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Causality and Axiomatic Probability Calculus Andrea L’Episcopo PhD at University of Catania - andrea.lepiscopo@libero.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 1.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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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). Practical possibility

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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. Kolmogorov’s 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 Dowe’s can’t. 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 L’Episcopo PhD at University of Catania - andrea.lepiscopo@libero.it

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