more on this theme
|
more from this text
Single Idea 10704
[filed under theme 5. Theory of Logic / A. Overview of Logic / 7. Second-Order Logic
]
Full Idea
In second-order logic only the formation rules are completely formalizable, not the inference rules.
Gist of Idea
We can formalize second-order formation rules, but not inference rules
Source
Michael Potter (Set Theory and Its Philosophy [2004], 01.2)
Book Ref
Potter,Michael: 'Set Theory and Its Philosophy' [OUP 2004], p.13
A Reaction
He cites Gödel's First Incompleteness theorem for this.
The
31 ideas
from Michael Potter
|
22273
|
Traditionally there are twelve categories of judgement, in groups of three
[Potter]
|
|
22279
|
Frege's sign |--- meant judgements, but the modern |- turnstile means inference, with intecedents
[Potter]
|
|
22281
|
A material conditional cannot capture counterfactual reasoning
[Potter]
|
|
22283
|
Compositionality should rely on the parsing tree, which may contain more than sentence components
[Potter]
|
|
22282
|
'Direct compositonality' says the components wholly explain a sentence meaning
[Potter]
|
|
22284
|
'Greater than', which is the ancestral of 'successor', strictly orders the natural numbers
[Potter]
|
|
22285
|
Impredicative definitions are circular, but fine for picking out, rather than creating something
[Potter]
|
|
22291
|
Deductivism can't explain how the world supports unconditional conclusions
[Potter]
|
|
22287
|
If 'concrete' is the negative of 'abstract', that means desires and hallucinations are concrete
[Potter]
|
|
22290
|
The phrase 'the concept "horse"' can't refer to a concept, because it is saturated
[Potter]
|
|
22295
|
Modern logical truths are true under all interpretations of the non-logical words
[Potter]
|
|
22296
|
Compositionality is more welcome in logic than in linguistics (which is more contextual)
[Potter]
|
|
22298
|
Why is fictional arithmetic applicable to the real world?
[Potter]
|
|
22301
|
The Identity Theory says a proposition is true if it coincides with what makes it true
[Potter]
|
|
22310
|
The formalist defence against Gödel is to reject his metalinguistic concept of truth
[Potter]
|
|
22324
|
It has been unfortunate that externalism about truth is equated with correspondence
[Potter]
|
|
22327
|
Knowledge from a drunken schoolteacher is from a reliable and unreliable process
[Potter]
|
|
10702
|
Set theory's three roles: taming the infinite, subject-matter of mathematics, and modes of reasoning
[Potter]
|
|
10703
|
Supposing axioms (rather than accepting them) give truths, but they are conditional
[Potter]
|
|
10704
|
We can formalize second-order formation rules, but not inference rules
[Potter]
|
|
10707
|
Mereology elides the distinction between the cards in a pack and the suits
[Potter]
|
|
13041
|
Collections have fixed members, but fusions can be carved in innumerable ways
[Potter]
|
|
10708
|
Nowadays we derive our conception of collections from the dependence between them
[Potter]
|
|
10709
|
Priority is a modality, arising from collections and members
[Potter]
|
|
13042
|
If dependence is well-founded, with no infinite backward chains, this implies substances
[Potter]
|
|
10712
|
If set theory didn't found mathematics, it is still needed to count infinite sets
[Potter]
|
|
10713
|
Usually the only reason given for accepting the empty set is convenience
[Potter]
|
|
13043
|
A relation is a set consisting entirely of ordered pairs
[Potter]
|
|
13044
|
Infinity: There is at least one limit level
[Potter]
|
|
17882
|
It is remarkable that all natural number arithmetic derives from just the Peano Axioms
[Potter]
|
|
13546
|
The 'limitation of size' principles say whether properties collectivise depends on the number of objects
[Potter]
|