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Single Idea 10260
[filed under theme 5. Theory of Logic / B. Logical Consequence / 1. Logical Consequence
]
Full Idea
Field defines logical consequence by taking the notion of 'logical possibility' as primitive. Hence q is a consequence of P if the conjunction of the items in P with the negation of q is not possible.
Gist of Idea
Logical consequence is defined by the impossibility of P and ¬q
Source
report of Hartry Field (Science without Numbers [1980]) by Stewart Shapiro - Philosophy of Mathematics 7.2
Book Ref
Shapiro,Stewart: 'Philosophy of Mathematics:structure and ontology' [OUP 1997], p.222
A Reaction
The question would then be whether it is plausible to take logical possibility as primitive. Presumably only intuition could support it. But then intuition will equally support natural and metaphysical possibilities.
The
21 ideas
from 'Science without Numbers'
10261
|
The application of mathematics only needs its possibility, not its truth
[Field,H, by Shapiro]
|
8959
|
Field presumes properties can be eliminated from science
[Field,H, by Szabó]
|
9570
|
In Field's Platonist view, set theory is false because it asserts existence for non-existent things
[Field,H, by Chihara]
|
10260
|
Logical consequence is defined by the impossibility of P and ¬q
[Field,H, by Shapiro]
|
8958
|
In Field's version of science, space-time points replace real numbers
[Field,H, by Szabó]
|
18212
|
Nominalists try to only refer to physical objects, or language, or mental constructions
[Field,H]
|
18215
|
It seems impossible to explain the idea that the conclusion is contained in the premises
[Field,H]
|
18213
|
Abstract objects are only applicable to the world if they are impure, and connect to the physical
[Field,H]
|
18214
|
Mathematics is only empirical as regards which theory is useful
[Field,H]
|
18216
|
Abstractions can form useful counterparts to concrete statements
[Field,H]
|
18218
|
Hilbert explains geometry, by non-numerical facts about space
[Field,H]
|
18220
|
Both philosophy and physics now make substantivalism more attractive
[Field,H]
|
18219
|
Relational space is problematic if you take the idea of a field seriously
[Field,H]
|
18222
|
Beneath every extrinsic explanation there is an intrinsic explanation
[Field,H]
|
18221
|
'Metric' axioms uses functions, points and numbers; 'synthetic' axioms give facts about space
[Field,H]
|
9623
|
Field needs a semantical notion of second-order consequence, and that needs sets
[Brown,JR on Field,H]
|
18223
|
In theories of fields, space-time points or regions are causal agents
[Field,H]
|
9917
|
'Abstract' is unclear, but numbers, functions and sets are clearly abstract
[Field,H]
|
8757
|
The Indispensability Argument is the only serious ground for the existence of mathematical entities
[Field,H]
|
18211
|
You can reduce ontological commitment by expanding the logic
[Field,H]
|
18210
|
Why regard standard mathematics as truths, rather than as interesting fictions?
[Field,H]
|