more on this theme
|
more from this thinker
Single Idea 10298
[filed under theme 5. Theory of Logic / A. Overview of Logic / 7. Second-Order Logic
]
Full Idea
Some authors argue that second-order logic (with standard semantics) is not logic at all, but is a rather obscure form of mathematics.
Gist of Idea
Some say that second-order logic is mathematics, not logic
Source
Stewart Shapiro (Higher-Order Logic [2001], 2.4)
Book Ref
'Blackwell Guide to Philosophical Logic', ed/tr. Goble,Lou [Blackwell 2001], p.50
The
12 ideas
from 'Higher-Order Logic'
|
10588
|
First-order logic is Complete, and Compact, with the Löwenheim-Skolem Theorems
[Shapiro]
|
|
10290
|
Second-order variables also range over properties, sets, relations or functions
[Shapiro]
|
|
10590
|
Up Löwenheim-Skolem: if natural numbers satisfy wffs, then an infinite domain satisfies them
[Shapiro]
|
|
10292
|
Downward Löwenheim-Skolem: if there's an infinite model, there is a countable model
[Shapiro]
|
|
10294
|
Second-order logic has the expressive power for mathematics, but an unworkable model theory
[Shapiro]
|
|
10591
|
Logicians use 'property' and 'set' interchangeably, with little hanging on it
[Shapiro]
|
|
10296
|
The Löwenheim-Skolem Theorems fail for second-order languages with standard semantics
[Shapiro]
|
|
10297
|
The Löwenheim-Skolem theorem seems to be a defect of first-order logic
[Shapiro]
|
|
10298
|
Some say that second-order logic is mathematics, not logic
[Shapiro]
|
|
10299
|
If the aim of logic is to codify inferences, second-order logic is useless
[Shapiro]
|
|
10300
|
Logical consequence can be defined in terms of the logical terminology
[Shapiro]
|
|
10301
|
The axiom of choice is controversial, but it could be replaced
[Shapiro]
|