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.
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.
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.