Exercises for the Reader

1. Construct two distinct parts which share two identical states.

2. Can two atomic parts share a state?

3. Construct two parts x and y, and a relation R on the states of parts such that

   R_x y iff R_y x iff x R y.

4. Show that if x and y are parts and Z is a state of affairs, then there is only one R such that x R y is a configuration of Z.

5. An object x is said to discriminate the pair of World states s₁ and s₂  if its state is different in the two World states, i. e. x( s₁)   is unequal to x( s₂) . Suppose that for a given pair of objects x  and y, every pair of World states is discriminated by x  or by y  (or by both.) Show that the join of the two objects x + y  is the maximal partition on World states I, i .e., no smaller object than the entire World contains both x  and y.

6. (Advanced but easy) Show that objects (parts), as defined in mathematical holism, cannot be individuals in the sense of Goodman's nominalist "calculus of individuals". (Hint: Goodman's axioms, in Structure of Appearance, particularly 2.43 , imply that the collection of individuals, partially ordered by the "is part of" relation and completed with an "empty" or least individual which is part of every individual , form a Boolean algebra.)

7. Philosophers sometimes use very small worlds as illustrative examples -- "imagine a world consisting of a black square and a white square." (See, for example, Hochberg, H., "Things and Descriptions" in Logic, Ontology, and Language, Philosophia Verlag: 1984). Show that a world with exactly two distinct parts is impossible, so there can be no world state consisting of just 'a black square and a white square'.
   
   Sometimes the philosopher's very small world contains not just two things, but two identical things (as for example in Black, M., "The identity of indiscernables", Universals and Particulars: readings in ontology, U. of Notre Dame: 1952.) We know from the previous exercise that this is impossible, but what about the weaker case -- a world with three objects, of which just two are in some suitable sense "identical"?

8. Could there be two practically identical objects -- two distinct objects a and b which share all their contingent properties, while retaining only their "quiddities", the (non-contingent) properties of being a and b, respectively? Show that if we mean by "share" that the each object's contingent properties are predicable of the other, then this is impossible. (Refer to the section Predication for definitions of contingent property and predicability.)

9. The Tractatus  (5.5262) asserts that if an elementary proposition is true, then there is at least one more  elementary proposition that is true; i. e., no elementary proposition can be incompatible with every other elementary proposition. Prove this, assuming that the World has at least two states.

10. Show that if the World has more than 2 states and T is not the identity, then some atom is non-deterministic.

11. Show that if X is a class of objects and every object in X defines the same B-type, then X = {x} for some object x.

12. Show that if A is an ESA and the sense of "P x" = A, then there is a World state s such that one of x's states is {s} and all the other states of x are elements of P.