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Full Idea
To get any advantage from moving to second-order logic, we need to assign to second-order variables a role different from merely ranging over collections made up of things the first-order variables range over.
Gist of Idea
Second-order variables need to range over more than collections of first-order objects
Source
Vann McGee (Logical Consequence [2014], 7)
Book Ref
'Bloomsbury Companion to Philosophical Logic', ed/tr. Horsten,L/Pettigrew,R [Bloomsbury 2014], p.47
A Reaction
Thus it is exciting if they range over genuine properties, but not so exciting if you merely characterise those properties as sets of first-order objects. This idea leads into a discussion of plural quantification.
Related Idea
Idea 18763 Basic variables in second-order logic are taken to range over subsets of the individuals [Anderson,CA]
| 18751 | Natural language includes connectives like 'because' which are not truth-functional [McGee] |
| 18754 | Logically valid sentences are analytic truths which are just true because of their logical words [McGee] |
| 18755 | Validity is explained as truth in all models, because that relies on the logical terms [McGee] |
| 18753 | An ontologically secure semantics for predicate calculus relies on sets [McGee] |
| 18757 | Soundness theorems are uninformative, because they rely on soundness in their proofs [McGee] |
| 18760 | The culmination of Euclidean geometry was axioms that made all models isomorphic [McGee] |
| 18761 | Second-order variables need to range over more than collections of first-order objects [McGee] |
| 18762 | A maxim claims that if we are allowed to assert a sentence, that means it must be true [McGee] |