Causality and Axiomatic Probability Calculus

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Causality and Axiomatic Probability Calculus

• Andrea LEpiscopo

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

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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

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• 1. Abstract probability and practical possibility

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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).
• .

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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).

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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.

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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.

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

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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.

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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).

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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.

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• 3. Problems for empirical and conceptual analysis
  

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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.

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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.

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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.

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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).

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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.

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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.

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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.

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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.

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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.

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• 4. Conclusions

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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.

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Causality and Axiomatic Probability Calculus

• Andrea LEpiscopo

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