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Single Idea 8708

[filed under theme 4. Formal Logic / E. Nonclassical Logics / 2. Intuitionist Logic ]

Full Idea

In intuitionist logic, if we do not know that we do not know A, it does not follow that we know A, so the inference (and, in general, double negation elimination) is not intuitionistically valid.

Gist of Idea

Double negation elimination is not valid in intuitionist logic

Source

Michèle Friend (Introducing the Philosophy of Mathematics [2007], 5.2)

Book Ref

Friend,Michèle: 'Introducing the Philosophy of Mathematics' [Acumen 2007], p.107

A Reaction

That inference had better not be valid in any logic! I am unaware of not knowing the birthday of someone I have never heard of. Propositional attitudes such as 'know' are notoriously difficult to explain in formal logic.

The 15 ideas with the same theme [logic which uses 'provable' in place of 'true']:

18832 Mathematical statements and entities that result from an infinite process must lack a truth-value [Dummett]
18073 Dummett says classical logic rests on meaning as truth, while intuitionist logic rests on assertability [Dummett, by Kitcher]
18122 Classical interdefinitions of logical constants and quantifiers is impossible in intuitionism [Bostock]
18074 Intuitionists rely on assertability instead of truth, but assertability relies on truth [Kitcher]
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]
13715 You can employ intuitionist logic without intuitionism about mathematics [Sider]
18789 Intuitionist logic looks best as natural deduction [Mares]
18790 Intuitionism as natural deduction has no rule for negation [Mares]
13249 (∀x)(A v B) |- (∀x)A v (∃x)B) is valid in classical logic but invalid intuitionistically [Beall/Restall]
8708 Double negation elimination is not valid in intuitionist logic [Friend]
17925 Showing a disproof is impossible is not a proof, so don't eliminate double negation [Colyvan]
17926 Rejecting double negation elimination undermines reductio proofs [Colyvan]
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]