more on this theme     |     more from this text

Single Idea 18837

[filed under theme 4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / b. Axiom of Extensionality I ]

Full Idea

Vagueness in respect of membership is consistent with determinacy of the set's identity, so long as a set's identity is taken to consist, not in its having such-and-such members, but in its being the extension of the concept A.

Gist of Idea

A set can be determinate, because of its concept, and still have vague membership

Source

Ian Rumfitt (The Boundary Stones of Thought [2015], 8.4)

Book Ref

Rumfitt,Ian: 'The Boundary Stones of Thought' [OUP 2015], p.241

A Reaction

To be determinate, it must be presumed that there is some test which will decide what falls under the concept. The rule can say 'if it is vague, reject it' or 'if it is vague, accept it'. Without one of those, how could the set have a clear identity?

The 56 ideas from Ian Rumfitt

11210 Standardly 'and' and 'but' are held to have the same sense by having the same truth table [Rumfitt]
11211 If a sound conclusion comes from two errors that cancel out, the path of the argument must matter [Rumfitt]
11212 The sense of a connective comes from primitively obvious rules of inference [Rumfitt]
11214 We learn 'not' along with affirmation, by learning to either affirm or deny a sentence [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]
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]
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]
18807 Monotonicity means there is a guarantee, rather than mere inductive support [Rumfitt]
18808 Normal deduction presupposes the Cut Law [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]
18813 Logical consequence is a relation that can extended into further statements [Rumfitt]
18814 'Absolute necessity' would have to rest on S5 [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]
9390 Logic guides thinking, but it isn't a substitute for it [Rumfitt]
9389 Vague membership of sets is possible if the set is defined by its concept, not its members [Rumfitt]
17461 Some 'how many?' answers are not predications of a concept, like 'how many gallons?' [Rumfitt]
17462 A single object must not be counted twice, which needs knowledge of distinctness (negative identity) [Rumfitt]
14532 A distinctive type of necessity is found in logical consequence [Rumfitt, by Hale/Hoffmann,A]
12193 Logical necessity is when 'necessarily A' implies 'not-A is contradictory' [Rumfitt]
12194 Contradictions include 'This is red and not coloured', as well as the formal 'B and not-B' [Rumfitt]
12195 Soundness in argument varies with context, and may be achieved very informally indeed [Rumfitt]
12198 Geometrical axioms in logic are nowadays replaced by inference rules (which imply the logical truths) [Rumfitt]
12199 There is a modal element in consequence, in assessing reasoning from suppositions [Rumfitt]
12201 We reject deductions by bad consequence, so logical consequence can't be deduction [Rumfitt]
12200 A logically necessary statement need not be a priori, as it could be unknowable [Rumfitt]
12202 Narrow non-modal logical necessity may be metaphysical, but real logical necessity is not [Rumfitt]
12203 If a world is a fully determinate way things could have been, can anyone consider such a thing? [Rumfitt]
12204 The logic of metaphysical necessity is S5 [Rumfitt]