10476
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The idea that groups of concepts could be 'implicitly defined' was abandoned [Hodges,W]
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Full Idea:
Late nineteenth century mathematicians said that, although plus, minus and 0 could not be precisely defined, they could be partially 'implicitly defined' as a group. This nonsense was rejected by Frege and others, as expressed in Russell 1903.
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From:
Wilfrid Hodges (Model Theory [2005], 2)
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A reaction:
[compressed] This is helpful in understanding what is going on in Frege's 'Grundlagen'. I won't challenge Hodges's claim that such definitions are nonsense, but there is a case for understanding groups of concepts together.
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10477
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|= in model-theory means 'logical consequence' - it holds in all models [Hodges,W]
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Full Idea:
If every structure which is a model of a set of sentences T is also a model of one of its sentences φ, then this is known as the model-theoretic consequence relation, and is written T |= φ. Not to be confused with |= meaning 'satisfies'.
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From:
Wilfrid Hodges (Model Theory [2005], 3)
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A reaction:
See also Idea 10474, which gives the other meaning of |=, as 'satisfies'. The symbol is ALSO used in propositional logical, to mean 'tautologically implies'! Sort your act out, logicians.
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10481
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Models in model theory are structures, not sets of descriptions [Hodges,W]
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Full Idea:
The models in model-theory are structures, but there is also a common use of 'model' to mean a formal theory which describes and explains a phenomenon, or plans to build it.
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From:
Wilfrid Hodges (Model Theory [2005], 5)
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A reaction:
Hodges is not at all clear here, but the idea seems to be that model-theory offers a set of objects and rules, where the common usage offers a set of descriptions. Model-theory needs homomorphisms to connect models to things,
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18431
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Internal relations combine some tropes into a nucleus, which bears the non-essential tropes [Simons, by Edwards]
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Full Idea:
Simons's 'nuclear' option blends features of the substratum and bundle theories. First we have tropes collected by virtue of their internal relations, forming the essential kernel or nucleus. This nucleus then bears the non-essential tropes.
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From:
report of Peter Simons (Particulars in Particular Clothing [1994], p.567) by Douglas Edwards - Properties 3.5
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A reaction:
[compression of Edwards's summary] This strikes me as being a remarkably good theory. I am not sure of the ontological status of properties, such that they can (unaided) combine to make part of an object. What binds the non-essentials?
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19382
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Abstracta are abbreviated ways of talking; there are just substances, and truths about them [Leibniz]
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Full Idea:
I consider abstracta not as real things but as abbreviated ways of talking ...and to that extent I am a nominalist, at least provisionally ...It suffices to posit only substances as real things, and, to assert truths about these.
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From:
Gottfried Leibniz (On the Reality of Accidents [1688]), quoted by Richard T.W. Arthur - Leibniz
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A reaction:
I am a modern nominalist, in my hostility to a serious ontological commitment to abstracta. You get into trouble, though, if you say there are only objects or substances. Physics says reality may all be 'fields', or something.... 'Truths' is good.
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