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Single Idea 13249
[filed under theme 4. Formal Logic / E. Nonclassical Logics / 2. Intuitionist Logic
]
Full Idea
The inference of 'distribution' (∀x)(A v B) |- (∀x)A v (∃x)B) is valid in classical logic but invalid intuitionistically. It is straightforward to construct a 'stage' at which the LHS is true but the RHS is not.
Gist of Idea
(∀x)(A v B) |- (∀x)A v (∃x)B) is valid in classical logic but invalid intuitionistically
Source
JC Beall / G Restall (Logical Pluralism [2006], 6.1.2)
Book Ref
Beall,J/Restall,G: 'Logical Pluralism' [OUP 2006], p.64
A Reaction
This seems to parallel the iterative notion in set theory, that you must construct your hierarchy. All part of the general 'constructivist' approach to things. Is some kind of mad platonism the only alternative?
The
31 ideas
from JC Beall / G Restall
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10688
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'Equivocation' is when terms do not mean the same thing in premises and conclusion
[Beall/Restall]
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10690
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Formal logic is invariant under permutations, or devoid of content, or gives the norms for thought
[Beall/Restall]
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10695
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Logical consequence is either necessary truth preservation, or preservation based on interpretation
[Beall/Restall]
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10689
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A step is a 'material consequence' if we need contents as well as form
[Beall/Restall]
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10691
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Logical consequence needs either proofs, or absence of counterexamples
[Beall/Restall]
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10693
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Models are mathematical structures which interpret the non-logical primitives
[Beall/Restall]
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10692
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Hilbert proofs have simple rules and complex axioms, and natural deduction is the opposite
[Beall/Restall]
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10696
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A 'logical truth' (or 'tautology', or 'theorem') follows from empty premises
[Beall/Restall]
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13233
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Propositions commit to content, and not to any way of spelling it out
[Beall/Restall]
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13232
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Logic studies arguments, not formal languages; this involves interpretations
[Beall/Restall]
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13235
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Logic studies consequence; logical truths are consequences of everything, or nothing
[Beall/Restall]
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13234
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The view of logic as knowing a body of truths looks out-of-date
[Beall/Restall]
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13236
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Logical truth is much more important if mathematics rests on it, as logicism claims
[Beall/Restall]
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13237
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Preface Paradox affirms and denies the conjunction of propositions in the book
[Beall/Restall]
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13239
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Judgement is always predicating a property of a subject
[Beall/Restall]
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13238
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Syllogisms are only logic when they use variables, and not concrete terms
[Beall/Restall]
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13240
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A sentence follows from others if they always model it
[Beall/Restall]
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13241
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The model theory of classical predicate logic is mathematics
[Beall/Restall]
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13243
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Excluded middle must be true for some situation, not for all situations
[Beall/Restall]
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13242
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It's 'relevantly' valid if all those situations make it true
[Beall/Restall]
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13244
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Relevant necessity is always true for some situation (not all situations)
[Beall/Restall]
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13245
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Relevant consequence says invalidity is the conclusion not being 'in' the premises
[Beall/Restall]
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13246
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Relevant logic does not abandon classical logic
[Beall/Restall]
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13247
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A truthmaker is an object which entails a sentence
[Beall/Restall]
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13248
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We can rest truth-conditions on situations, rather than on possible worlds
[Beall/Restall]
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13249
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(∀x)(A v B) |- (∀x)A v (∃x)B) is valid in classical logic but invalid intuitionistically
[Beall/Restall]
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13250
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Free logic terms aren't existential; classical is non-empty, with referring names
[Beall/Restall]
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13252
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Some truths have true negations
[Beall/Restall]
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13254
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A doesn't imply A - that would be circular
[Beall/Restall]
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13255
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Relevant logic may reject transitivity
[Beall/Restall]
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13253
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There are several different consequence relations
[Beall/Restall]
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