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Title: Causality and Axiomatic Probability Calculus


1
Causality and Axiomatic Probability Calculus
  • Andrea LEpiscopo

PhD at University of Catania - andrea.lepiscopo_at_li
bero.it
2
Abstract
  • Theses
  • There is one intuitive notion of causality, but
    there can be highly specialized versions of it,
    such as the physical notion of causality
  • Probabilistic causality is a conceptual analysis
    of a constrained version of the intuitive notion
    of causality.
  • Aims
  • To illustrate some relevant features of
    axiomatic probability calculus
  • To distinguish between the conceptual analysis
    of the intuitive notion of causality, and the
    empirical analysis of the physical notion of
    causality
  • To re-examine some criticisms against empirical
    and conceptual analyses of causality, in light of
    the two points above
  • To propose a pluralistic and pragmatic approach
    to causality.

3
Index
  • Abstract probability and practical possibility
  • 2. Conceptual/empirical analysis of
    intuitive/physical notion of causality
  • 3. Problems for empirical and conceptual
    analysis
  • 4. Conclusions

4
  • 1. Abstract probability and practical possibility

5
Abstract probability
  • Probability calculus exhibits important features
    proper to formal theories
  • its starting points are undefined primitive
    terms and axioms, which are the statements in
    which these terms are involved
  • the symbols in the theory do not stand for
    objects, they just are signs to be manipulated
    according to theory-specific rules
  • more generally, ...in the formal approach
    there seems to be no appeal to intuition, because
    definitions, axioms and rules of transformation
    are clearly laid out from the beginning, and the
    proof produced appeals only to the meaning of the
    axioms, definitions and rules of transformation
    (G. Oliveri, Do We Really Need Axioms in
    Mathematics?, in C. Cellucci and D. Gillies
    (eds.), Mathematical Reasoning and Heuristics,
    King's College Publications, London 2005, p.
    122).
  • .

6
Practical possibility
  • The definition of a probability field in
    empirical applications presupposes qualitative
    judgements about a priori possibility of events
    and their variants.
  • In order to define a field of probability, we
    have in fact first of all to form the set E,
    which includes ...all the variants which we
    regard a priori as possible (A. N. Kolmogorov,
    Foundations of Probability, Chelsea Publishing
    Company, New York 1950, p. 11. FoP, from now on).

7
Probability and frequencies
  • In applications to experimental data, the
    approximate equality between P (A) and m/n is
    a practical certainty, and it is at least
    possible that it is not the case ...that in a
    very large number of series of n tests each, in
    each the ratio m/n will differ only slightly from
    P (A) (FoP, p. 5).
  • Probability theory deals with the logical
    notion of possibility, whereas frequencies are
    concerned with contingent possibilities.
    Logically, possibility is a redundant attribute
    of all that is not a contradiction contingent
    possibilities are instead a seemingly irreducible
    characteristic of actual events, and they render
    very hard even the task of identifying sets of
    possible events.

8
Probability calculus and experimental data
  • To an impossible event (an empty set)
    corresponds, in accordance with our axioms, the
    probability P (0) 0, but the converse is not
    true P (A) 0 does not imply the impossibility
    of A ( FoP, p. 5).
  • Zero-probability events are then practically
    impossible only a posteriori.
  • Kolmogorovs definition of conditional
    probability requires that the probability value
    of the event on which to condition has to be more
    than zero.
  • In light of what precedes, such a requirement is
    tenable only with respect to the logical notion
    of impossibility when we instead are dealing
    with experimental data, the probability of an
    event equals zero only a posteriori, and the fact
    that the repetitions of the conditions have shown
    such an event to be practically impossible is
    presumably meaningful with respect to other
    events in the field of probability.
  • Probability calculus is an abstract theory, and
    it works properly only when it deals with the
    abstract basic elements it has been built on.

9
  • 2. Conceptual/empirical analysis of
    intuitive/physical notion of causality

10
The intuitive and the physical notion of
causality
  • The intuitive notion of causality applies to a
    wide variety of objects it is a representational
    tool and it deals with causality as a semantical
    matter. It can be either subjective or objective.
  • While being a special case of the intuitive
    notion of causality, the physical notion of
    causality only applies to objects from the world
    of physics. It deals with causality as a matter
    of fact. It is an explanatory and predictive
    tool. It can only be objective.

11
Empirical and conceptual analyses of causality
  • Conceptual analysis is not just dictionary
    writing. It is concerned to spell out the logical
    consequences and to propose a plausible and
    illuminating explication of the concept. Here,
    logical coherence and philosophical plausibility
    will also count. The analysis is a priori, and if
    true, will be necessary true.
  • ...empirical analysis seeks to establish what
    causality in fact is in the actual world.
    Empirical analysis aims to map the objective
    world, not our concepts. Such an analysis can
    only proceed a posteriori
  • (P. Dowe, Physical Causation, Cambridge
    University Press, New York, 2000, pp. 2-3. PC,
    from now on).

12
Empirical and conceptual analyses of causality
  • The conceptual analysis applies to the intuitive
    notion of causality, while the empirical analysis
    applies to the physical one.
  • The application of probability calculus to the
    physical notion of causality is problematic
    anyway, probability calculus cannot be an
    analysis of such a notion of causality.
  • Probability calculus can be a conceptual
    analysis, but of a notion of causality which is a
    probabilistic constrained version of the
    intuitive one.

13
  • 3. Problems for empirical and conceptual analysis

14
The CQ theory of causality
  • In PC, the declared goal is to formulate an
    empirical analysis of causality, the CQ theory of
    causality
  • CQ1. A causal process is a world line of an
    object that possesses a conserved quantity.
  • CQ2. A causal interaction is an intersection of
    world lines that involves exchange of a conserved
    quantity.

15
The CQ theory of causality
  • The CQ theory of causality is empirical,
    contingent with respect to the identity of causal
    processes, and particularist. It does not take
    position with respect to the direction of
    causality, and it is noncommittal with respect to
    probabilistic causality. Conserved quantities
    being the quantities typically associated with
    causality is claimed to be just a plausible
    conjecture.

16
Causation
  • Causation, as defined by Dowe, is causation by
    prevention and/or omission A causes not-B not-A
    causes B, respectively.
  • Dowe develops a counterfactual theory to deal
    with causation, because obviously no set of
    causal processes and interactions can link A to
    not-B, or not-A to B, and so the CQ theory cannot
    handle causation.

17
Counterfactual theory of causation and CQ
  • The counterfactual theory of causation cannot
    be seen as an extension of the CQ theory it is a
    conceptual analysis of the intuitive notion of
    causality, and so it diverges from the CQ theory
  • Dowe describes his account of causation as a
    cross-platform solution in that virtually any
    account of causation can be plugged in. But
    Dowes cant. Since Dowe has only offered a
    contingent specification of how causation
    operates in the actual world, he has yet to say
    how causation operates in those nonactual worlds
    that his counterfactual take us to (here a
    conceptual analysis is needed) (Schaffer J.,
    Phil Dowe. Physical Causation, Review article,
    Brit. J. Phil. Sci., 2001, n. 52, pp. 809-813).

18
Causation is not causation
  • No event such as not-A can be involved in causal
    processes and interactions, as they occur in
    physical world then no physical notion of
    causation is conceivable, and an empirical
    analysis of the physical notion of causality,
    i.e. the CQ theory, must deny that causation has
    to be seen as causation.

19
Against probabilistic causality
  • The existence of a probabilistic relation between
    two events is not a necessary condition for
    singular causation between those events
  • Chance-lowering causation.

20
Probabilistic singular causation
  • If probability calculus is applied to empirical
    data, then probabilistic relations are de facto
    quantitative relations between actual events, and
    so they hold only a posteriori.
  • The existence of such probabilistic relations
    cannot be a necessary condition for singular
    causation, especially in the absence of a widely
    accepted objective interpretation for single-case
    probabilities.
  • The impossibility, by probabilistic causality,
    to provide a necessary condition for singular
    causation, is then not an objection against
    probabilistic causality its target is instead
    the expectation of probabilistic causality being
    well suited even for singular causation.

21
Chance-lowering and PSR
  • PSR theories propose a conceptual analysis of
    the intuitive notion of causality. Given this
    conceptual analysis, PSR theories of causality
    rule out chance-lowering causation by the same
    definition of causation, quite similarly to what
    the probability calculus does with conditional
    probabilities with zero-probability antecedents.
  • Excluding by definition some features of its
    object, being it causality or probability, can
    surely be a drawback of a theory, but if we claim
    it is, we have to say why it is so in some but
    not in all cases.

22
Physical causation and conceptual analysis
  • Lewis and Menzies chains are proposals
    intended to handle cases of chance-lowering
    causation. In PC, Dowe contends such proposals
    are successful by means of a decay example.
  • Probabilistic theories of causality are a
    particular kind of conceptual analysis of the
    intuitive notion of causality, so they are not
    always able to deal with causation as it takes
    place within the world of physics, particularly
    when such cases of causation are too far from the
    intuitive notion of causality.
  • Cases like the decay example are not
    counterexamples against probabilistic causality.
    They have instead to be directed against the
    pretension that probabilistic theories of
    causality can provide even an empirical analysis
    of the physical notion of causality.

23
  • 4. Conclusions

24
Conclusions
  • Given what precedes, with respect to the
    possibility to formalize causality, we think it
    would be better
  • to assume a pluralistic stance, driven by
    pragmatic considerations
  • to make reference, case by case, to one of the
    many theories which formalize the many aspects of
    causality
  • to exploit the intuitive notion of causality, as
    an heuristics, in building models representing
    actual causal processes.

25
Causality and Axiomatic Probability Calculus
  • Andrea LEpiscopo

PhD at University of Catania - andrea.lepiscopo_at_li
bero.it
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