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Single Idea 15411
[filed under theme 5. Theory of Logic / J. Model Theory in Logic / 1. Logical Models
]
Full Idea
In practice there is no need to consider any but mathematical models, models whose universes consist of mathematical objects, since every model is isomorphic to one of these.
Clarification
'isomorphic' means maps onto
Gist of Idea
We only need to study mathematical models, since all other models are isomorphic to these
Source
John P. Burgess (Philosophical Logic [2009], 1.8)
Book Ref
Burgess,John P.: 'Philosophical Logic' [Princeton 2009], p.10
A Reaction
The crucial link is the technique of Gödel Numbering, which can translate any verbal formula into numerical form. He adds that, because of the Löwenheim-Skolem theorem only subsets of the natural numbers need be considered.
The
34 ideas
from John P. Burgess
15404
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Technical people see logic as any formal system that can be studied, not a study of argument validity
[Burgess]
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15403
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Philosophical logic is a branch of logic, and is now centred in computer science
[Burgess]
|
15405
|
Classical logic neglects the non-mathematical, such as temporality or modality
[Burgess]
|
15407
|
Formalising arguments favours lots of connectives; proving things favours having very few
[Burgess]
|
15406
|
'Induction' and 'recursion' on complexity prove by connecting a formula to its atomic components
[Burgess]
|
15408
|
'Tautologies' are valid formulas of classical sentential logic - or substitution instances in other logics
[Burgess]
|
15409
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All occurrences of variables in atomic formulas are free
[Burgess]
|
15411
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We only need to study mathematical models, since all other models are isomorphic to these
[Burgess]
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15412
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Models leave out meaning, and just focus on truth values
[Burgess]
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15413
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With four tense operators, all complex tenses reduce to fourteen basic cases
[Burgess]
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15415
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The temporal Barcan formulas fix what exists, which seems absurd
[Burgess]
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15414
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The denotation of a definite description is flexible, rather than rigid
[Burgess]
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15416
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We aim to get the technical notion of truth in all models matching intuitive truth in all instances
[Burgess]
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15418
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Validity (for truth) and demonstrability (for proof) have correlates in satisfiability and consistency
[Burgess]
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15417
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Logical necessity has two sides - validity and demonstrability - which coincide in classical logic
[Burgess]
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15419
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General consensus is S5 for logical modality of validity, and S4 for proof
[Burgess]
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15420
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De re modality seems to apply to objects a concept intended for sentences
[Burgess]
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15421
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Classical logic neglects counterfactuals, temporality and modality, because maths doesn't use them
[Burgess]
|
15422
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Three conditionals theories: Materialism (material conditional), Idealism (true=assertable), Nihilism (no truth)
[Burgess]
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15423
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It is doubtful whether the negation of a conditional has any clear meaning
[Burgess]
|
15424
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Asserting a disjunction from one disjunct seems odd, but can be sensible, and needed in maths
[Burgess]
|
15426
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We can build one expanding sequence, instead of a chain of deductions
[Burgess]
|
15425
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The sequent calculus makes it possible to have proof without transitivity of entailment
[Burgess]
|
15427
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The Cut Rule expresses the classical idea that entailment is transitive
[Burgess]
|
15428
|
The Liar seems like a truth-value 'gap', but dialethists see it as a 'glut'
[Burgess]
|
15429
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Relevance logic's → is perhaps expressible by 'if A, then B, for that reason'
[Burgess]
|
15430
|
Is classical logic a part of intuitionist logic, or vice versa?
[Burgess]
|
15431
|
It is still unsettled whether standard intuitionist logic is complete
[Burgess]
|
10186
|
If set theory is used to define 'structure', we can't define set theory structurally
[Burgess]
|
10187
|
Abstract algebra concerns relations between models, not common features of all the models
[Burgess]
|
10185
|
Set theory is the standard background for modern mathematics
[Burgess]
|
10184
|
Structuralists take the name 'R' of the reals to be a variable ranging over structures, not a structure
[Burgess]
|
10189
|
There is no one relation for the real number 2, as relations differ in different models
[Burgess]
|
10188
|
How can mathematical relations be either internal, or external, or intrinsic?
[Burgess]
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