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Single Idea 18392
[filed under theme 6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / a. Units
]
Full Idea
Classes, because they have a particular cardinality, are essentially a certain number of ones, things that, within the particular class, are each taken as a unit.
Gist of Idea
Classes have cardinalities, so their members must all be treated as units
Source
David M. Armstrong (Truth and Truthmakers [2004], 09.1)
Book Ref
Armstrong,D.M.: 'Truth and Truthmakers' [CUP 2004], p.114
A Reaction
[Singletons are exceptions] So units are basic to set theory, which is the foundations of technical analytic philosophy (as well as, for many, of mathematics). If you can't treat something as a unit, it won't go into set theory. Vagueness...
Related Idea
Idea 18396
The set theory brackets { } assert that the member is a unit [Armstrong]
The
133 ideas
from David M. Armstrong
8505
|
Refusal to explain why different tokens are of the same type is to be an ostrich
[Armstrong]
|
8507
|
Some think of reality as made of things; I prefer facts or states of affairs
[Armstrong]
|
8506
|
Particulars and properties are distinguishable, but too close to speak of a relation
[Armstrong]
|
3900
|
Maybe experience is not essential to perception, but only to the causing of beliefs
[Armstrong, by Scruton]
|
4253
|
Externalism says knowledge involves a natural relation between the belief state and what makes it true
[Armstrong]
|
15547
|
Negative existentials have 'totality facts' as truthmakers
[Armstrong, by Lewis]
|
15542
|
All possibilities are recombinations of properties in the actual world
[Armstrong, by Lewis]
|
5690
|
A mental state without belief refutes self-intimation; a belief with no state refutes infallibility
[Armstrong, by Shoemaker]
|
5493
|
If pains are defined causally, and research shows that the causal role is physical, then pains are physical
[Armstrong, by Lycan]
|
4600
|
Armstrong and Lewis see functionalism as an identity of the function and its realiser
[Armstrong, by Heil]
|
6498
|
Armstrong suggests secondary qualities are blurred primary qualities
[Armstrong, by Robinson,H]
|
14330
|
To be realists about dispositions, we can only discuss them through their categorical basis
[Armstrong]
|
11003
|
The best version of reductionist actualism around is Armstrong's combinatorial account
[Armstrong, by Read]
|
4032
|
The problem of universals is how many particulars can all be of the same 'type'
[Armstrong]
|
7429
|
Causal Functionalism says mental states are apt for producing behaviour
[Armstrong]
|
7432
|
One mental role might be filled by a variety of physical types
[Armstrong]
|
7431
|
The identity of mental states with physical properties is contingent, because the laws of nature are contingent
[Armstrong]
|
7434
|
Behaviourism is false, but mind is definable as the cause of behaviour
[Armstrong]
|
7436
|
The manifestations of a disposition need never actually exist
[Armstrong]
|
7437
|
Consciousness and experience of qualities are not the same
[Armstrong]
|
7438
|
A causal theory of mentality would be improved by a teleological element
[Armstrong]
|
7440
|
Secondary qualities are microscopic primary qualities of physical things
[Armstrong]
|
5492
|
How can essences generate the right powers to vary with distance between objects?
[Armstrong]
|
4036
|
What matters is not how many entities we postulate, but how many kinds of entities
[Armstrong, by Mellor/Oliver]
|
8533
|
Predicates need ontological correlates to ensure that they apply
[Armstrong]
|
4035
|
There must be some explanation of why certain predicates are applicable to certain objects
[Armstrong]
|
8529
|
Deniers of properties and relations rely on either predicates or on classes
[Armstrong]
|
8532
|
Resemblances must be in certain 'respects', and they seem awfully like properties
[Armstrong]
|
8530
|
Change of temperature in objects is quite independent of the predicates 'hot' and 'cold'
[Armstrong]
|
8536
|
We want to know what constituents of objects are grounds for the application of predicates
[Armstrong]
|
8531
|
In most sets there is no property common to all the members
[Armstrong]
|
4034
|
Whether we apply 'cold' or 'hot' to an object is quite separate from its change of temperature
[Armstrong]
|
8535
|
To the claim that every predicate has a property, start by eliminating failure of application of predicate
[Armstrong]
|
8537
|
Tropes fall into classes, because exact similarity is symmetrical and transitive
[Armstrong]
|
8538
|
Trope theory needs extra commitments, to symmetry and non-transitivity, unless resemblance is exact
[Armstrong]
|
8539
|
Universals are required to give a satisfactory account of the laws of nature
[Armstrong]
|
8541
|
Regularities theories are poor on causal connections, counterfactuals and probability
[Armstrong]
|
8540
|
The introduction of sparse properties avoids the regularity theory's problem with 'grue'
[Armstrong]
|
15753
|
Essences might support Resemblance Nominalism, but they are too coarse and ill-defined
[Armstrong]
|
15754
|
Without properties we would be unable to express the laws of nature
[Armstrong]
|
7024
|
Properties are universals, which are always instantiated
[Armstrong, by Heil]
|
9478
|
Even if all properties are categorical, they may be denoted by dispositional predicates
[Armstrong, by Bird]
|
10729
|
Universals explain resemblance and causal power
[Armstrong, by Oliver]
|
15544
|
If what is actual might have been impossible, we need S4 modal logic
[Armstrong, by Lewis]
|
10728
|
A thing's self-identity can't be a universal, since we can know it a priori
[Armstrong, by Oliver]
|
4031
|
It doesn't follow that because there is a predicate there must therefore exist a property
[Armstrong]
|
10024
|
The type-token distinction is the universal-particular distinction
[Armstrong, by Hodes]
|
18467
|
Truth-making can't be entailment, because truthmakers are portions of reality
[Armstrong]
|
18468
|
Armstrong says truthmakers necessitate their truth, where 'necessitate' is a primitive relation
[Armstrong, by MacBride]
|
18368
|
For all being, there is a potential proposition which expresses its existence and nature
[Armstrong]
|
18370
|
A realm of abstract propositions is causally inert, so has no explanatory value
[Armstrong]
|
18371
|
The class of similar things is much too big a truthmaker for the feature of a particular
[Armstrong]
|
18372
|
We need properties, as minimal truthmakers for the truths about objects
[Armstrong]
|
18373
|
If tropes are non-transferable, then they necessarily belong to their particular substance
[Armstrong]
|
18374
|
Truthmaking needs states of affairs, to unite particulars with tropes or universals.
[Armstrong]
|
18375
|
General truths are a type of negative truth, saying there are no more ravens than black ones
[Armstrong]
|
18377
|
Negative truths have as truthmakers all states of affairs relevant to the truth
[Armstrong]
|
18378
|
Length is a 'determinable' property, and one mile is one its 'determinates'
[Armstrong]
|
18379
|
The determinates of a determinable must be incompatible with each other
[Armstrong]
|
18380
|
Negative causations supervene on positive causations plus their laws?
[Armstrong]
|
18381
|
Necessitating general truthmakers must also specify their limits
[Armstrong]
|
18382
|
The nature of arctic animals is truthmaker for the absence of penguins there
[Armstrong]
|
18384
|
One truthmaker will do for a contingent truth and for its contradictory
[Armstrong]
|
18385
|
Logical atomism builds on the simple properties, but are they the only possible properties?
[Armstrong]
|
18386
|
What is the truthmaker for 'it is possible that there could have been nothing'?
[Armstrong]
|
18387
|
The truthmakers for possible unicorns are the elements in their combination
[Armstrong]
|
18388
|
Possible worlds don't fix necessities; intrinsic necessities imply the extension in worlds
[Armstrong]
|
18389
|
When entities contain entities, or overlap with them, there is 'partial' identity
[Armstrong]
|
18390
|
All metaphysical discussion should be guided by a quest for truthmakers
[Armstrong]
|
18393
|
For 'there is a class with no members' we don't need the null set as truthmaker
[Armstrong]
|
18392
|
Classes have cardinalities, so their members must all be treated as units
[Armstrong]
|
18391
|
'Naturalism' says only the world of space-time exists
[Armstrong]
|
18394
|
In mathematics, truthmakers are possible instantiations of structures
[Armstrong]
|
18396
|
The set theory brackets { } assert that the member is a unit
[Armstrong]
|
18400
|
Properties are not powers - they just have powers
[Armstrong]
|
18397
|
Powers must result in some non-powers, or there would only be potential without result
[Armstrong]
|
18399
|
How does the power of gravity know the distance it acts over?
[Armstrong]
|
18401
|
The pure present moment is too brief to be experienced
[Armstrong]
|
4437
|
'Mereological Nominalism' sees whiteness as a huge white object consisting of all the white things
[Armstrong]
|
4438
|
'Mereological Nominalism' may work for whiteness, but it doesn't seem to work for squareness
[Armstrong]
|
4435
|
'Class Nominalism' cannot explain co-extensive properties, or sets with random members
[Armstrong]
|
4434
|
'Class Nominalism' says that properties or kinds are merely membership of a set (e.g. of white things)
[Armstrong]
|
4436
|
'Class Nominalism' may explain properties if we stick to 'natural' sets, and ignore random ones
[Armstrong]
|
4431
|
'Predicate Nominalism' says that a 'universal' property is just a predicate applied to lots of things
[Armstrong]
|
4433
|
Concept and predicate nominalism miss out some predicates, and may be viciously regressive
[Armstrong]
|
4432
|
'Concept Nominalism' says a 'universal' property is just a mental concept applied to lots of things
[Armstrong]
|
4439
|
'Resemblance Nominalism' says properties are resemblances between classes of particulars
[Armstrong]
|
4440
|
'Resemblance Nominalism' finds that in practice the construction of resemblance classes is hard
[Armstrong]
|
4446
|
It is claimed that some universals are not exemplified by any particular, so must exist separately
[Armstrong]
|
4445
|
If properties and relations are particulars, there is still the problem of how to classify and group them
[Armstrong]
|
4444
|
One moderate nominalist view says that properties and relations exist, but they are particulars
[Armstrong]
|
4448
|
Should we decide which universals exist a priori (through words), or a posteriori (through science)?
[Armstrong]
|
4442
|
Most thinkers now reject self-predication (whiteness is NOT white) so there is no Third Man problem
[Armstrong]
|
12677
|
Armstrong holds that all basic properties are categorical
[Armstrong, by Ellis]
|
16246
|
Rather than take necessitation between universals as primitive, just make laws primitive
[Maudlin on Armstrong]
|
9480
|
Armstrong has an unclear notion of contingent necessitation, which can't necessitate anything
[Bird on Armstrong]
|
8582
|
Regularities are lawful if a second-order universal unites two first-order universals
[Armstrong, by Lewis]
|
17662
|
Science depends on laws of nature to study unobserved times and spaces
[Armstrong]
|
17664
|
Each subject has an appropriate level of abstraction
[Armstrong]
|
17663
|
If you know what it is, investigation is pointless. If you don't, investigation is impossible
[Armstrong]
|
17666
|
Actualism means that ontology cannot contain what is merely physically possible
[Armstrong]
|
17667
|
Dispositions exist, but their truth-makers are actual or categorical properties
[Armstrong]
|
17668
|
It is likely that particulars can be individuated by unique conjunctions of properties
[Armstrong]
|
17669
|
Realist regularity theories of laws need universals, to pick out the same phenomena
[Armstrong]
|
17671
|
A naive regularity view says if it never occurs then it is impossible
[Armstrong]
|
17670
|
Newton's First Law refers to bodies not acted upon by a force, but there may be no such body
[Armstrong]
|
17672
|
A good reason for something (the smoke) is not an explanation of it (the fire)
[Armstrong]
|
17674
|
The raven paradox has three disjuncts, confirmed by confirming any one of them
[Armstrong]
|
17675
|
Unlike 'green', the 'grue' predicate involves a time and a change
[Armstrong]
|
17676
|
Best explanations explain the most by means of the least
[Armstrong]
|
17677
|
Past, present and future must be equally real if universals are instantiated
[Armstrong]
|
17680
|
The identity of a thing with itself can be ruled out as a pseudo-property
[Armstrong]
|
17678
|
Universals are just the repeatable features of a world
[Armstrong]
|
17679
|
All instances of some property are strictly identical
[Armstrong]
|
17681
|
The laws of nature link properties with properties
[Armstrong]
|
17682
|
A universe couldn't consist of mere laws
[Armstrong]
|
17684
|
To explain observations by a regular law is to explain the observations by the observations
[Armstrong]
|
17685
|
Induction aims at 'all Fs', but abduction aims at hidden or theoretical entities
[Armstrong]
|
17683
|
Science suggests that the predicate 'grue' is not a genuine single universal
[Armstrong]
|
17687
|
If everything is powers there is a vicious regress, as powers are defined by more powers
[Armstrong]
|
17688
|
Negative facts are supervenient on positive facts, suggesting they are positive facts
[Armstrong]
|
17689
|
Absences might be effects, but surely not causes?
[Armstrong]
|
17690
|
Oaken conditional laws, Iron universal laws, and Steel necessary laws
[Armstrong, by PG]
|
17691
|
Nothing is genuinely related to itself
[Armstrong]
|
17692
|
We can't deduce the phenomena from the One
[Armstrong]
|
17693
|
The necessary/contingent distinction may need to recognise possibilities as real
[Armstrong]
|
17686
|
Universals are abstractions from states of affairs
[Armstrong]
|
15442
|
Universals are abstractions from their particular instances
[Armstrong, by Lewis]
|
15550
|
Properties are contingently existing beings with multiple locations in space and time
[Armstrong, by Lewis]
|
4798
|
In recent writings, Armstrong makes a direct identification of necessitation with causation
[Armstrong, by Psillos]
|
9497
|
Without modality, Armstrong falls back on fictionalism to support counterfactual laws
[Bird on Armstrong]
|
4743
|
The truth-maker for a truth must necessitate that truth
[Armstrong]
|
4742
|
Correspondence may be one-many or many one, as when either p or q make 'p or q' true
[Armstrong]
|