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Single Idea 18800

[filed under theme 5. Theory of Logic / H. Proof Systems / 4. Natural Deduction ]

Full Idea

'Introduction rules' state the conditions under which one may deduce a conclusion whose dominant logical operator is the connective. 'Elimination rules' state what may be deduced from some premises, where the major premise is dominated by the connective.

Gist of Idea

Introduction rules give deduction conditions, and Elimination says what can be deduced

Source

Ian Rumfitt (The Boundary Stones of Thought [2015], 1.1)

Book Ref

Rumfitt,Ian: 'The Boundary Stones of Thought' [OUP 2015], p.4

A Reaction

So Introduction gives conditions for deduction, and Elimination says what can actually be deduced. If my magic wand can turn you into a frog (introduction), and so I turn you into a frog, how does that 'eliminate' the wand?

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]
18799 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]
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]
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 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]