The Kolmogorov probability axioms and objective chance

Philosophy of Probability and Statistical Modelling by Mauricio Suárez [1] provides a historical overview of the philosophy of probability argues for the significance of this philosophy for well founded statistical modelling.  Within this wider scope, the book discusses, the themes: objective probability, propensities, and measurements. So, it has much in common with the themes of this blog.  Suarez defends a model of objective probability that disentangles propensity from single-case chance and the observed sequences of outputs that result in relative frequencies of outcomes.  This is close to what I argued for in Potentiality and probability, but there are differences that I will return to them in a future post.

Suarez's main theme is objective probability, but he expends a lot of effort on examining subjective probabilities. I have no doubt that subjective probability has its place alongside objective probability but ad hoc mixture of the two is to be avoided. Subjective probability has been zealously defended by de Finetti and Jaynes to the point of them attempting to eliminate objective probability altogether. I believe, however, the posts in this blog make a case for objective probability as an aspect of ontology. In this post I will focus on the Kolmogorov axioms of probability. Curiously, it is only in examining subjective probability that Suarez discusses the axioms of probability.

The axioms that Suarez states are as follows.

Let \(\{E_1 , E_2, …, E_n\}\) be the set of events over which an agent´s degrees of belief range; and let \(\Omega\) be an event which occurs necessarily. The axioms of probability may be expressed as follows:

Axiom 1: \(0 \le P (E) \le 1\), for any \(P (E)\): In other words, all probabilities lie in the real unit number interval.

Axiom 2: \(P (\Omega) =1\): The tautologous proposition, or the necessary event has probability one.

Axiom 3: If \(\{E_1, E_2, …, E_n\}\) are exhaustive and exclusive events, then \(P (E_1) + P(E_2) + … + P(E_n) = P (\Omega) = 1\): This is known as the addition law and is sometimes expressed equivalently as follows: If \(\{E_1, E_2, …, E_n \}\) is a set of exclusive (but not necessarily exhaustive) events then: \(P (E_1 \vee E_2 \vee … E_n) = P (E_1) + P(E_2) + … + P(E_n)\).

Axiom 4: \(P (E_1 \& E_2) = P (E_1 | E_2) P (E_2)\). This is sometimes known as the multiplication axiom, the axiom of conditional probability, or the ratio analysis of conditional probability since it expresses the conditional probability of \(E_1\) given \(E_2\).

According to Suarez, the Kolmogorov axioms are essentially equivalent to those above. The axioms that Kolmogorov published it in 1933 have become the standard formulation. The axioms themself form a short passage near the start of the book. 

 Nathan Morrison has translated the Kolmogorov axioms [2] as:

Let \(E\) be a collection of elements ξ, η, ζ, ..., which we shall call elementary events, and \(\mathfrak{F}\) a set of subsets of \(E\); the elements of the set \(\mathfrak{F}\) will be called random events. 

I. \(\mathfrak{F}\) is a field of sets. 

II. \(\mathfrak{F}\) contains the set \(E\).

III. To each set \(A\) in \(\mathfrak{F}\) is assigned a non-negative real number \(P(A)\). This number \(P(A)\) is called the probability of the event \(A\).

IV. \(P(E)\) equals \(1\).  

V. If \(A\) and \(B\) have no element in common, then 

$P(A+B) =P(A)+P(B)$

A system of sets \(\mathfrak{F}\), together with a definite assignment of numbers \(P(A)\), satisfying Axioms I-V, is called a field of probability. 

Terminology has moved on and it would now be usual to identify the tiple \((E, \mathfrak{F}. P)\) as the probability space with the term field reserved for the Borel field of sets \(\mathfrak{F}\). In addition, it is preferable not to use the same symbol for the addition of numbers and the union of sets. It is also important to note that ξ, η, ζ, ..., indicating the elements of \(E\) does not constrain the set of elementary events to be a countable set. This is important for applications to statistical and quantum physics where, for example, particle positions position and momentum can take a continuum of values. 

It is a shorthand when Kolmogorov writes in Axiom III that the number \(P(A)\) is the probability of the event \(A\). The full interpretation is that \(P(A)\) is the probability that the outcome will be an element in \(A\). This does seem to be the standard, if often implicit, understanding of the situation. In the same Axiom III, the use of the term "assigned" is also deceptive. The probabilities are more properly assigned to the singleton set with each element of \(E\) as the sole member and from that the probability for each set in\(\mathfrak{F}\) is constructed.

The salient differences between the axioms provided by Suarez and those of Kolmogorov are:

• Suarez provides axioms only for a discrete finite set of events
• What he calls events are, within the restriction in the bullet above are Kolmogorovs elementary events.
• There is therefore no need to introduce the field of sets
• Apart from event, Suarez uses the language and symbols of propositional logic  
• As he is dealing with probabilities as credences (degrees of belief) in this section of his book it would have been better to consistently employ the language and symbols of propositional logic.
• This would give a mapping

                                Logical                                    Set theoretical

                    elementary propositions                       elementary events

                    Logical 'and', \(\land\) or \(\&\)          Set intersection \(\cap\)

                      Logical 'or' \(\vee\)                           Set union \(\cup\) 

                    Tautology \(\Omega\)                        Set of elementary events

• Axiom 1 should read:  \(0 \le P (E) \le 1\), for any \(E\). This axiom is not one of Kolmogorov's but can be derives from them.
• Caution: in the Suarez version \(E\) is an arbitrary 'event' whereas in Kolmogorov this symbol is used for the set of elementary events.
• Kolmogorov has no equivalent to Axiom 4 in his list but introduces the equivalent formula for conditional probability later, as a definition. I think it is better to list it as one of the axioms.

As a formal structure, it would have been even better if Kolmogorov had used rigorously the language of sets and not used a term like 'event'. A set theoretic formulation with potential possibilities and numerical probability assignment would then be:

Let \(E\) be a collection of elements ξ, η, ζ, ..., which we shall call elementary possibilities and \(\mathfrak{F}\) a set of subsets of \(E\). 

I. \(\mathfrak{F}\) is a field of sets. 

II. \(\mathfrak{F}\) contains the set \(E\).

III. To each set \(A\) in \(\mathfrak{F}\) is assigned a non-negative real number \(P(A)\). This number \(P(A)\) is called the probability of \(A\), an element of (\mathfrak{F}\).

IV. \(P(E)\) equals \(1\).  

V. If \(A\) and \(B\) in \(\mathfrak{F}\) have no element in common, then 

$P(A \cup B) =P(A)+P(B)$

                    VI. For any \(A\) and \(B\) in \(\mathfrak{F}\) 

                                    \(P (A | B) = \frac{P (A \cap B)}{P (B)}\), 

                          is the conditional probability

Such a system of sets \(\mathfrak{F}\), \(E\), together with a definite assignment of numbers \(P(A)\) for all \(A \in \mathfrak{F}\), satisfying Axioms I-V, is called a probability space.

I propose that this is a neutral set of axioms for a probability theory. It has one advantage that it can be applied to eventless situations such as in the quatum description of a free particle. The formualtion is based on the mathematics of measure theory plus a numerical probability assignment and the identification of a set of possibilities. I have added, as is often done, the definition of conditional probability as an axiom.  It is possible to map this formulation to one in terms of propositions and logical operators. This move would not restrict application and interpretation to subjective probability. That would be governed by the meaning given to the numerical probability assignment. The axiom system considered here take the numerical probability assignment as fundamental and the conditional probability is then added. It is also possible to take conditional probability as fundamental, as done by Renyi. I will discuss this formulation in a future post.

In adopting these axioms for objective chance, the collection of elements can again be called elementary events or possible events, and \(P\) is a numerical assignment furnished by a theory or estimated through experiment. The elements of (\mathfrak{F}\) are not, in general, outcomes in the way that the elements of \(E\) are. An element of (\mathfrak{F}\) is a set of some elements of \(E\). That this can useful can be made clear through two examples.

A simple physical example is provided by a die with six sides and in normal game playing circumstances this provides six possibilities for which face will face upwards when thrown. These six possibilities are the elementary events \(E\) in the probability space for a dice throwing game with one die. I will also call these elementary events outcomes. These outcomes are not in \(\mathfrak{F}\), the set of subsets of \(E\) that Kolmogorov calls events. For example, the outcome "face 5 faces upwards" is not in set of random events but the subset {"face 5 faces upwards"} is. On one throw of the die we can only obtain an element of \(E\) so what are the random events \(\mathfrak{F}\)? Consider \(P(E)\) that is equal to one. It is usual to interpret this as saying that \(E\) is the sure event. But in one throw of the die we will never get \(E\) but only one element of it. What is therefore sure is that any outcome will be in \(E\).  

So far, I have said little about \(P\) itself.  In the die example \(P(\){"face 5 faces upwards"}\()\)  can be estimated as the proportion of times it occurs in a long run of repetitions. For anyone familiar with the relative frequency interpretation of probability, I emphasise that this in not such an interpretation. Here the relative frequencies are an estimate of the numerical probability assignment. \(P(\){"face 5 faces upwards"}\()\) itself is the relative strength of the tendency for "face 5 faces upwards" to occur, otherwise known as the single-case chance for that event. If we consider and elements of \(\mathfrak{F}\) such that \(A = \{\)"face 1 faces upwards", "face 3 faces upwards", "face 5 faces upwards"\(\}\) then \(A\) can be interpreted as "an odd valued face faces upwards".  This illustrates that even in the application to objective chance that it is difficult to avoid the use of propositions to give meaning to useful subsets of \(\mathfrak{F}\). 

A strength of the Kolmogorov axioms and my reformulation is the application to continuous infinite sets. An example is the case of an observation of an electron governed by the Schrödinger equation. According to quantum mechanics the probability of it being observed at any pre-designated spot is zero. In this example, all the elements of the set of elementary events have numerical probability assignment zero. This is where the field of events \(\mathfrak{F}\) is useful in providing sets with non-zero probability in which the position of the electron may be observed.

As discussed above, it can be convenient to use proposition to give meaning to relevant elements of \(\mathfrak{F}\) in a specific application. However, this need not introduce any subjectivity. The subjectivity enters through interpreting \(P\) as credence or degree of belief.  In applications with objective probability, \(P(A)\) is a numerical assignment of the strength of the single-chance tendency for an elementary event to appear in set \(A\).  In objective probability \(P\) is ontological but in subjective probability it is epistemic.

1. Mauricio Suárez, Philosophy of Probability and Statistical Modelling, Elements in the Philosophy of Science, Cambridge University Press, 2021
2. Kolmogorov, AN. (2018) Foundations of the Theory of Probability. [edition unavailable]. Dover Publications. Available at: https://www.perlego.com/book/823610/foundations-of-the-theory-of-probability-second-english-edition-pdf (Accessed: 18 December 2022).

Causation and chance in quantum mechanics

The competing and complementary concepts of causality within the dispositional or powers approach is already quite intricate, as indicated by a previous post.  There are also concepts of causality that do not follow a dispositional approach and are in fact better known.

For a concise and balanced overview of the status see Anjum and Mumford Causality, Oxford 2014. My previous post provided a commentary on Chapter 3 of "What Tends to Be the Philosophy of Dispositional Modality", which examined a dispositional ontological for objective probability. In Chapter 4 of the same book Anjum, Mumford and Andersen examine how a dispositional theory of causality stands up against the ontological challenges of quantum mechanics. 

 The understanding of causality that is still the most influential can be traced back to David Hume's sceptical analysis and Emmanuel Kant's response. In A Treatise of Human Nature (1973, Book I, Part III, Section VI) Hume argued that all that can be observed in nature is a series of events. One thing happens and then another, and then another, and so on. Whether any of those events are causally connected is not itself part of experience of events. For example, a match is struck and then almost immediately that same match lights, but what is not observed is that the striking of the match caused it to light. If there is cause and effect, then it is not known through direct observation of events. But what do our physical theories tells us? In Newtonian physics one thing happens followed by another but that following is determined by the laws of physics. The state of the world at one time determines the state of the world later. That this seems not to be the case in a world described by quantum mechanics leads to claims that causality no longer holds. In standard quantum mechanics a later state does follow deterministically from an earlier state but measurement breaks the causal link.

Heisenberg and Bohr (Walter Heitler and Léon Rosenfeld in the background), in the mid 1930s 

Now let's look at those features of classical causation that were explicitly discussed by Bohr and Heisenberg: necessitation, determinism, predictability, and separability.

Classical causation

Hume was sceptical about a necessary connection between cause and effect because none can be known from experience. However, the intuition that a cause necessitates an effect is strong and led Kant to formulate causal necessity as a precondition for science.  It may seem strange that the radical empiricist, Hume, is seen to be undermining the scientific enterprise but his analysis, according to Kant, denies the possibility of scientific explanation. Kant, in the Critique of Pure Reason, states:

[T]he very concept of a cause so obviously contains the concept of a necessity of connection with an effect and a strict universality of rule that it would be entirely lost if one sought, as Hume did, to derive it from a frequent association of that which happens with that which precedes and a habit (thus a merely subjective necessity) of connecting representations arising from that association.

Kant's influence was strong, especially in Germany, and Heisenberg felt compelled to respond given the empirical success of quantum mechanics. In the lectures "Physics and Philosophy" he uses the radioactive decay of the radium atom as an example in which there is no predictability of the timing of a decay event.  There is no decay event that follows through necessity from the prior state of the atom. There is no quantum law that necessarily connects the state of the radium atom with the decay event. However, quantum theory does predict the probability of a decay event per unit time. There is no determinacy. If necessity and determinism are characteristic of a cause, then causality does not hold in quantum mechanics. 

Heisenberg, in agreement with Bohr argues that causality is needed but only as a classical concept for interpreting experiments. This is because we do not directly observe the decay but use a detector and it is taken as a rule that the decay causes a chain of events that result in what is detected or measured. 

The question remains whether necessitation and determinism are characteristic of all theories of causality. A further question is whether it is coherent to have classical causal laws holding in interpreting measurements but not in atomic physics.

Without determinism and necessitation at the fundamental level, the same event would not always follow from two identical sets of initial conditions or states. Predictions in the quantum realm, prior to detection, are probabilistic in general and this provides a reason for rejecting a classical causal interpretation of quantum mechanics.

The connection between determinism and prediction is that the former provides the metaphysical ground for the later.

In addition, the classical theory of causation has the cause and the effect as two distinct things. How distinct must these two things be? The predictions of quantum mechanics, in addition to being indeterministic, do not, according to Bohr (Causality and Complementarity, Philosophy of Science, Vol. 4, No. 3, 1937), allow the separation of the microscopic quantum system and the measuring apparatus. This gives rise to the notion that the act of measurement is in part responsible for the measured result. Cause and effect cannot be separated and so there is no clear demarcation between them. Classical causality breaks down again.

In the same paper Bohr insists that "the concept of causality underlies the very interpretation of each result of experiment, and that even in the coordination of experience one can never, in the nature of

things, have to do with well-defined breaks in the causal chain."

So, for Bohr there is a theory of atomic and sub-atomic physics, quantum mechanics, but there is no separable cause and effect in that theory. After an event is detected by the measuring apparatus then classical cause and effect come into play.

Potentiality

Heisenberg was not content to leave atomic physics inexplicable and so proposed an ontology at the atomic level with substance in isolation understood as pure potentiality

All the elementary particles are made of the same substance, which we may call energy or universal matter; they are just different forms in which matter can appear. If we compare this situation with the Aristotelian concepts of matter and form, we can say that the matter of Aristotle, which is mere ‘potentia’, should be compared to our concept of energy, which gets into ‘actuality’ by means of the form, when the elementary particle is created. (Heisenberg 1959)

Every time a potency gets actualised, causation happens and that is due to isolated substance encountering actualised matter. In mentioning particle creation Heisenberg has moved from quantum mechanics to field theory and his claim 'poentia' are energy like can only be sustained in an Aristotelian sense that has little to do with the concept in physics. As will be explained below, potentiality can be introduced within quantum mechanics using the theory outlined in the previous post. The idea is that objects behave the way they do, not because of some external laws that determine what happens to them, but because of their own intrinsic dispositions and their interactions. Let's follows Anjum and Mumford and call such a theory of causality neo-Aristotelian.

Neo-Aristotelian causality does not invoke necessitation. Instead, there are irreducible tendencies. It adopts dispositional modality rather than conditional necessity. A tendency is less than necessity, so the effect is not guaranteed by its cause.

As described in the previous post, a typical example of a probabilistic disposition is the 50:50 propensity of a fair coin to land heads or tails if tossed, while a non-probabilistic disposition could be the propensity of a vase to break if dropped onto a firm surface.

Neo-Aristotelian causation is not a relation between two separate events or objects, but is a continuous, unified process that typically takes time to unfold. One way to understand an event as stochastic is that there is some objective probabilistic element involved. In other words, a chance event. This is an ontological interpretation of probability, which contrasts with the purely epistemic notion of credence or subjective probability. An individual event could therefore still be caused, in the neo-Aristotelian sense, even if it is random to some degree and not predictable. Typically, this type of causation takes place when the possibilities exist, when the effect is enabled by the right stimulus and under the right conditions. Neo-Aristotelian causation happens once the disposition manifests itself. 

An object with the potentiality to manifest as possible events will find itself in a situation where there are enabling dispositions and interfering dispositions. For example, firewood is disposed to ignite but the process leading up to it burning requires the presence of manifestation partners: a suitable site, proper ignition, enough oxygen, and so on, and that inhibitors such as dampness are not too strong. The firewood, its enablers and its inhibitors are all active in the cause of the effect. No ontological distinction is drawn between properties belonging to the object undergoing change and the contextual properties in this process. They are all, in a general sense, causes of the specific outcome. Depending on the balance the firewood may

• Burn brightly and sustainably
• Burn but go out quickly
• Smolder and smoke
• Fail to light at all.

In summary, neo-Aristotelian causation

• Involves irreducible tendencies
• Is not deterministic, although some process can be close to deterministic
• Supports predictions of what tends to happen, not what will happen with certainty
• Is a unified process, not a relationship between two distinct events.

A sketch of quantum causation

Take the electron as a typical quantum object. We need to identify what possibilities it can potentially manifest and what is needed to enable that manifestation.

The possible manifestations of the electron are as values of position, momentum, spin, charge, and mass. At this level of description charge and mass are classical properties. The others are potential properties and from quantum theory there is structure to and constraints on how the other properties can appear. This is captured in the formulation of quantum mechanics favoured in this blog that uses the generalisation of probability to a \(\sigma\)-complex.

As indicated, at this stage I only propose a sketch of how this may work. Developing the detail and confirming whether the proposed process is correct will require much more work.

For the potentiality to manifest the electron cannot be isolated. It therefore interacts with other objects. It is proposed that the interaction selects one \(\sigma\)-algebra from the complex. That is, the context will be for certain spin values or position or momentum to manifest. This provides a preliminary selection of a \(\sigma\)-algebras from the complex and therefore standard probability description of the tendency of the electron properties to manifest. Now this manifestation takes place via a Monte-Carlo selection process from the probability distribution associated with the selected \(sigma\)-algebra. This provides a model of the causal chain from potentiality to actuality in the quantum domain.

To make this more concrete consider the analysis of the double slit experiment.  In the initial version discussed previously the slits split the quantum state to produce an interference effect. The context is already selecting position and therefore one probability distribution from those possible. At the detecting screen the position of the electron is made actual through Monte-Carlo selection from the probability distribution.

Now consider the addition of a pointer immediately after the double slit. The pointer tends to point towards the electron. The pointer description is purely quantum and is not a measurement apparatus. As my earlier analysis shows the presence of the pointer eliminates the interference effect and selects a probability distribution for the electron position that is a normalised sum of the distributions that would obtain if the electron only passed through one slit. Again, the context has selected the probability distribution over position and at the detecting screen the position of the electron is made actual with a Monte-Carlo selection from the probability distribution.

Potentiality and probability

As outlined in the previous post, Barbara Vetter (Potentiality from Dispositions to Modality Oxford University Press, 2015) developed the concept of potentiality in her theory of dispositional powers. In that theory potentials are dispositions that are responsible for the manifestation of possibilities. The possibilities then tend to become actual events or states of affairs. The concept of 'potential' in philosophy, in a sense close to that discussed here, goes back at least to Aristotle in Metaphysics Book \(\Theta\).

In contrast probabilities are weightings summing to one that describe in what proportion the possibilities tend to appear. I propose that the potential underpins the actual appearance of the possibilities while probability shapes it. This will be discussed further in this post. 

Barbara Vetter proposed a formal definition of possibility in terms of potentiality:

POSSIBILITY:  It is possible that \(p =_{def}\) Something has an iterated potentiality for it to be the case that \(p\).

 So, it is further proposed that the probabilities are the weights that can be measured through this iteration using the frequency of appearances of each possibility. Note that this indicates how probabilities can be measured but it is not a definition of probability.

In the field of disposition research there is an unfortunate proliferation of terms meaning roughly the same thing. The concept of 'power' brings out a disposition's causal role but so does 'potential'. As technical terms in the field both are dispositions. Now 'tendency' will also be introduced, and it is often used as yet another flavour of disposition.

 Tendencies

Barbara Vetter mentions tendencies in passing in her 2015 book on potentiality and, although she discusses graded dispositions, tendencies are not a major topic in that work. In "What Tends to Be the Philosophy of Dispositional Modality" Rani Lill Anjum and Stephen Mumford (2018) provide an examination of the relationship between dispositions and probabilities while developing a substantial theory of dispositional tendency. In their treatment powers are understood as disposing towards their manifestations, rather than necessitating them. This is consistent with Vetter's potentials. Tendencies are powers that do not necessitate manifestations but nonetheless the power will iteratively cause the possibility to be the case.

In common usage a contingency is something that might possibly happen in the future.  That is, it is a possibility. A more technical but still common view is that contingency is something that could be either true or false. This captures an aspect of possibility, but not completely because there is no role for potentially; something responsible for the possibilities.  There is also logical possibility in which anything that does not imply a contradiction is logically possible. This concept may be fine for logic but in this discussion, it is possibilities that can appear in the world that are under consideration. Here an actual possibility needs a potentiality to tend to produce it.

Example (adapted from Anjum and Mumford)

Struck matches tend to light. Although disposed to light when struck, we all know that there is no guarantee that they will light as there are many times a struck match fails to light. But there is an iterated potentiality for it to be the case that the match lights. The lighting of a struck match is more than a mere possibility or a logical possibility. There are many mere possibilities towards which the struck match has no disposition - that is no potential in the match towards struck matches melting, for instance.

Iterated potentiality provides the tendency for possible outcomes to show some patterns in their manifestation. In very controlled cases the number of cases of success in iterations of match striking could provide a measure of the strength of the power that is this potentiality. This would require a collection of matched that are essentially the same.

Initial discussion of probability

Anjum and Mumford introduce their discussion of probability through a simple example that builds on a familiar understanding of dispositional tendencies associated with fragility.

"The fragility of a wine glass, for instance, might be understood to be a strong disposition towards breakage with as much as 0.8 probability, whereas the fragility of a car windscreen probabilities its breaking to the lesser degree 0.3. Furthermore, it is open to a holder of such a theory to state that the probability of breakage can increase or decrease in the circumstances and, indeed, that the manifestation of the tendency occurs when and only when its probability reaches one." 

This example is merely an introduction and needs further development but already the claim that "the manifestation of the tendency occurs when and only when its probability reaches one" shows that it is not a model for objective probability. What is needed is a theory of dispositions that explains stable probability distributions. Of course, if the glass is broken then the probability of it being broken is \(1\). However, this has nothing to do with the dispositional tendency to break. What is needed is a systemic understanding of the relationship between the strength of a dispositional tendency and the values or, in the continuous case, shape of a probability distribution.

In the quoted example above each power is to be understood in terms of a probability of the occurrence of a certain effect, which is given a specific value. The fragility of a wine glass, for instance, might be understood to be a strong disposition towards breakage with as much as 0.8 probability, whereas the fragility of a car windscreen is less, and the probability of its breaking is a lesser degree 0.3. But given a wine glass or windscreen produced to certain norms and standards it would be expected that the disposition towards breakage would be quite stable. A glass with a different disposition would be a different glass.

Anjum and Mumford claim, that in some understandings the manifestation of a possibility occurs when and only when its probability reaches one (see Popper, "A World of Propensities", 1990: 13, 20). This is a misunderstanding of how probability works. Popper distinguished clearly between the mathematical probability measure and what he called the physical propensity, which is more like a force, but Popper does limit a propensity to have a strength of at most \(1\).   As I will attempt to show below, Popper in proposing propensity interpretation of objective probabilities oversimplifies the relationship between dispositions and probabilities. This confusion led Humphreys to draft a paper (The Philosophical Review, Vol. XCIV, No. 4 (October 1985)) to show that propensities cannot be probabilities. As indeed they are not. They are dispositions. That would leave open the proposition that probabilities are dispositional tendencies, but that also will turn out to be untenable.

The proposal by Anjum and Mumford that powers can over dispose does seem to be sound. Over disposing is where there is a stronger magnitude than what is minimally needed to bring about a particular possibility. This indicates that there is a difference between the notion of having a power to some degree and the probability of the power’s manifestation occurring. Among other conclusions, this also shows that the dispositional tendency does not reduce to probability, preserving its status as a potential. 

 Anjum and Mumford continue the discussion using 'propensity' as having a tendency to some degree, where degree is non-probabilistically defined.  Anjum and Mumford use the notions of ‘power’, ‘disposition’ and ‘tendency’ more or less interchangeably, whereas an object may have a power to a degree there are powers that are simply properties. In what follows I try to will eliminate the use of 'propensity', except where commenting of the usage of others, and use 'tendency' to qualify either 'power', 'potential' or 'disposition' rather than let it stand on its own.

A probability always takes a value within a bounded inclusive range between zero and one. If probability is \(1\) then probability theory stipulates that it is almost certain (occurs except for a set of cases of measure zero). In contrast to what Anjum and Mumford claim it is not natural to interpret this as necessity because there are exceptions. For cases where there are only a finite set of possibilities then probability \(1\) does mean that there are no exceptions. But as this is a special case in applied probability theory there is no justification in equating it with logical or metaphysical necessity.

A power must be strong enough to reach the threshold to make the possibilities actual.  Once the power is strong enough then the probability distribution over the possibilities may be stable or affected by other aspects of the situation. So, instead of understanding powers and their degrees of strength as probabilistic, powers and their tendencies towards certain manifestations are the underpinning grounds of probabilities.  Consider the example of tossing a coin.

A coin when tossed has the potential to fall either heads or tails. This tendency to fall either way can be made symmetric and then the coin is 'fair'. From which probability weightings of \(1/2\) for each outcome (taking account of the tossing mechanism) can be assumed and then confirmed by measuring the proportion of outcomes on iteration. The reason why the head and the tail events are equally probable statistically, when a fair coin is tossed, is that the coin is equally disposed towards those two outcomes due to its physical constitution. The probability weightings derive, in this example, from a symmetry in the potentiality, which in turn derives from the physical composition and detailed geometry of the coin.

https://en.m.wikipedia.org/wiki/Rai_stones)

Consider a society that uses very large stone discs as currency.  On examination of the disc, it would be possible to conject that if it were tossed then there would be two possible outcomes and that those outcomes are equally likely. But this disposition is not realised because of the effort required to construct the tossing mechanism, as such a stone may weigh several metric tons. The enabling disposition that would give rise to the iteration of possibilities would have been this missing tossing mechanism. It is not a property of the disc. The manifestation of the dispositional tendency of the disc to come to lie in one of two states needs an external mechanism that is disposed through design to toss the coin in a certain way. If the mechanism is constructed it may be too weak. It may tend to only flip the coin once giving a sequence such as 

... T H T H T H H T H T H T H T H T H T H ...

that would give a frequency of T close to \(1/2\) but the sequence does not exhibit the potential for random outcomes to which the coin disposed. 

 Probabilities and chance

From the above: potential and possibility are more fundamental than (or prior to) probability. Both are needed to construct and explain objective probability. The alternative, subjective probability, is based on beliefs about possibilities but that is not the same thing as what is actually possible and how things will appear independently of anyone's beliefs or judgements.

In this blog I have already referred to a dispositional tendency begin to explain objective probabilities in quantum mechanics. The term propensity has been used to describe these probabilities. I now think that was wrong. Propensity should be reserved for the dispositional tendency that is responsible for the probabilities to avoid this term merging the underpinning dispositional elements and probability structure. Anjum and Mumford claim that they have made a key contribution to clarifying the relationship between dispositional tendencies and probability through their analysis of over disposition. 

Anjum and Mumford claim "information is lost in the putative conversion of propensities to probabilities" but only if the dispositional grounding of probabilities is forgotten.  Their discussion is strongly influences by their interest in application to medical evidence where a major goal is reduction of uncertainty.  Anjum and Mumford propose two rules on how dispositions and probability relate.

1. The more something disposes towards an effect \(e\), the more probable is \(e\), ceteris paribus; and the more something over disposes \(e\), the closer we approach probability \(P(e) =1\).
2. There is a nonlinear ‘diminishing return’ in over disposing. E.g., if over disposing \(e\) by a magnitude \(x2\) produces a probability \(P(e) =0.98\), over disposing \(x3\) might increase that probability ‘only’ to \(P(e) =0.99\), and over disposing \(x4\) ‘only’ to \(P(e) =0.995\), and so on.

While these rules are fine as propositions, they miss the mark in explaining the relationship between dispositions and probability. In the coin tossing example strengthening the mechanism is not about strengthening one outcome. Over disposing does provide support for the distinction between the strength of the disposition and value of the probability but the relationship between underpinning potentials, dispositional mechanisms, and the iterated outcomes needs to be made clear.

Anjum and Mumford also discuss coin tossing and make substantially the same points as I made above. However, having clarified the distinction between propensity and probability, they revert to using the term propensity in a way that risks confusing the concepts of dispositional tendency and probabilities with random outcomes. They say "50% propensity" rather than "50% probability". They then introduce the term "chance" that they relate to outcomes in some specified situations. Propensity is then reserved by them for potential probability while chance is the probability of an outcome in a situation. This is more confusing than helpful.

Anjum and Mumford go on to a discussion of radioactive decay that is known to be described by quantum theory. They make no mention of quantum theory (this will be corrected by them in Chapter 4) and strangely claim that radioactive decay is not probabilistic.   The probability distributions derived from quantum mechanics unambiguously give the probability of decay per unit time. There are, per unit time, two possibilities "decay" or "no decay". Their error is to claim, "only one manifestation type" (decay) and from this that there is only one possibility. Ignoring quantum mechanics, they write:

 "The reason it is tempting to think of radioactive decay as probabilistic is that there is certainly a distinct tendency to decay that varies in strength for different kinds of particles, where that strength is specified in terms of a half-life (the time at which there is a 50/50 chance of decay having occurred)."

 But no, the reason to think that radioactive decay is probabilistic is that our best theory of nuclear phenomena explains it in terms of probabilities. This misunderstanding leads them to introduce the concept of indeterministic propensities. However, they have arrived at the concept it is left open as to whether there are non-probabilistic indeterminate powers, but radioactive decay is not an example.

The examples of the concept chance provided by Anjum and Mumford can be derived in their examples from a correct application of probability theory. Chance is often used as a term for 'objective probability', and I have done so in previous posts. I will continue to follow that usage and exploit the clarification obtained from the analysis above that shows that objective probability depends on the possibilities that are properties of an object. The manifestation of these possibilities may require an enabling mechanism.    The statistical regularities displayed by these manifestations on iteration are due primarily to the physical properties of the object unless the enabling mechanism is badly designed.

Thet term 'propensity' has given rise to much confusion in the literature. Now that we are reaching an explanation of objective probability the term 'propensity' might better be avoided. 

I propose that the model of objective probability is that:

OBJECTIVE PROBABILITY An object has probabilistic properties if it is physically constituted so that it has a potential to manifest possibilities that show statistical regularities.

It it possible to describe statistical regularities without invoking the term 'probability'.

Although some criticism of Anjum and Mumford is implied here, I recognise that their contribution has done much to disentangle considerations about the strength of dispositions that describe tendencies form a direct interpretation as probabilities. However, the value of the three distinctions they have identified is mixed

• Chance and probability are not fundamentally distinct and just require a correct application of probability theory
• Probabilistic dispositional tendencies are distinct from non-probabilistic dispositional tendencies: this is a real and fruitful distinction
• Deterministic and indeterministic dispositional tendencies also provide a useful distinction but it remains to be seen whether there are fundamental non-probabilistic indeterministic dispositions.

The next post will continue this theme with a discussion of dispositional tendencies in causality and quantum mechanics, engaging once more with the 2018 book by Anjum and Mumford.

 

Concepts of causal powers

The concept of dispositions has played a key role in the formulation of objective quantum chance in this blog. However, there is ambiguity about what these are as powers. Ruth Porter Groff has helpfully addressed this issue and identified four senses of the term. She identifies dispositional power to be conceptualised as an:

• Activity
• Capacity
• Essence
• Necessitation.

While Groff indicates that there is a further task to work out which is correct in a given context. That is, depending on what is (what the world is like), there could be powers of distinct types. For example, at the various levels of reality (Hartmann) different concepts could play their part. The intention is that clarity on which concept of power applies will strengthen any theory of quantum chance.

These concepts of powers need to be contrasted with what is called the Humean view that there are no necessary connections or causes in nature. That is, there are no powers.

Activity

Consider a film in which each frame is static. Playing the film gives the impression of movement or activity. If activity in the world is like the film, then activity is an illusion or a metaphor. In which case activity is not an aspect of how things are, and the ontology can be called passive. If activity is not just a sequence of static configurations, then activity may not be an illusion. We can follow Groff and use the term anti-passivist to refer to the opinion that activity is a real and irreducible component of the world.

Activity is taken to cover a range of things. Movement, deliberation (moving away from the visual film example), inquiry and chemical reaction (to take an inorganic example). Any instance of causation is an activity.

The view that there is activity in the world has common sense on its side. For example, action as captured by verbs as part of the deep structure of language.

From this perspective, to say that things in the world have causal powers is to say that things engage in activity and are able to do. Reality is in this sense genuinely, irreducibly, non-metaphorically dynamic. In contrast Esfeld's [1] primitive ontology, that is in favour with some who defend the Bohmian version of Quantum Mechanics, is passive and at best kinematic. It is an example of an extreme (or to use Esfeld's own term Super) Humean ontology.

 Real activity contrasts with the Humean view that inanimate matter is essentially passive and never intrinsically active. In real activity the action of things depends on their causal powers. Examples of activity, from Cartwright [2] Hunting causes and using them: Approaches in philosophy and economics, are:

1. The carburettor feeds gasoline and air to a car’s engine ...

2. The pistons suck air in through the chamber...

3. The low-pressure air sucks gasoline out of a nozzle ...

4. The throttle valve allows air to flow through the nozzle ...

5. Pressing the pedal opens the throttle valve more, speeding the airflow and sucking in more gasoline...

6. …

These examples indicate that mechanisms can undertake activity.

Capacity

A capacity is a way that something presently is, such that it could be a way that it presently is not. The phenomenon of a capacity is thus inherently modal, invoking possibility. That is capacity is the potential to be in a possible state for that thing.  This type of potentiality is attached to the nature of a thing and is sometimes called real, or metaphysical possibility, in contrast to logical possibility.

A capacity may be engaged in activity but there is nothing about the concept of a real possibility that requires realism about activity to be either built into it or entailed by it. A power, as a capacity, is a property that need not be fully actualised in the present in order to exist.

Capacities are properties that include, as part of their identity in the present, non-actualized but nevertheless real potential manifestations. By contrast, for the Humean the only properties that exist are categorical properties; properties that are fully occurrent or actual.

Essence

Essentialism is when things (or some kinds of things) are such that they could not be, in part or in full, otherwise without ceasing to be what they are.  Among dispositionalists, Bird [3] defines powers, or potencies, as fundamental properties whose identities are not just dispositional, but fixed. A power as an essence is to be a property whose identity is essential to it.

Necessitation

Necessitation is when one thing is the case, some other thing must be the case. A necessary connection is a power in this sense. Equating powers with necessary connections is proposed by Armstrong [4].

It is not clear that even those anti-passivists who are most focused on defending the reality of necessary connections (in the name of defending the notion of a law) do believe in metaphysical necessitation.

What is meant by the term ‘metaphysical necessitation’? To answer this, we need to know

1. whether one who affirms it believes
1. that things of a given kind necessarily tend to behave in one way or another, or
2. that they must behave in one way or another; and also
2. whether or not one who affirms the existence of metaphysical necessitation holds that given behaviours necessarily bring about assigned outcomes.

Accepting the above would tend someone strongly towards metaphysical determinism and intuitively this would be a natural consequence of necessitation. However, it may be that a version of metaphysical necessity that commits only to the existence of necessary tendencies does not translate into a commitment to what may be called ‘causal necessitarianism,’ or even hard determinism.

Conclusion

It is conceivable that all the concepts described above have role to play in understanding the role of powers in the world. However, from the posts in this blog on quantum physics and in particular quantum chance it is capacity that seems to be the best fit. This is because the potential to have a physical property is attached to a physical object and possibilities are captured by the set of possible values that can be made actual. For example, an electron has the property of spin with the potential to take certain value. The possible values that can be actualised and with whichever probabilities depends on this potential and the context the particle find itself in.

The concept of activity also has causal force but seems better suited to powers of designed mechanism or psychological or social situations. If a theory can be developed of what gives rise to the occurrence of an actual event in Quantum Mechanics, then activity may be a valid concept for the power in question.

Essence plays a role in quantum chance in that the power that governs the tendency for an object, such as an electron, to take particular values of spin is an essential property of an electron. That is, an electron would not be an electron if spin and quantum chance were not aspects of its state.

Necessary connections have a role to play in a physical theory even in the presence of objective chance. The evolution of the wavefunction governed by the Schrödinger equation is deterministic making the state of the object necessarily connected to its state at an earlier time.

However, it is proposed that potentiality and possibility are the key concepts, equivalent to capacity, which play the key role in the developing theory of quantum chance. The most complete treatment to date of powers as potentiality and possibility is by Barbara Vetter [ 5] whose classification of potentiality as a localised modality and possibility as a non-localised modality looks promising. We may then have the quantum state of the electron representing the chance potential to take certain spin values while the possible spin values that can be actualised will depend on the non-localised situation that constrains the quantum state.

 

[1] Esfeld, M., & Deckert, D.-A. (2017). A Minimalist Ontology of the Natural World (1st ed.). Routledge.

[2] Cartwright, N. (2007). Hunting causes and using them: Approaches in philosophy and economics. Cambridge University Press.

[3] Bird, A. (2007). Nature’s metaphysics: Laws and properties. Oxford University Press.

[4] Armstrong, D. M. (2005). Four disputes about properties. Synthese, 144(3), 309-320.

[5] Vetter, B. (2015). Potentiality: From dispositions to modality. Oxford University Press.

 

Review: October 2022

This blog is intended to

• Help organise and develop my thinking on quantum mechanics, the role of probability, and the ontological status of particles and states.
• Examine and develop ontologies in other areas.

 My guiding principle is that entities have properties and interactions that are independent of whatever anyone or anything knows about them. Experiments are for finding out about the physical world not for instantiating it. That is, the physical world is there even when no one is looking. So far, the discussion has engaged with ontologies that deal with an autonomous physical world independent of considerations of what is known about the system or who is interfering with it. There are quantum theories that focus on what can be known about a physical system rather than what the systems behaviour is per se. The original Copenhagen interpretation of quantum mechanics falls into this category, but a more recent approach is Quantum Bayesianism. Commonly shortened to QBism, it interprets the quantum state as capturing a degree of belief. There are questions about the ontological assumptions associated with the theory, with options ranging from a form of idealism to "participatory realism". I will return to this in future posts.  

This blog has inherited a lot of material from a draft paper I prepared on ontology and quantum mechanics. The substance of that paper has now been captured as a set of blog posts. One reason for putting aside the draft paper was the recognition of the further reading on ontology that I needed to do. At the time of moving from the academic paper format to the blog, I had only started to delve into the New Ontology by Nicolai Hartman [1]. What I won from Hartmann was a structure of ontological strata and spheres of being. This structure is much richer than Karl Popper's three-world ontology [2]. An academic paper is appropriate once all the problems with concepts have been addressed or at least clarified, whereas a blog is more flexible and informal and that should help in developing my ideas.

Critical ontology

I have taken the title of the blog from a earlier paper by Hartmann: Wie ist kritische Ontologie überhaupt möglich?. Critical ontology implies, in this blog, a constructive but forensic attitude to concepts and theories in science and philosophy whether I am initially sympathetic to them or not. This is an attitude that I think is consistent with both Hartmann's and Popper's approach to philosophy.

Quantum chance

A major challenge, and focus so far, is how to include objective probabilities in an ontology for quantum mechanics. Popper's proposal for a propensity interpretation introduced a dispositional model for objective chance but its ontological status remains ambiguous with its presentation strongly dependent on artificial experimental arrangements rather than the situations in which physical entities mostly find themselves. Popper's intent seems to be to reduce quantum indeterminacy to classical probability with a dispositional interpretation. I do not think that can be directly achieved. 

In previous posts not enough emphasis was given to explaining the special status of quantum chance. The use of the letter p to represent the quantum state should not be taken to imply that quantum mechanics has been reduced to classical probability. There are major differences, such as interference terms, as shown in the mathematical presentation.

Here I want to explain another major difference. Classical probability theory has its origins in the analysis of games of chance and statistics was initially developed to deal with the vast quantities of data associated with entities of interest to the state.  In games of chance the numbers on dice or patterns of playing cards are actual but hidden. Similarly in the use of statistics by the state, members of the population are actual, and while not hidden, the state needs to work with averages and distributions. In quantum chance the probabilities are not merely a means of dealing with hidden or irrelevant variables. It is known from the work of Kochen and Specker [3] that in general the quantum variables are not actual. The electron does not have an actual spin value that is unknown because in general its spin value is only a potential value. This means that the dispositional powers that lead to quantum chance are fundamental.

A consideration that requires further work is the mechanism for property values to become actual that is not tied to a measurement. There are proposals for spontaneous wavefunction reduction but if they only decrease the variance (or some other measure of the spread) of the wavefunction this only reduces the number of possibilities without selecting one to be actualised. This seems to me to indicate a major open problem.

These considerations indicate the need for a theory that recognises dispositions as fundamental properties of physical entities. The concepts developed by Alexander Bird [4] show promise although it is a still open to investigation how widely ranging dispositional properties are in nature. 

Ontological status of mathematics and physical theories

 The ontological status of mathematical representations in physical theories also needs further development. In Hartmann's ontological structure pure mathematics belongs to the ideal rather than the real sphere. Physical theories seem to fit better objectivated mode belonging to the spirit stratum of the real sphere, but they apply structures with an origin in pure mathematics and provide a description of aspects of the inorganic stratum of the real sphere. This means it is necessary to understand to what extent mathematical entities belong to the ideal rather than the real sphere and what the interaction is between the ideal sphere, the spirit stratum, and the inorganic stratum in the real sphere. Whereas Hartmann goes back to Aristotle to develop an ontology that includes ideal and real spheres of existence, an Aristotelean ontology that places mathematics in the real sphere is also a possibility. Franklin [5] has developed a version of this.

Next steps

My next task is to absorb the content of books like those of by Franklin and Bird, and capture what I learn in future posts. It is hoped that my earlier posts in this blog will then be developed, clarified, and improved. In addition, I will examine and discuss the proposals for gravity induced state reduction [6], event-oriented theories [7], and quantum Bayesian approaches.

[1] Hartmann, N., New Ways of Ontology, Taylor and Francis, 2017 (translated from Hartmann, N., Neue Wege der Ontologie, W Kohlhammer, 1949)

[2] Popper, K. R., Objective Knowledge, Oxford: Clarendon Press, 1972

[3] Kochen, S. & Specker, E. P., The Problem of Hidden Variables in Quantum Mechanics, 

J. Math. & Mech., 1967, 17, 59 

[4] Bird, A., Nature’s Metaphysics, Oxford University Press, 2007

[5] Franklin, J., An Aristotelian Realist Philosophy of Mathematics, Palgrave Macmillan UK, 2014

[6] Penrose, R., Shadows of the Mind, Oxford University Press, 1994

[7] Fröhlich, J. & Pizzo, A., The Time-Evolution of States in Quantum Mechanics according to the ETH-Approach, Communications in Mathematical Physics, 2021