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Single Idea 18827
[filed under theme 5. Theory of Logic / A. Overview of Logic / 6. Classical Logic
]
Full Idea
If we specify the senses of the connectives by way of the standard truth-tables, then we must justify classical logic only by appeal to the Principle of Bivalence.
Gist of Idea
If truth-tables specify the connectives, classical logic must rely on Bivalence
Source
Ian Rumfitt (The Boundary Stones of Thought [2015], 7)
Book Ref
Rumfitt,Ian: 'The Boundary Stones of Thought' [OUP 2015], p.184
A Reaction
Rumfitt proposes to avoid the truth-tables, and hence not to rely on Bivalence for his support of classical logic. He accepts that Bivalence is doubtful, citing the undecidability of the Continuum Hypothesis as a problem instance.
Related Idea
Idea 18830
Most set theorists doubt bivalence for the Continuum Hypothesis, but still use classical logic [Rumfitt]
The
37 ideas
from 'The Boundary Stones of Thought'
18803
|
Semantics for propositions: 1) validity preserves truth 2) non-contradition 3) bivalence 4) truth tables
[Rumfitt]
|
18802
|
In specifying a logical constant, use of that constant is quite unavoidable
[Rumfitt]
|
18798
|
It is the second-order part of intuitionistic logic which actually negates some classical theorems
[Rumfitt]
|
18799
|
Intuitionists can accept Double Negation Elimination for decidable propositions
[Rumfitt]
|
18805
|
Classical logic rules cannot be proved, but various lines of attack can be repelled
[Rumfitt]
|
18804
|
The case for classical logic rests on its rules, much more than on the Principle of Bivalence
[Rumfitt]
|
18800
|
Introduction rules give deduction conditions, and Elimination says what can be deduced
[Rumfitt]
|
18808
|
Normal deduction presupposes the Cut Law
[Rumfitt]
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18807
|
Monotonicity means there is a guarantee, rather than mere inductive support
[Rumfitt]
|
18809
|
Logical truths are just the assumption-free by-products of logical rules
[Rumfitt]
|
18815
|
Logic is higher-order laws which can expand the range of any sort of deduction
[Rumfitt]
|
18814
|
'Absolute necessity' would have to rest on S5
[Rumfitt]
|
18813
|
Logical consequence is a relation that can extended into further statements
[Rumfitt]
|
18816
|
Metaphysical modalities respect the actual identities of things
[Rumfitt]
|
18817
|
We understand conditionals, but disagree over their truth-conditions
[Rumfitt]
|
18819
|
The idea that there are unrecognised truths is basic to our concept of truth
[Rumfitt]
|
18820
|
In English 'evidence' is a mass term, qualified by 'little' and 'more'
[Rumfitt]
|
18821
|
Possibilities are like possible worlds, but not fully determinate or complete
[Rumfitt]
|
18824
|
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
|
'True at a possibility' means necessarily true if what is said had obtained
[Rumfitt]
|
18827
|
If truth-tables specify the connectives, classical logic must rely on Bivalence
[Rumfitt]
|
18828
|
If two possibilities can't share a determiner, they are incompatible
[Rumfitt]
|
18829
|
The truth grounds for 'not A' are the possibilities incompatible with truth grounds for A
[Rumfitt]
|
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]
|
18836
|
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]
|
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]
|
18840
|
When faced with vague statements, Bivalence is not a compelling principle
[Rumfitt]
|
18842
|
Maybe an ordinal is a property of isomorphic well-ordered sets, and not itself a set
[Rumfitt]
|
18843
|
The iterated conception of set requires continual increase in axiom strength
[Rumfitt]
|
18845
|
If the totality of sets is not well-defined, there must be doubt about the Power Set Axiom
[Rumfitt]
|
18846
|
Cantor and Dedekind aimed to give analysis a foundation in set theory (rather than geometry)
[Rumfitt]
|