Ideas of Harold Noonan, by Theme

[British, fl. 1989, At Birmingham University.]

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6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / c. Counting procedure
It is controversial whether only 'numerical identity' allows two things to be counted as one
     Full Idea: 'Numerical identity' implies the controversial view that it is the only identity relation in accordance with which we can properly count (or number) things: x and y are to be properly counted as one just in case they are numerically identical.
     From: Harold Noonan (Identity [2009], 1)
     A reaction: Noonan cites Geach, presumably to remind us of relative identity, where two things may be one or two, depending on what they are relative to. The one 'guard on the gate' may actually be two men.
9. Objects / E. Objects over Time / 4. Four-Dimensionalism
I could have died at five, but the summation of my adult stages could not
     Full Idea: Persons have different modal properties from the summations of person-stages. I might have died when I was five. But the maximal summation of person-stages which perdurantists say is me could not have had a temporal extent of a mere five years.
     From: Harold Noonan (Identity [2009], 5)
     A reaction: Thus the summation of stages seems to fail Leibniz's Law, since truths about a part are not true of the whole. But my foot might be amputated without me being amputated. The objection is the fallacy of composition?
9. Objects / E. Objects over Time / 5. Temporal Parts
Stage theorists accept four-dimensionalism, but call each stage a whole object
     Full Idea: Stage theorists, accepting the ontology of perdurance, modify the semantics to secure the result that fatness is a property of a cat. Every temporal part of a cat (such as Tabby-on-Monday) is a cat. (but they pay a price over the counting of cats).
     From: Harold Noonan (Identity [2009], 5)
     A reaction: [Noonan cites Hawley and Sider for this view. The final parenthesis compresses Noonan] I would take the difficulty over counting cats to be fatal to the view. It produces too many cats, or too few, or denies counting altogether.
9. Objects / F. Identity among Objects / 2. Defining Identity
Problems about identity can't even be formulated without the concept of identity
     Full Idea: If identity is problematic, it is difficult to see how the problem could be resolved, since it is difficult to see how a thinker could have the conceptual resources with which to explain the concept of identity whilst lacking that concept itself.
     From: Harold Noonan (Identity [2009], 1)
     A reaction: I don't think I accept this. We can comprehend the idea of a mind that didn't think in terms of identities (at least for objects). I suppose any relation of a mind to the world has to distinguish things in some way. Does the Parmenidean One have identity?
Identity definitions (such as self-identity, or the smallest equivalence relation) are usually circular
     Full Idea: Identity can be circularly defined, as 'the relation everything has to itself and to nothing else', or as 'the smallest equivalence relation'.
     From: Harold Noonan (Identity [2009], 2)
     A reaction: The first one is circular because 'nothing else' implies identity. The second is circular because it has to quantify over all equivalence relations. (So says Noonan).
Identity is usually defined as the equivalence relation satisfying Leibniz's Law
     Full Idea: Numerical identity is usually defined as the equivalence relation (or: the reflexive relation) satisfying Leibniz's Law, the indiscernibility of identicals, where everything true of x is true of y.
     From: Harold Noonan (Identity [2009], 2)
     A reaction: Noonan says this must include 'is identical to x' among the truths, and so is circular
Identity can only be characterised in a second-order language
     Full Idea: There is no condition in a first-order language for a predicate to express identity, rather than indiscernibility within the resources of the language. Leibniz's Law is statable in a second-order language, so identity can be uniquely characterised.
     From: Harold Noonan (Identity [2009], 2)
     A reaction: The point is that first-order languages only refer to all objects, but you need to refer to all properties to include Leibniz's Law. Quine's 'Identity, Ostension and Hypostasis' is the source of this idea.
9. Objects / F. Identity among Objects / 8. Leibniz's Law
Indiscernibility is basic to our understanding of identity and distinctness
     Full Idea: Leibniz's Law (the indiscernibility of identicals) appears to be crucial to our understanding of identity, and, more particularly, to our understanding of distinctness.
     From: Harold Noonan (Identity [2009], 2)
     A reaction: True, but indiscernibility concerns the epistemology, and identity concerns the ontology.
Leibniz's Law must be kept separate from the substitutivity principle
     Full Idea: Leibniz's Law must be clearly distinguished from the substitutivity principle, that if 'a' and 'b' are codesignators they are substitutable salva veritate.
     From: Harold Noonan (Identity [2009], 2)
     A reaction: He gives a bunch of well-known problem cases for substitutivity. The Morning Star, Giorgione, and the number of planets won't work. Belief contexts, or facts about spelling, may not be substitutable.