to say that numbers are level zero (i.e. they are objects). We are free, it seems, to rewrite sentences containing number terms to suit whatever logical form appeals. Numbers are just quantifiers?
| 10011 | Identity is a level one relation with a second-order definition [Hodes] |
| 10027 | Mathematics is higher-order modal logic [Hodes] |
| 10021 | It is claimed that numbers are objects which essentially represent cardinality quantifiers [Hodes] |
| 10016 | When an 'interpretation' creates a model based on truth, this doesn't include Fregean 'sense' [Hodes] |
| 10017 | Truth in a model is more tractable than the general notion of truth [Hodes] |
| 10015 | Higher-order logic may be unintelligible, but it isn't set theory [Hodes] |
| 10018 | Truth is quite different in interpreted set theory and in the skeleton of its language [Hodes] |
| 10022 | Numerical terms can't really stand for quantifiers, because that would make them first-level [Hodes] |
| 10023 | Talk of mirror images is 'encoded fictions' about real facts [Hodes] |
| 10026 | Arithmetic must allow for the possibility of only a finite total of objects [Hodes] |