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Single Idea 10225
[filed under theme 5. Theory of Logic / A. Overview of Logic / 7. Second-Order Logic
]
Full Idea
Boolos has proposed an alternative understanding of monadic, second-order logic, in terms of plural quantifiers, which many philosophers have found attractive.
Gist of Idea
Monadic second-order logic might be understood in terms of plural quantifiers
Source
report of George Boolos (To be is to be the value of a variable.. [1984]) by Stewart Shapiro - Philosophy of Mathematics 3.5
Book Ref
Shapiro,Stewart: 'Philosophy of Mathematics:structure and ontology' [OUP 1997], p.105
The
31 ideas
from George Boolos
10482
|
The logic of ZF is classical first-order predicate logic with identity
[Boolos]
|
10483
|
Mathematics and science do not require very high orders of infinity
[Boolos]
|
10484
|
The iterative conception says sets are formed at stages; some are 'earlier', and must be formed first
[Boolos]
|
10485
|
Naïve sets are inconsistent: there is no set for things that do not belong to themselves
[Boolos]
|
10488
|
It is lunacy to think we only see ink-marks, and not word-types
[Boolos]
|
10487
|
I am a fan of abstract objects, and confident of their existence
[Boolos]
|
10489
|
We deal with abstract objects all the time: software, poems, mistakes, triangles..
[Boolos]
|
10491
|
Infinite natural numbers is as obvious as infinite sentences in English
[Boolos]
|
10490
|
Mathematics isn't surprising, given that we experience many objects as abstract
[Boolos]
|
10492
|
A few axioms of set theory 'force themselves on us', but most of them don't
[Boolos]
|
8693
|
An 'abstraction principle' says two things are identical if they are 'equivalent' in some respect
[Boolos]
|
13547
|
Limitation of Size is weak (Fs only collect is something the same size does) or strong (fewer Fs than objects)
[Boolos, by Potter]
|
18192
|
Do the Replacement Axioms exceed the iterative conception of sets?
[Boolos, by Maddy]
|
14249
|
Boolos reinterprets second-order logic as plural logic
[Boolos, by Oliver/Smiley]
|
13841
|
Why should compactness be definitive of logic?
[Boolos, by Hacking]
|
10829
|
A sentence can't be a truth of logic if it asserts the existence of certain sets
[Boolos]
|
10830
|
Second-order logic metatheory is set-theoretic, and second-order validity has set-theoretic problems
[Boolos]
|
10832
|
'∀x x=x' only means 'everything is identical to itself' if the range of 'everything' is fixed
[Boolos]
|
10833
|
Many concepts can only be expressed by second-order logic
[Boolos]
|
10834
|
Weak completeness: if it is valid, it is provable. Strong: it is provable from a set of sentences
[Boolos]
|
7806
|
Boolos invented plural quantification
[Boolos, by Benardete,JA]
|
13671
|
Second-order quantifiers are just like plural quantifiers in ordinary language, with no extra ontology
[Boolos, by Shapiro]
|
10267
|
We should understand second-order existential quantifiers as plural quantifiers
[Boolos, by Shapiro]
|
10225
|
Monadic second-order logic might be understood in terms of plural quantifiers
[Boolos, by Shapiro]
|
7785
|
The use of plurals doesn't commit us to sets; there do not exist individuals and collections
[Boolos]
|
10736
|
Boolos showed how plural quantifiers can interpret monadic second-order logic
[Boolos, by Linnebo]
|
10780
|
Any sentence of monadic second-order logic can be translated into plural first-order logic
[Boolos, by Linnebo]
|
10697
|
Identity is clearly a logical concept, and greatly enhances predicate calculus
[Boolos]
|
10698
|
Plural forms have no more ontological commitment than to first-order objects
[Boolos]
|
10699
|
Does a bowl of Cheerios contain all its sets and subsets?
[Boolos]
|
10700
|
First- and second-order quantifiers are two ways of referring to the same things
[Boolos]
|