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Single Idea 19296
[filed under theme 5. Theory of Logic / A. Overview of Logic / 7. Second-Order Logic
]
Full Idea
Contrary to what Quine supposes, it is neither necessary nor desirable to interpret bound higher-order variables as ranging over sets. Sets are a species of object. They should range over entities of a completely different type: properties and relations.
Gist of Idea
If second-order variables range over sets, those are just objects; properties and relations aren't sets
Source
Bob Hale (Necessary Beings [2013], 08.2)
Book Ref
Hale,Bob: 'Necessary Beings' [OUP 2013], p.182
A Reaction
This helpfully clarifies something which was confusing me. If sets are objects, then 'second-order' logic just seems to be the same as first-order logic (rather than being 'set theory in disguise'). I quantify over properties, but deny their existence!
The
23 ideas
from 'Necessary Beings'
19275
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You cannot understand what exists without understanding possibility and necessity
[Hale]
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19276
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The big challenge for essentialist views of modality is things having necessary existence
[Hale]
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19278
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There is no gap between a fact that p, and it is true that p; so we only have the truth-condtions for p
[Hale]
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19279
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What are these worlds, that being true in all of them makes something necessary?
[Hale]
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19281
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Interesting supervenience must characterise the base quite differently from what supervenes on it
[Hale]
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19282
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It seems that we cannot show that modal facts depend on non-modal facts
[Hale]
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19286
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'Absolute necessity' is when there is no restriction on the things which necessitate p
[Hale]
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19285
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Logical necessity is something which is true, no matter what else is the case
[Hale]
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19287
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Maybe each type of logic has its own necessity, gradually becoming broader
[Hale]
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19288
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Logical and metaphysical necessities differ in their vocabulary, and their underlying entities
[Hale]
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19289
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Maybe conventionalism applies to meaning, but not to the truth of propositions expressed
[Hale]
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19290
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Absolute necessities are necessarily necessary
[Hale]
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19291
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A canonical defintion specifies the type of thing, and what distinguish this specimen
[Hale]
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19293
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Essentialism doesn't explain necessity reductively; it explains all necessities in terms of a few basic natures
[Hale]
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19294
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If necessity derives from essences, how do we explain the necessary existence of essences?
[Hale]
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19295
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Add Hume's principle to logic, to get numbers; arithmetic truths rest on the nature of the numbers
[Hale]
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19296
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If second-order variables range over sets, those are just objects; properties and relations aren't sets
[Hale]
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19297
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The two Barcan principles are easily proved in fairly basic modal logic
[Hale]
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19298
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Unlike axiom proofs, natural deduction proofs needn't focus on logical truths and theorems
[Hale]
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19299
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Possible worlds make every proposition true or false, which endorses classical logic
[Hale]
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19300
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The molecules may explain the water, but they are not what 'water' means
[Hale]
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19301
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With a negative free logic, we can dispense with the Barcan formulae
[Hale]
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19302
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If a chair could be made of slightly different material, that could lead to big changes
[Hale]
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