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Single Idea 18956
[filed under theme 5. Theory of Logic / A. Overview of Logic / 5. First-Order Logic
]
Full Idea
The natural understanding of first-order logic is that in writing down first-order schemata we are implicitly asserting their validity, that is, making second-order assertions. ...Thus even quantification theory involves reference to classes.
Gist of Idea
Asserting first-order validity implicitly involves second-order reference to classes
Source
Hilary Putnam (Philosophy of Logic [1971], Ch.3)
Book Ref
Putnam,Hilary: 'Philosophy of Logic' [Routledge 1972], p.32
A Reaction
If, as a nominalist, you totally rejected classes, presumably you would get by in first-order logic somehow. To say 'there are no classes so there is no logical validity' sounds bonkers.
Related Idea
Idea 18951
For scientific purposes there is a precise concept of 'true-in-L', using set theory [Putnam]
The
23 ideas
with the same theme
[logic where variables only refer to objects]:
22873
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Liberalism should improve the system, and not just ameliorate it
[Dewey]
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21495
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Theoretical and practical politics are both concerned with the best lives for individuals
[Russell]
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18956
|
Asserting first-order validity implicitly involves second-order reference to classes
[Putnam]
|
10767
|
Elementary logic is complete, but cannot capture mathematics
[Tharp]
|
13840
|
First-order logic is the strongest complete compact theory with Löwenheim-Skolem
[Hacking]
|
13844
|
A limitation of first-order logic is that it cannot handle branching quantifiers
[Hacking]
|
18108
|
First-order logic is not decidable: there is no test of whether any formula is valid
[Bostock]
|
18109
|
The completeness of first-order logic implies its compactness
[Bostock]
|
10478
|
Since first-order languages are complete, |= and |- have the same meaning
[Hodges,W]
|
19315
|
In quantified language the components of complex sentences may not be sentences
[Kirkham]
|
17788
|
First-order logic only has its main theorems because it is so weak
[Mayberry]
|
13624
|
The 'triumph' of first-order logic may be related to logicism and the Hilbert programme, which failed
[Shapiro]
|
13660
|
Maybe compactness, semantic effectiveness, and the Löwenheim-Skolem properties are desirable
[Shapiro]
|
13662
|
First-order logic was an afterthought in the development of modern logic
[Shapiro]
|
13673
|
The notion of finitude is actually built into first-order languages
[Shapiro]
|
10588
|
First-order logic is Complete, and Compact, with the Löwenheim-Skolem Theorems
[Shapiro]
|
10897
|
A first-order 'sentence' is a formula with no free variables
[Zalabardo]
|
10986
|
Not all validity is captured in first-order logic
[Read]
|
13534
|
In first-order logic syntactic and semantic consequence (|- and |=) nicely coincide
[Wolf,RS]
|
13535
|
First-order logic is weakly complete (valid sentences are provable); we can't prove every sentence or its negation
[Wolf,RS]
|
20100
|
Classical liberalism seeks freedom of opinion, of private life, of expression, and of property
[Micklethwait/Wooldridge]
|
21122
|
Liberal Nationalism says welfare states and democracy needed a shared sense of nationality
[Shorten]
|
22609
|
Philosophers accepted first-order logic, because they took science to be descriptive, not explanatory
[Ingthorsson]
|