The objective–subjective dichotomy and its use in describing probability

ABSTRACT

This article reviews the nature of the objective–subjective dichotomy, first from a general historical point of view, and then with regard to the use of these terms over time to describe theories of probability. The different (metaphysical and epistemological) meanings of ‘objective’ and ‘subjective’ are analysed, and then used to show that all probability theories can be divided into three broad classes. A formal definition of (epistemological) objectivity is given and used to clarify this division.

KEYWORDS:

• Probability
• objective
• subjective
• dichotomy
• objectivity
• subjectivity

The words ‘objective’ and ‘subjective’ have a long history and a number of different meanings, as shown by their lengthy entries in the Oxford English Dictionary (OED). This article summarizes their use as contrasting concepts, first from a general historical viewpoint (section 1), and then as applied to theories of probability from about 1700 to the present day (section 2). In addition, by using the metaphysical and epistemological meanings of the objective-subjective terms, and by defining objectivity, any interpretation of probability put forward over the years is seen to belong to one of three general classes (sections 3 and 4).

1. The dichotomy historically

Consider first the modern understanding of the dichotomy from a metaphysical point of view. Metaphysics deals with the fundamental nature of what exists, and in this regard ‘objective’ refers to something that has a real existence in the world independent of being thought of, whereas ‘subjective’ refers to anything that depends for its existence on consciousness. Thus an individual has objective attributes such as weight and height, whereas the person’s honesty and intelligence are subjective. To regard these subjective qualities as attributes that can be measured is the fallacy of reification.

It is an interesting fact that scholars of the Middle Ages used ‘objective’ and ‘subjective’ with meanings that are the exact reverse of those that are used today. The OED (2019) illustrates this with an untranslated quote in Latin (c. 1325) from the work of the nominalist William of Ockham. A modern translation is as follows (Spade 1994, 218):

A universal is not anything real having subjective being [existence], either in the soul [mind] or outside the soul [mind]. Instead it only has objective being [existence] in the soul [mind]. It is a kind of fictum [mental picture] having being [existence] in objective being [existence] like what the external thing has in subjective being [existence].¹

Over the succeeding centuries, there was a gradual shift from the scholastic distinction to its modern form. Lorraine Daston (1994) has described these changes as reflected in philosophical works and various dictionaries of the time, writing that ‘the meanings of the terms had, however, already branched and crisscrossed in the seventeenth century in both Latin and in various vernaculars, although ‘objective’ still generally modified thoughts rather than external objects’ (333). She points out that mid-nineteenth-century German and French dictionaries traced the origin of the newer meanings to Immanuel Kant, although this has been questioned by Machiel Karskens (1992) in his detailed study of the use of the dichotomy in the eighteenth century. He writes that the work of philosopher Adolph Hoffmann² ‘strongly accentuates in subjective the internal, particular state of mind in (the consciousness of) the knowing person or subject,’ and that Hoffmann used ‘object(ive) to refer to the status of the real existence of the thing or event in itself, apart from intentional acts of understanding’ (247). Karskens suggests these ideas influenced Kant and his followers, and led to the ‘final equation of subject with human beings and of object with extra-mental entities’ (251).

The OED (2019) gives examples from the seventeenth century of both the scholastic and modern meanings of the words, but the scholastic use was gradually displaced. The dictionary gives a good example of the modern sense from the late eighteenth century (Anonymous 1793, 498): ‘Have the objects, which we consider as really existing, in fact, a real objective existence, independent of our mode of perceiving them?’ This usage became more common in English after 1800, owing perhaps to the work of poet and philosopher Samuel Taylor Coleridge, who studied in Germany and was influenced by the ideas of Kant and his followers. In his Biographia Literaria ([1817] 1967, 174) he writes:

Now the sum of all that is merely OBJECTIVE we will henceforth call NATURE, confining the term to its passive and material sense, as comprising all the phenomena by which its existence is made known to us. On the other hand the sum of all that is SUBJECTIVE, we may comprehend in the name of the SELF or INTELLIGENCE. Both conceptions are in necessary antithesis.

In addition to this metaphysical view of the objective-subjective terms, there is also an epistemological dichotomy, which refers basically to whether one is impartial (or not) when making judgments. The OED (2019) shows that this usage in English dates from the nineteenth century, with some examples for ‘subjective’ from the eighteenth century. This dichotomy is discussed further in section 3.

2. The dichotomy and probability

The systematic treatment of probability is usually regarded as beginning in the second half of the seventeenth century. This period saw the famous correspondence between Pascal and Fermat in 1654, the publication of Christiaan Huygens’s book on games of chance (1657), and John Graunt’s analysis of the data in weekly bills of mortality (1662). At the same time there emerged the idea of the dual nature of probability, as described by Hacking (2006, 12): ‘On the one side it is statistical, concerning itself with stochastic laws of chance processes. On the other side it is epistemological, dedicated to assessing reasonable degrees of belief in propositions quite devoid of statistical background.’ With hindsight one can identify this difference, but probability theorists up to the mid-nineteenth century did not distinguish explicitly between the two approaches.

Various descriptions have been used for this duality, such as objective versus epistemic, or frequency-type versus belief-type; in this article it is referred to as the metaphysical objective-subjective dichotomy. Thus if probability is regarded as an actual physical property of a system, existing independent of consciousness and estimated by measurements of some kind, then it is ‘metaphysically objective,’ otherwise it is ‘metaphysically subjective,’ that is, dependent on consciousness and having a value assigned (not measured) to represent a state of knowledge. The two probability classes are mutually exclusive and exhaustive.

In the 1680s Jacob Bernoulli worked on his important book Ars Conjectandi, but it was published posthumously only in 1713. In it he writes ([1713] 2006, 315, emphasis in original):

The certainty of anything is considered either objectively and in itself or subjectively and in relation to us. Objectively, certainty means nothing else than the truth of the present or future existence of the thing. Subjectively, certainty is the measure of our knowledge concerning this truth. In themselves and objectively, all things under the sun, which are, were, or will be, always have the highest certainty.

Bernoulli was a determinist; he believed that what happens in the future ‘will occur with certainty’ as a result of ‘the highest Creator’s omniscience and omnipotence’ (315). This is what Bernoulli called objective certainty, whereas the certainty of we mortals is subjective, limited by our incomplete knowledge of the future. This leads to the concept of probability (315, emphasis in original): ‘Probability, indeed, is degree of certainty, and differs from the latter as a part differs from the whole.’ Although he did not describe probability as objective or subjective, his approach is generally in accord with the metaphysical distinction introduced above: that ‘objective’ means independent of human cognition, whereas ‘subjective’ refers to one’s state of knowledge. Regarding Bernoulli’s analysis, Hacking (2006, 145) writes that ‘for the first time a ‘subjective’ conception of probability is explicitly avowed.’

One hundred years later one finds Laplace ([1814] 1995, 2) expressing a similar determinism:

All events, even those that on account of their rarity seem not to obey the great laws of nature, are as necessary a consequence of these laws as the revolutions of the sun. … We ought then to consider the present state of the universe as the effect of its previous state and as the cause of that which is to follow.

Thus Laplace believed that the laws of physics determine the future of the physical world. It is only because of our lack of all the knowledge required to apply these laws that the future cannot be predicted, hence the need for probability, a measure of uncertainty: ‘Probability is relative in part to this ignorance and in part to our knowledge’ (3). Laplace did not use the words ‘objective’ or ‘subjective’ in his writings, but he and Bernoulli shared a similar subjective view, their views therefore being classified as metaphysically subjective, using the terminology introduced earlier. As Daston (1994, 330) has commented, the possibility of an objective probability ‘would have been an oxymoron for classical probabilists such as Jakob Bernoulli and Pierre Simon Laplace.’

By the early 1800s the modern version of the dichotomy had replaced the scholastic view, and the words began to enter the language of probability. Daston (1994) has shown that around 1840 at least six probability theorists, apparently independently, began emphasizing the dual nature of probability, although they did not all use the objective-subjective terminology. Thus Poisson, writing in 1837, used the French words chance and probabilité to describe the duality, but it was Cournot (1843, v) who first introduced the familiar terms, when he wrote that it was necessary for him to use ‘les deux épithètes d’objective et de subjective’ in order to discuss the meaning of probability³ (emphasis in original). Cournot’s work led to the development of the frequentist theory, although Daston (336) writes that neither he nor the other probabilists ‘went so far as to identify probabilities baldly with frequencies.’ Hacking (1990, 97) has pointed out that this frequentist emphasis coincided with the increasing availability of statistical data, that ‘the world teemed with frequencies, and the ‘objective’ notion would come to seem more important than the ‘subjective’ one for the rest of the century – simply because there were so many more frequencies to be known.’

The logician John Venn certainly described probability in terms of a frequency in his influential The Logic of Chance, first published in 1866. A probability was obtained in principle from an infinitely long series of proportions: ‘As we keep on taking more terms of the series we shall find the proportion still fluctuating a little, but its fluctuations will grow less. The proportion, in fact, will gradually approach towards some fixed numerical value’ (1888, 164). Venn clearly has in mind here that probability is a property that one measures, although he doesn’t use the word ‘objective’ to describe it.⁴ He uses that term, however, in his criticism of Bernoulli’s well-known limit theorem (Bernoulli [1713] 2006, 328–335), which he writes ‘is generally expressed somewhat as follows: in the long run all events will tend to occur with a relative frequency proportional to their objective probabilities’ (91). But Venn adds that the basis on which the theorem depends for its proof is faulty, since there is ‘really nothing which we can with propriety call an objective probability’ (91). Here Venn is criticizing the idea that one can obtain knowledge of probability without reference to experience, see the comment by Wesley Salmon (1981, 132): ‘Since Venn is convinced that the only kinds of probabilities are frequencies, he objects to the notion of ‘objective probability’ as another manifestation of unwarranted a priorism.’⁵

Frequentists in the early twentieth century, on the other hand, were explicit in regarding probability as being ‘objective.’ One finds, for example, Richard von Mises (1964, 332) claiming the following (emphasis in original): ‘Bayesian approach is often connected with a ‘subjective’ or ‘personal’ probability concept. Our probability concept is and remains ‘objective’.’ In addition, he writes that this means probability is a measurable property (1981, 14, emphasis in original):

The probability of a 6 is a physical property of a given die and is a property analogous to its mass, specific heat, or electrical resistance. Similarly, for a given pair of dice (including of course the total setup) the probability of a ‘double 6’ is a characteristic property, a physical constant belonging to the experiment as a whole and comparable with all its other physical properties. The theory of probability is only concerned with relations existing between physical quantities of this kind.

R. A. Fisher was the pre-eminent frequentist of this era and he agreed with the Mises view. For example, see the critical comment he made regarding the work of Harold Jeffreys: ‘It will be noticed that the idea that a probability can have an objective value, independent of the state of our information, in the sense that the weight of an object, and the resistance of a conductor have objective values, is here completely abandoned’ (1934, 4).

Mises and Fisher are the best-known advocates of the metaphysically objective view. The opposing metaphysically subjective view that was developed at this time branched in two directions. These are associated mainly with John Maynard Keynes and Harold Jeffreys on the one hand, and with Frank Ramsey, Bruno de Finetti, and Leonard J. Savage on the other. They are often referred to as logical and personalist theories respectively. In this paper they are called epistemologically objective and epistemologically subjective; these terms are discussed further in section 3.

The logical view was described by Keynes in A Treatise on Probability, published in 1921 but written before World War I. He regarded probability as a degree of belief, but stressed the need to avoid personal bias. For him, probability was ‘concerned with the degree of belief which it is rational to entertain in given conditions, and not merely with the actual beliefs of particular individuals, which may or may not be rational’ (1921, 4, emphasis in original). Furthermore, probability was interpreted as a logical relation between propositions, written (in current notation) as a conditional probability P(A|E), which is the extent to which proposition A is logically implied by the evidence E. Keynes stressed that ‘when once the facts are given which determine our knowledge, what is probable or improbable in these circumstances has been fixed objectively, and is independent of our opinion’ (4). So for Keynes ‘objectivity’ meant that, given prior information E, the probability of A would be the same for everyone who is rational.

Harold Jeffreys’s book Theory of Probability, first published in 1939, was a defense of Bayesian methods in probability, and his views were similar to those of Keynes. Thus he regarded probability as a reasonable degree of belief, which should always be considered as conditional on some prior information (1961, 406):

It is argued that because P(p|q) depends on both p and q it cannot be an objective statement, since different persons with different knowledge would assess different probabilities of p. … The probability of a proposition irrespective of the data has no meaning and is simply an unattainable ideal. On the other hand, two people both following the rules would arrive at the same value of P(p|q). It is a fact that the probabilities of a proposition with respect to different data will in general differ, and people with different data will make different assessments. But this is no contradiction, but merely the recognition of an obvious fact. They will arrive at consistent assessments if they tell each other their data and follow the rules.

E. T. Jaynes was a strong supporter of the Keynes-Jeffreys approach, although he regarded objectivity as a goal to be reached: ‘Our goal is that inferences are to be completely ‘objective’ in the sense that two persons with the same prior information must assign the same prior probabilities’ (2003, 373). But Jaynes understood the difficulty involved, since he also wrote: ‘There is no single universal rule for assigning priors – the conversion of verbal prior information into numerical prior probabilities is an open-ended problem of logical analysis’ (88). As a result, his goal of achieving objectivity is surely unlikely to be realized. This, however, is an incorrect view of what objectivity means, as discussed in section 3.

An alternative to the logical theory is the personalist approach, where probability is also regarded as a degree of belief, but a purely personal one. It is a measure of ‘the confidence that a particular individual has in the truth of a particular proposition’ (Savage 1972, 3). On this view, two people ‘faced with the same evidence may have different degrees of confidence in the truth of the same proposition’ (3), since a person determines a probability ‘by interrogating himself, not by reference to the external world’ (51). This is an epistemologically subjective view, in direct contrast to the logical view. The probabilities assigned cannot be completely arbitrary, of course, since they must be coherent, that is, obey the standard rules of probability.

The probability theories discussed above are obviously important in data analysis, but they also feature prominently in modern physics, especially in statistical mechanics and in quantum mechanics. Application of statistical mechanics to a gas, for example, assumes that its constituent atoms or molecules obey the laws of classical physics, so that their behaviour is, in principle, predictable (see the discussion of Laplace in section 2). Quantum mechanical systems, on the other hand, are regarded as inherently non-deterministic, with experimental outcomes that are not completely predictable. This situation is clearly summarized by physicist J. M. Jauch (1974, 131, emphasis in original):

The probabilities which occur in classical physics are interpreted as being due to an incomplete specification of the systems under consideration, caused by the limitations of our knowledge of the detailed structure and development of these systems. Thus these probabilities should be interpreted as being of a subjective nature.

In quantum mechanics this interpretation of the probability statements has failed to yield any useful insight, because it has not been possible to define an infrastructure whose knowledge would yield an explanation for the occurrence of probabilities on the observational level. Although such theories with ‘hidden variables’ have been envisaged by many physicists, no useful result has come from such attempts. I therefore take here the opposite point of view which holds that the probabilities in quantum mechanics are of a fundamental nature deeply rooted in the objective structure of the real world. We may therefore call them objective probabilities.⁶

Comparing Jauch’s analysis with that given in this paper, one would classify the probability used in statistical mechanics as metaphysically subjective, and the quantum mechanical probability as metaphysically objective.⁷ Further detailed discussions of these probabilities can be found in the volume edited by Beisbart and Hartmann (2011).

The sometimes confusing use of the objective and subjective terms has led some writers to suggest avoiding them altogether. In his introductory textbook on probability and induction, Hacking (2001, 131) writes: ‘If you read a lot about probability, you will often read about ‘objective’ and ‘subjective’ probabilities. These are terrible terms, loaded with ideology.’ Jeffreys avoided using the dichotomy when discussing probability, and he wrote that the terms ‘have been used in so many senses that they have become a source of confusion,’ and ‘the less the words are used the better’ (1973, 210). But the words continue to be used, and numerous attempts have been made to analyse their different meanings, see for example Gillies (2000) and Zabell (2011). In this connection Gillies (20) has commented as follows: ‘The analysis of the notion of objectivity is a difficult matter. Indeed, one could say that it is one of the most fundamental problems in philosophy.’ What appears to be lacking is a formal definition of objectivity, an issue addressed in the next section.

3. Definition of objectivity

In recent years there have been several uses of the term ‘objective’ to describe a form of Bayesian analysis. For example, the nature of the prior probability one should use has been emphasized by Berger (2006, 387), who writes that ‘the most familiar element of the objective Bayesian school is the use of objective prior distributions, designed to be minimally informative in some sense.’ Another example is that of Williamson (2010, 1), who gives a number of criteria that qualify his approach as a ‘version of objective Bayesianism,’ namely ‘the view that an agent’s degrees of belief should be probabilities, should respect constraints imposed by empirical evidence, but should otherwise equivocate between basic propositions.’ But without a definition of the word ‘objective’ being given, it is not clear why the approaches of these authors deserve that description.

In the metaphysical dichotomy used in section 2 to characterize the dual nature of probability, ‘subjective’ refers to a state of knowledge, and the question then arises as to how one acquires this knowledge. This gives rise to another (epistemological) dichotomy: one acquires knowledge either objectively or subjectively. Subjective knowledge (or, more correctly, subjective belief) is generally described as being partial or biased or prejudiced, or based on arbitrary assumptions or personal feelings; objective knowledge is then inferred by contrast, as impartial or unbiased or not prejudiced, and so on. Whenever possible, however, one should define a concept by what it is rather than by what it is not, but one will be hard-pressed to find such a definition of objectivity. One writer on objectivity (Gaukroger 2012, 3) declines to define it, since ‘many difficulties are generated in the search for a definition, because ‘objectivity’ can be understood in different ways.’

For a positive analysis of objectivity, consider the work of novelist-philosopher Ayn Rand, whose philosophy relies strongly on this concept (1990, 18):

Objectivity is both a metaphysical and an epistemological concept. It pertains to the relationship of consciousness to existence. Metaphysically, it is the recognition of the fact that reality exists independent of any perceiver’s consciousness. Epistemologically, it is the recognition of the fact that a perceiver’s (man’s) consciousness must acquire knowledge of reality by certain means (reason) in accordance with certain rules (logic).

In addition, she defined ‘reason’ as ‘the faculty that identifies and integrates the material provided by man’s senses’ (Peikoff 1991, 152), and stressed that the exercise of reason is volitional and not automatic. So her description of epistemological objectivity can be rephrased as a formal definition (Baise 2017, 231): Objectivity is a method of thought by means of which one acquires knowledge of reality by the volitional use of reason in accordance with the rules of logic. Having defined objectivity in this way, it is not unreasonable to violate the rule given above by defining ‘subjective’ as ‘not objective,’ that is, by what it is not, since objectivity involves a specific way of thinking, whereas it is subjectivity that ‘can be understood in different ways.’ In short, if one’s method of thought is not objective then it is subjective; the two methods can be thought of as dividing the metaphysically subjective class of probability theories into two mutually exclusive and exhaustive subclasses, namely epistemologically objective and epistemologically subjective.

One can see now the problem with the interpretation of objectivity by writers such as Keynes, Jeffreys and Jaynes, in which a unique probability is supposedly obtainable from given prior information, although how one does it is not made clear. Objectivity, however, is properly understood as a method of thought, in which reason and logic are used to reach a conclusion. So two individuals, given the same information, can both be objective yet assign different prior probabilities for an event or a hypothesis by making rationally and logically justified different assumptions. Jaynes was a leading developer of methods for estimating prior probabilities, and he used, for example, maximum entropy calculations and also transformation groups, but he acknowledged (1983, 129) that these procedures ‘are not necessarily applicable to all problems, and so it remains an open question whether other approaches may be as good or better.’ But they surely qualify as objective probability assignments.

As an example of the importance of the assignment of Bayesian prior probabilities, consider a widely-used area of artificial intelligence (AI), namely Bayesian networks. Probability was little used in AI prior to the 1980s, but that changed with the development of Bayesian networks, which are now the standard for handling reasoning with uncertainty. For example, one important development of this approach is the creation of expert systems, such as those used in aiding medical diagnosis.

Briefly, a Bayesian network is represented graphically by a set of nodes joined by links. Nodes represent variables of interest, and the links generally represent causal relationships between the variables of any two nodes. A simple example of a two-node network is a root node consisting of a variable V representing a virus that is (or is not) present in a patient, and a non-root node consisting of a variable T that indicates whether a test for the virus is positive or negative. Bayes’s theorem is used to relate the two nodes by calculating, for example, the probability that the virus is present given that the test is positive. A Bayesian network can consist of hundreds of nodes, and whenever new information is obtained the whole network of probabilities can be updated using Bayes’s theorem. For an introductory discussion of such networks, see Neapolitan and Jiang (2016).

The important point here is that for every root node one needs to assign a prior probability in order to use Bayes’s theorem initially, so the process of choosing priors is important in network construction. It is often claimed that the problem with Bayesian calculations is that the choice of a prior is subjective. As shown above, however, prior probabilities can be objective provided one understands what is meant by that term. Since objectivity as a method of thought is volitional and not automatic, any probability analysis may not be consistently objective, but the means of achieving objectivity is clear, namely by rational and logical analysis rather than by subjective introspection.

4. Conclusion

This article presented a review of the objective-subjective dichotomy, which has been used historically in a number of different ways to describe the nature of probability. In addition to giving the historical data, it has been shown that the various interpretations of probability can be classified by considering the dichotomy from two points of view: metaphysical and epistemological. The result is that probability theories can be divided into the following three classes (along with some prominent advocates): metaphysically objective (von Mises, Fisher), epistemologically objective (Keynes, Jeffreys, Jaynes), and epistemologically subjective (de Finetti, Savage), the last two classes being mutually exclusive and exhaustive subclasses of the metaphysically subjective class.

An advantage of this classification is that it eliminates any confusion sometimes seen in the literature regarding the use of the objective-subjective terms. For example, ‘objective’ has been used to describe the work of frequentists like von Mises and the work of Bayesians like Jeffreys, while ‘subjective’ has been used for classical probabilists like Laplace and for Bayesians like de Finetti and Savage. The classification also shows that a theory of probability can be both subjective and objective by being subjective metaphysically but objective epistemologically. Furthermore, by giving a formal definition of (epistemological) objectivity, the proper meaning of objective Bayesianism is suggested, overriding the unsupported claims of Bayesians referred to in the paper (Section 3).

Disclosure statement

No potential conflict of interest was reported by the author.

Notes on contributor

Arnold Baise is an independent scholar. Formerly a research chemist, he obtained a PhD in Physical Chemistry from the University of Wales before moving to the USA. There he worked for IBM, doing research on the materials used in making microelectronic components. After retiring, he became interested in studying probability and statistics, in particular the historical development of theories of probability.

Notes

1 The word ‘fictum’ was not translated by Spade. Words added in brackets are suggested replacements taken from alternative translations, see Ockham (1990, 41) and Karskens (1992, 214).

2 Adolph Friedrich Hoffmann (1707–1741) taught at the University of Leipzig and was a follower of the influential German philosopher Christian Thomasius (1655–1728). Their ideas are regarded as anticipating in some ways the philosophy of Kant. See Klemme and Kuehn (2010).

3 For details of Cournot’s analysis of probability, see Zabell (2011, 1154–1157).

4 Fisher (1956, 25) writes that Venn ‘was developing the concept of probability as an objective fact, verifiable by observations of frequency.’

5 An anonymous reviewer has supplied a relevant reference dealing with objective and subjective probability, in particular with regard to the British probabilists de Morgan, Mill, Boole, Ellis, and Venn (Verburgt 2015). Verburgt discusses in detail the difficulties involved in classifying their work from an objective or subjective point of view, which arise partly from their varying use of these terms. With regard to Venn, he argues that Venn was a frequentist but should not be regarded as advocating an objective probability.

6 The ‘hidden variables’ referred to by Jauch are variables that would allow one, in principle, to predict the future of a quantum system if only they could be accessed experimentally. In other words, a quantum system would then be regarded as deterministic and predictable, i.e. described by a metaphysically subjective probability.

7 This description of objective probability resembles Karl Popper’s propensity theory, which was intended, in part, to handle the interpretation of probability in quantum theory, see Gillies (2000, 113–136).