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Single Idea 18813
[filed under theme 5. Theory of Logic / B. Logical Consequence / 1. Logical Consequence
]
Full Idea
Logical consequence, I argue, is distinguished from other implication relations by the fact that logical laws may be applied in extending any implication relation so that it applies among some complex statements involving logical connectives.
Gist of Idea
Logical consequence is a relation that can extended into further statements
Source
Ian Rumfitt (The Boundary Stones of Thought [2015], 3.3)
Book Ref
Rumfitt,Ian: 'The Boundary Stones of Thought' [OUP 2015], p.74
A Reaction
He offers implication in electronics as an example of a non-logical implication relation. This seems to indicate that logic must be monotonic, that consequence is transitive, and that the Cut Law always applies.
The
37 ideas
from 'The Boundary Stones of Thought'
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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18799
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Intuitionists can accept Double Negation Elimination for decidable propositions
[Rumfitt]
|
18798
|
It is the second-order part of intuitionistic logic which actually negates some classical theorems
[Rumfitt]
|
18804
|
The case for classical logic rests on its rules, much more than on the Principle of Bivalence
[Rumfitt]
|
18805
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Classical logic rules cannot be proved, but various lines of attack can be repelled
[Rumfitt]
|
18802
|
In specifying a logical constant, use of that constant is quite unavoidable
[Rumfitt]
|
18800
|
Introduction rules give deduction conditions, and Elimination says what can be deduced
[Rumfitt]
|
18808
|
Normal deduction presupposes the Cut Law
[Rumfitt]
|
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
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Logic is higher-order laws which can expand the range of any sort of deduction
[Rumfitt]
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18814
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'Absolute necessity' would have to rest on S5
[Rumfitt]
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18813
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Logical consequence is a relation that can extended into further statements
[Rumfitt]
|
18816
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Metaphysical modalities respect the actual identities of things
[Rumfitt]
|
18817
|
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
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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]
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18825
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S5 is the logic of logical necessity
[Rumfitt]
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18826
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'True at a possibility' means necessarily true if what is said had obtained
[Rumfitt]
|
18827
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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
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Medieval logicians said understanding A also involved understanding not-A
[Rumfitt]
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18830
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Most set theorists doubt bivalence for the Continuum Hypothesis, but still use classical logic
[Rumfitt]
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18834
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Infinitesimals do not stand in a determinate order relation to zero
[Rumfitt]
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18835
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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]
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18837
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A set can be determinate, because of its concept, and still have vague membership
[Rumfitt]
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18839
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An object that is not clearly red or orange can still be red-or-orange, which sweeps up problem cases
[Rumfitt]
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18838
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The extension of a colour is decided by a concept's place in a network of contraries
[Rumfitt]
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18840
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When faced with vague statements, Bivalence is not a compelling principle
[Rumfitt]
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18842
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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]
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18845
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If the totality of sets is not well-defined, there must be doubt about the Power Set Axiom
[Rumfitt]
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18846
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Cantor and Dedekind aimed to give analysis a foundation in set theory (rather than geometry)
[Rumfitt]
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