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Single Idea 13504
[filed under theme 3. Truth / F. Semantic Truth / 1. Tarski's Truth / b. Satisfaction and truth
]
Full Idea
We explain truth for sentences in terms of satisfaction of formulae. The crux here is that for a sentence, either all sequences satisfy it or none do (with no middle ground). For formulae, some sequences may satisfy it and others not.
Gist of Idea
Truth for sentences is satisfaction of formulae; for sentences, either all sequences satisfy it (true) or none do
Source
William D. Hart (The Evolution of Logic [2010], 4)
Book Ref
Hart,W.D.: 'The Evolution of Logic' [CUP 2010], p.107
A Reaction
This is the hardest part of Tarski's theory of truth to grasp.
The
45 ideas
from 'The Evolution of Logic'
13456
|
Set theory articulates the concept of order (through relations)
[Hart,WD]
|
13443
|
∈ relates across layers, while ⊆ relates within layers
[Hart,WD]
|
13442
|
Without the empty set we could not form a∩b without checking that a and b meet
[Hart,WD]
|
13458
|
A partial ordering becomes 'total' if any two members of its field are comparable
[Hart,WD]
|
13460
|
'Well-ordering' must have a least member, so it does the natural numbers but not the integers
[Hart,WD]
|
13461
|
We can choose from finite and evident sets, but not from infinite opaque ones
[Hart,WD]
|
13462
|
With the Axiom of Choice every set can be well-ordered
[Hart,WD]
|
13441
|
Naïve set theory has trouble with comprehension, the claim that every predicate has an extension
[Hart,WD]
|
13457
|
A 'partial ordering' is irreflexive and transitive; the sets are ordered, but not the subsets
[Hart,WD]
|
13459
|
The less-than relation < well-orders, and partially orders, and totally orders the ordinal numbers
[Hart,WD]
|
13463
|
There are at least as many infinite cardinals as transfinite ordinals (because they will map)
[Hart,WD]
|
13446
|
19th century arithmetization of analysis isolated the real numbers from geometry
[Hart,WD]
|
13516
|
If we accept that V=L, it seems to settle all the open questions of set theory
[Hart,WD]
|
13515
|
To study abstract problems, some knowledge of set theory is essential
[Hart,WD]
|
13469
|
Tarski showed how we could have a correspondence theory of truth, without using 'facts'
[Hart,WD]
|
13466
|
We are all post-Kantians, because he set the current agenda for philosophy
[Hart,WD]
|
13477
|
The problems are the monuments of philosophy
[Hart,WD]
|
13471
|
Mathematics makes existence claims, but philosophers usually say those are never analytic
[Hart,WD]
|
13481
|
Maybe sets should be rethought in terms of the even more basic categories
[Hart,WD]
|
13474
|
Euclid has a unique parallel, spherical geometry has none, and saddle geometry has several
[Hart,WD]
|
13476
|
The failure of key assumptions in geometry, mereology and set theory throw doubt on the a priori
[Hart,WD]
|
13475
|
The Fregean concept of GREEN is a function assigning true to green things, and false to the rest
[Hart,WD]
|
13488
|
Mass words do not have plurals, or numerical adjectives, or use 'fewer'
[Hart,WD]
|
13494
|
The iterative conception may not be necessary, and may have fixed points or infinitely descending chains
[Hart,WD]
|
13492
|
Von Neumann's ordinals generalise into the transfinite better, because Zermelo's ω is a singleton
[Hart,WD]
|
13491
|
The axiom of infinity with separation gives a least limit ordinal ω
[Hart,WD]
|
13496
|
First-order logic is 'compact': consequences of a set are consequences of a finite subset
[Hart,WD]
|
13484
|
Berry's Paradox: we succeed in referring to a number, with a term which says we can't do that
[Hart,WD]
|
13482
|
The Burali-Forti paradox is a crisis for Cantor's ordinals
[Hart,WD]
|
13497
|
Nowadays ZFC and NBG are the set theories; types are dead, and NF is only useful for the whole universe
[Hart,WD]
|
13493
|
In the modern view, foundation is the heart of the way to do set theory
[Hart,WD]
|
13495
|
Foundation Axiom: an nonempty set has a member disjoint from it
[Hart,WD]
|
13490
|
Von Neumann defines α<β as α∈β
[Hart,WD]
|
13504
|
Truth for sentences is satisfaction of formulae; for sentences, either all sequences satisfy it (true) or none do
[Hart,WD]
|
13503
|
A first-order language has an infinity of T-sentences, which cannot add up to a definition of truth
[Hart,WD]
|
13500
|
Conditional Proof: infer a conditional, if the consequent can be deduced from the antecedent
[Hart,WD]
|
13502
|
∃y... is read as 'There exists an individual, call it y, such that...', and not 'There exists a y such that...'
[Hart,WD]
|
13507
|
The machinery used to solve the Liar can be rejigged to produce a new Liar
[Hart,WD]
|
13506
|
The universal quantifier can't really mean 'all', because there is no universal set
[Hart,WD]
|
13505
|
Model theory studies how set theory can model sets of sentences
[Hart,WD]
|
13509
|
We can establish truths about infinite numbers by means of induction
[Hart,WD]
|
13511
|
Model theory is mostly confined to first-order theories
[Hart,WD]
|
13513
|
Models are ways the world might be from a first-order point of view
[Hart,WD]
|
13512
|
Modern model theory begins with the proof of Los's Conjecture in 1962
[Hart,WD]
|
13480
|
Fregean self-evidence is an intrinsic property of basic truths, rules and definitions
[Hart,WD]
|