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LMPS 2007, Beijing
INDIVIDUALS IN BRANCHING THEORIES
Tomasz Placek
Department of Philosophy, Jagiellonian UniversityKraków, Poland
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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)
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Objectives
• to address Lewis’s objection to branching individuals
• to discuss some properties of branching individuals
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Controversy
• Possible worlds, histories, scenarios….
• Do they overlap or not?
• Do they branch or diverge ?
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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.
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Pictures
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Branching: every two histories share an initial segment
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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 ..)
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Hubert Humphrey (1911-1978)
U.S. vice-president 1965-1969
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real-possibility vs. conceivability
At same stage of Humphrey’s development it was really possible that six fingers would grow on his left hand
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Branching individuals
• Idealizations: • individuals are spatially infinitely thin (point-like)• no attempt to give a sufficient condition for individuals
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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
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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.
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Branching individuals in BT
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Branching individuals in BT
an individual is (spatially) as thick as a history
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Branching individuals in BST
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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.
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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).
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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
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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 HE of histories that contain E
Proposition HE is true in history h iff h HE
H5 and H6 are incompatible, i.e., H5 H6 = 0. Hence, there is no history in which both H5 and H6 are true.
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Back to Lewis’s objection: sentences
Is there in a BT semantic model a point of evaluation e/h such that the two sentences:
p5 HH has (now) 5 fingers on his left hand, and
p6 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.
