Leibniz and the propensity interpretation of probability

The point of focus here are the propensity interpretation of probability theory, in which probabilities are physical tendencies that cause events. Contemporary interest in the interpretation is down to Karl Popper and been picked up by Mellor. It is now playing a role in the dispositional metaphysics of objective chance. The origins and initial philosophical discussion of probability can be traced to Pascal and Leibniz and, it is argued, something close to the propensity interpretation attributable to Leibniz too. This role of Leibniz came as surprise on reading The Emergence of Probability by Ian Hacking. Hacking presents Leibniz as the first philosopher of probability and principal guide to the early development of the theory. In addition, as is often the case, the thinking of Leibniz was ahead of his own epoch and many of his points can best be appreciate only following developments in the 20th century.

The origin of probability as a useful science is primarily attributed to Blaise Pascal (1623-1662) and Pierre de Fermat (1601-1665) in a correspondence motivated by a request from Chevalier de Méré for mathematical guidance on games of chance. The answer that Pascal and Fermat developed is that Probability Theory is built upon a fundamental set of equally likely outcomes. This approach is somewhat circular but can be interpreted as an argument based on symmetry and this leads Leibniz naturally to the to the argument from indifference in the interpretation of the theory of probability.

The principle of indifference can take various forms:

• If there is no reason that one event or outcome will happen more often than an other then they are equiprobable
• If there is no reason to prefer the outcome of one event over another then they are equiprobable
• If it is believed that one event will be no more likely to happen than another then they are equiprobable.

The interpretations of probability that derive from Pascal’s principle of symmetry (or equally likely cases) must be distinguished from the logical interpretation. Like so much of his work, most of Leibniz's thoughts on the relationship between probability and logic were not published in his lifetime, however there are important letters. Here as a representative and published quotation from Nouveaux Essais sur l’entendement humain

J’ay dit plus d’une fois qu’il faudroit une nouvelle espece de Logique, qui traiteroit des degrés de probabilité, . . .(I have said more than once that there is need of new type of logic, which will deal with the degrees of probability ...)

Leibniz was more optimistic that this can be done than Locke, who viewed it as “impossible to reduce to precise rules the various degrees wherein men give their assent.” Leibniz believed that a logical analysis of conditional implication would yield such rules, however, this is still considered problematic. The relationship that he saw here was that probability is useful when there is insufficient knowledge to make a rigorous deduction. Leibniz and his logical approach began from legal considerations (he trained as a lawyer), where there is uncertainty in the determination of a question of right (e.g., to property) or guilt. His approach is also important for the emphasis that conditional or relational probabilities are fundamental.

As a young man of 19, Leibniz published a paper proposing a numerical measure of proof for legal cases: “degrees of probability.” His goal was to render jurisprudence into an axiomatic deductive system akin to Euclidean geometry. So, the goal was to transform evidence (a legal notion), into something to be measured by some allocation of weight that will make calculation of justice possible. However, he was also convinced that there had to be an objective and correct situation. From this he developed a dual interpretation of probability:

• Epistemic - dealing with uncertainty due to lack of knowledge
• Objective - dealing with the degree feasibility for possibilities to be physically realised.

The epistemic view came first, initially dominated and its conditional character was inherited through its emergence from legal considerations. A bridge is provided from the legal term cases or casu in latin which also means events. Events happen and are part of the standard terminology in modern probability theory. Another term, important across Leibniz's philosophy, is possibility. Leibniz associated equi-possibility with probability and asserted that probability is a degree of possibility. Here he means by possibility the power to achieve various events. In a letter to Bourguet (in Die Philosophischen Schriften von Gottfried Wilhelm Leibniz: Band 3 ed Gerhardt) Leibniz states:

L'art de conjecturer est fondée sûr ce qui est plus ou moins facile, ou bien plus ou moins faisable ... (The art of conjecture is founded on that which is more or less easy or, better, more or less feasible ...)

So, there are now degrees of feasibility that are not dependent on any state of knowledge. However, these degrees of feasibility, propensities, objective possibilities can be themselves objects of knowledge.

Leibniz distinguishes epistemic probability that some possibility is realised and the physical, objective or ontological propensity for some possibility to exist. The relationship between the two is still problematic. For Leibniz, every possibility tends to exist and so every possible world has its tendency to exist to a degree that depends on its feasibility. Leibniz had access to a metaphysical synthesis that provides important insights even if we cannot subscribe to it.