more on this theme
|
more from this thinker
Single Idea 11020
[filed under theme 4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / c. Logical sets
]
Full Idea
Hard-headed realism tends to embrace the full Comprehension Principle, that every well-defined concept determines a set.
Gist of Idea
Realisms like the full Comprehension Principle, that all good concepts determine sets
Source
Stephen Read (Thinking About Logic [1995], Ch.8)
Book Ref
Read,Stephen: 'Thinking About Logic' [OUP 1995], p.214
A Reaction
This sort of thing gets you into trouble with Russell's paradox (though that is presumably meant to be excluded somehow by 'well-defined'). There are lots of diluted Comprehension Principles.
The
42 ideas
from 'Thinking About Logic'
10966
|
A proposition objectifies what a sentence says, as indicative, with secure references
[Read]
|
10987
|
Three traditional names of rules are 'Simplification', 'Addition' and 'Disjunctive Syllogism'
[Read]
|
10985
|
We should exclude second-order logic, precisely because it captures arithmetic
[Read]
|
10970
|
A theory of logical consequence is a conceptual analysis, and a set of validity techniques
[Read]
|
10984
|
Logical consequence isn't just a matter of form; it depends on connections like round-square
[Read]
|
10973
|
A theory is logically closed, which means infinite premisses
[Read]
|
10978
|
In second-order logic the higher-order variables range over all the properties of the objects
[Read]
|
10971
|
A logical truth is the conclusion of a valid inference with no premisses
[Read]
|
10988
|
Any first-order theory of sets is inadequate
[Read]
|
10974
|
Compactness is when any consequence of infinite propositions is the consequence of a finite subset
[Read]
|
10975
|
Compactness does not deny that an inference can have infinitely many premisses
[Read]
|
10977
|
Compactness blocks the proof of 'for every n, A(n)' (as the proof would be infinite)
[Read]
|
10976
|
Compactness makes consequence manageable, but restricts expressive power
[Read]
|
10986
|
Not all validity is captured in first-order logic
[Read]
|
10972
|
The non-emptiness of the domain is characteristic of classical logic
[Read]
|
10979
|
Although second-order arithmetic is incomplete, it can fully model normal arithmetic
[Read]
|
10980
|
Second-order arithmetic covers all properties, ensuring categoricity
[Read]
|
10983
|
Knowledge of possible worlds is not causal, but is an ontology entailed by semantics
[Read]
|
10982
|
How can modal Platonists know the truth of a modal proposition?
[Read]
|
10981
|
A possible world is a determination of the truth-values of all propositions of a domain
[Read]
|
10989
|
The standard view of conditionals is that they are truth-functional
[Read]
|
10992
|
The point of conditionals is to show that one will accept modus ponens
[Read]
|
10996
|
Actualism is reductionist (to parts of actuality), or moderate realist (accepting real abstractions)
[Read]
|
11000
|
If worlds are concrete, objects can't be present in more than one, and can only have counterparts
[Read]
|
10998
|
The mind abstracts ways things might be, which are nonetheless real
[Read]
|
10997
|
Von Neumann numbers are helpful, but don't correctly describe numbers
[Read]
|
10995
|
A haecceity is a set of individual properties, essential to each thing
[Read]
|
11001
|
Equating necessity with truth in every possible world is the S5 conception of necessity
[Read]
|
11004
|
Necessity is provability in S4, and true in all worlds in S5
[Read]
|
11011
|
Same say there are positive, negative and neuter free logics
[Read]
|
11007
|
Quantifiers are second-order predicates
[Read]
|
11012
|
A 'supervaluation' gives a proposition consistent truth-value for classical assignments
[Read]
|
11005
|
Negative existentials with compositionality make the whole sentence meaningless
[Read]
|
11013
|
Identities and the Indiscernibility of Identicals don't work with supervaluations
[Read]
|
11014
|
Self-reference paradoxes seem to arise only when falsity is involved
[Read]
|
11018
|
There are fuzzy predicates (and sets), and fuzzy quantifiers and modifiers
[Read]
|
11017
|
Some people even claim that conditionals do not express propositions
[Read]
|
11019
|
Supervaluations say there is a cut-off somewhere, but at no particular place
[Read]
|
11016
|
Would a language without vagueness be usable at all?
[Read]
|
11020
|
Realisms like the full Comprehension Principle, that all good concepts determine sets
[Read]
|
11025
|
Infinite cuts and successors seems to suggest an actual infinity there waiting for us
[Read]
|
11024
|
Semantics must precede proof in higher-order logics, since they are incomplete
[Read]
|