1. 1 LMPS 2007, Beijing INDIVIDUALS IN BRANCHING THEORIES Tomasz Placek Department of Philosophy, Jagiellonian University Kraków, Poland Tomasz.Placek@uj.edu.pl

2. 2 Individuals and their modal aspects […W]e insisted that real things have a temporal dimension: they are not mere slices. I think that real things also have a modal dimension. A real thing would be less real if its possibilities were not included. A chair is what it is partly because it could not possibly be a desk. […] I claim that for something to be real, it must have a modal „history” as well as a temporal one. Garson (2006)

3. 3 Objectives to address Lewis’s objection to branching individuals to discuss some properties of branching individuals

4. 4 Controversy Possible worlds, histories, scenarios…. Do they overlap or not? Do they branch or diverge ?

5. 5 Lewis: worlds do not overlap, though some worlds diverge Worlds w and w’ diverge if they have initial segments i and i’, resp, such that i and i’ are duplicates, yet, their complements, w/i and w’/i’ are not duplicates Result: a real individual lives in the actual world; its counterparts live in other possible worlds.

6. 6 Pictures

7. 7 Branching: every two histories share an initial segment

8. 8 David Lewis' argument against branching [Hubert Humphrey] could have had six fingers on his left hand. There is some other world that so represents him. [...] So, Humphrey, who is a part of this world and here has five fingers on the left hand, is also a part of some other world and there has six fingers on his left hand. Qua part of this world he has five fingers; qua part of that world he has six. He himself [...] has five fingers on the left hand, and has not five, but six. How can this be? (On the Plurality..)

9. 9 Hubert Humphrey (1911-1978) U.S. vice-president 1965-1969

10. 10 real-possibility vs. conceivability At same stage of Humphrey’s development it was really possible that six fingers would grow on his left hand

11. 11 Branching individuals Idealizations: individuals are spatially infinitely thin (point-like) no attempt to give a sufficient condition for individuals

12. 12 Branching individuals If A is an individual in a branching model W =  W, ≤ >, then A  W and if x, y  A then there is in A an interval I such that x, y  I (interval is a dense chain), and for every two maximal chains in A, there is no history containing them both, where W is a model of branching-time (BT) or branching space-times (BST), and histories are defined in BT as maximal chains in W, and in BST as maximal upward directed subsets of W, Maximal chains in A represent A’s alternative lives

13. 13 Some optional postulates: A must have originated sometime: every maximal chain in A is lower bounded. A will have to die sometime: every maximal chain in A is upper bounded. A could not have originated differently (Kripkean intuition): A has a single minimal element A could not have originated at a different time / different space-time point: definable in BT +Instants and BST+Space-Time Points, resp.

14. 14 Branching individuals in BT

15. 15 Branching individuals in BT an individual is (spatially) as thick as a history

16. 16 Branching individuals in BST

17. 17 Branching individuals in BST: some properties (1) For every maximal chain c in A, if c is lower bounded, then it has an infimum (proper or improper) Comment: if A comes to being in history h, then there is in h either the first event in its life, or the last event before it comes to being. For every maximal chain c in A, if c is upper bounded, then in every history h such that c  h, c has a supremum. Comment: tricky, allows for a chancy factor responsible for the individual’s dying.

18. 18 Branching individuals in BST: some properties (2) If c and c’ are two maximal chains in A and c  h and c’  h’, then there is a maximal element e in the intersection h ∩h’ such that e < c/h’. Comment: a possible cause of why A continued to live in history h rather than h’ is located earlier than that part of A’s life which takes place in h but not in h’. e is cause-like wrt the choice of c rather than c’. Beware: although e < c/h’ is true, e < c’/h might be false. Belnap’s cause-like-locus-not-in-the-past. Equivalent to modal funny business, see Belnap (2002 and 2003).

19. 19 Branching individuals in BST: some properties (3) Let e be cause-like wrt the choice of c rather than c’, where c and c’ are maximal chains in A. Then two cases: (1) e  A: the choice is up to A (2) e  A: the choice is not made by A

20. 20 Back to Lewis’s objection: propositions Is there a history in which both the propositions, i.e., that HH has 5 fingers, and that HH has 6 fingers on his left hand, are true? Occurence proposition that event E occurs: the set H E of histories that contain E Proposition H E is true in history h iff h  H E H 5 and H 6 are incompatible, i.e., H 5  H 6 = 0. Hence, there is no history in which both H 5 and H 6 are true.

21. 21 Back to Lewis’s objection: sentences Is there in a BT semantic model a point of evaluation e/h such that the two sentences: p 5  HH has (now) 5 fingers on his left hand, and p 6  HH has (now) 6 fingers on his left hand, are true at e/h? In constructing a BT semantical model, we will see to it that there is no such a point in the model.