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Single Idea 14532
[filed under theme 10. Modality / A. Necessity / 3. Types of Necessity
]
Full Idea
Rumfitt argues that there is a distinctive notion of necessity implicated in the notion of logical consequence.
Gist of Idea
A distinctive type of necessity is found in logical consequence
Source
report of Ian Rumfitt (Logical Necessity [2010]) by Bob Hale/ Aviv Hoffmann - Introduction to 'Modality' 2
Book Ref
'Modality', ed/tr. Hale,B/Hoffman,A [OUP 2010], p.10
The
56 ideas
from Ian Rumfitt
11210
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Standardly 'and' and 'but' are held to have the same sense by having the same truth table
[Rumfitt]
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11211
|
If a sound conclusion comes from two errors that cancel out, the path of the argument must matter
[Rumfitt]
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11212
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The sense of a connective comes from primitively obvious rules of inference
[Rumfitt]
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11214
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We learn 'not' along with affirmation, by learning to either affirm or deny a sentence
[Rumfitt]
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18800
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Introduction rules give deduction conditions, and Elimination says what can be deduced
[Rumfitt]
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18802
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In specifying a logical constant, use of that constant is quite unavoidable
[Rumfitt]
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18798
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It is the second-order part of intuitionistic logic which actually negates some classical theorems
[Rumfitt]
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18799
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Intuitionists can accept Double Negation Elimination for decidable propositions
[Rumfitt]
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18803
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Semantics for propositions: 1) validity preserves truth 2) non-contradition 3) bivalence 4) truth tables
[Rumfitt]
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18805
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Classical logic rules cannot be proved, but various lines of attack can be repelled
[Rumfitt]
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18804
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The case for classical logic rests on its rules, much more than on the Principle of Bivalence
[Rumfitt]
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18807
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Monotonicity means there is a guarantee, rather than mere inductive support
[Rumfitt]
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18808
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Normal deduction presupposes the Cut Law
[Rumfitt]
|
18809
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Logical truths are just the assumption-free by-products of logical rules
[Rumfitt]
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18815
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Logic is higher-order laws which can expand the range of any sort of deduction
[Rumfitt]
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18813
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Logical consequence is a relation that can extended into further statements
[Rumfitt]
|
18814
|
'Absolute necessity' would have to rest on S5
[Rumfitt]
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18816
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Metaphysical modalities respect the actual identities of things
[Rumfitt]
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18817
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We understand conditionals, but disagree over their truth-conditions
[Rumfitt]
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18819
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The idea that there are unrecognised truths is basic to our concept of truth
[Rumfitt]
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18820
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In English 'evidence' is a mass term, qualified by 'little' and 'more'
[Rumfitt]
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18821
|
Possibilities are like possible worlds, but not fully determinate or complete
[Rumfitt]
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18824
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Since possibilities are properties of the world, calling 'red' the determination of a determinable seems right
[Rumfitt]
|
18825
|
S5 is the logic of logical necessity
[Rumfitt]
|
18826
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'True at a possibility' means necessarily true if what is said had obtained
[Rumfitt]
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18827
|
If truth-tables specify the connectives, classical logic must rely on Bivalence
[Rumfitt]
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18828
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If two possibilities can't share a determiner, they are incompatible
[Rumfitt]
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18829
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The truth grounds for 'not A' are the possibilities incompatible with truth grounds for A
[Rumfitt]
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18831
|
Medieval logicians said understanding A also involved understanding not-A
[Rumfitt]
|
18830
|
Most set theorists doubt bivalence for the Continuum Hypothesis, but still use classical logic
[Rumfitt]
|
18834
|
Infinitesimals do not stand in a determinate order relation to zero
[Rumfitt]
|
18835
|
Logic doesn't have a metaphysical basis, but nor can logic give rise to the metaphysics
[Rumfitt]
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18836
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A set may well not consist of its members; the empty set, for example, is a problem
[Rumfitt]
|
18837
|
A set can be determinate, because of its concept, and still have vague membership
[Rumfitt]
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18839
|
An object that is not clearly red or orange can still be red-or-orange, which sweeps up problem cases
[Rumfitt]
|
18838
|
The extension of a colour is decided by a concept's place in a network of contraries
[Rumfitt]
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18840
|
When faced with vague statements, Bivalence is not a compelling principle
[Rumfitt]
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18842
|
Maybe an ordinal is a property of isomorphic well-ordered sets, and not itself a set
[Rumfitt]
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18843
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The iterated conception of set requires continual increase in axiom strength
[Rumfitt]
|
18845
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If the totality of sets is not well-defined, there must be doubt about the Power Set Axiom
[Rumfitt]
|
18846
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Cantor and Dedekind aimed to give analysis a foundation in set theory (rather than geometry)
[Rumfitt]
|
9390
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Logic guides thinking, but it isn't a substitute for it
[Rumfitt]
|
9389
|
Vague membership of sets is possible if the set is defined by its concept, not its members
[Rumfitt]
|
17461
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Some 'how many?' answers are not predications of a concept, like 'how many gallons?'
[Rumfitt]
|
17462
|
A single object must not be counted twice, which needs knowledge of distinctness (negative identity)
[Rumfitt]
|
14532
|
A distinctive type of necessity is found in logical consequence
[Rumfitt, by Hale/Hoffmann,A]
|
12193
|
Logical necessity is when 'necessarily A' implies 'not-A is contradictory'
[Rumfitt]
|
12194
|
Contradictions include 'This is red and not coloured', as well as the formal 'B and not-B'
[Rumfitt]
|
12195
|
Soundness in argument varies with context, and may be achieved very informally indeed
[Rumfitt]
|
12198
|
Geometrical axioms in logic are nowadays replaced by inference rules (which imply the logical truths)
[Rumfitt]
|
12199
|
There is a modal element in consequence, in assessing reasoning from suppositions
[Rumfitt]
|
12201
|
We reject deductions by bad consequence, so logical consequence can't be deduction
[Rumfitt]
|
12202
|
Narrow non-modal logical necessity may be metaphysical, but real logical necessity is not
[Rumfitt]
|
12200
|
A logically necessary statement need not be a priori, as it could be unknowable
[Rumfitt]
|
12203
|
If a world is a fully determinate way things could have been, can anyone consider such a thing?
[Rumfitt]
|
12204
|
The logic of metaphysical necessity is S5
[Rumfitt]
|