Inference Calculus Research Articles

Lattice duality: The origin of probability and entropy

Bayesian probability theory is an inference calculus, which originates from a generalization of inclusion on the Boolean lattice of logical assertions to a degree of inclusion represented by a real number. Dual to this lattice is the distributive lattice of questions constructed from the ordered set of down-sets of assertions, which forms the foundation of the calculus of inquiry—a generalization of information theory. In this paper we introduce this novel perspective on these spaces in which machine learning is performed and discuss the relationship between these results and several proposed generalizations of information theory in the literature.

An Overlooked Type of Inference

A Detailed analysis of single statements corroborates more and more our conviction that the elements of Hindu mentality (viz. philosophy, religion, and fine arts) are subject to certain fixed and common rules of thinking. If we acknowledge any one analytical method—of course the most general possible—as sufficient and adequate, we can presume that whatever may at any time be the object of our analysis must follow the method adopted. If we accept as a principle for the veracity of all judgments that they must be subjected to the rule of sapakṣe sattva and vipakṣe asattva, then the analysis will show that in reality all statements are measurable under the aspect of those two criteria. The only breach in this principle was made by a Jaina logical school which, while anticipating the principles of our “implication”, admits the syllogism fulfilling the anyathānupapannatva condition, i.e. it accepts as true conditional sentences of which the protasis does not reach beyond the sphere of the predicated subject (pakṣa). In other words: the argument, when predicating a fact, forms a true sentence, even if it does not predicate the class to which the fact belongs. Whilst the Buddhist syllogism oscillates between class inference and sentence calculus (sapakṣa and vipakṣa being the necessary conditions), the Jaina syllogism advances exclusively the sentence calculus, and the validity of the predication is confined merely to the implication in question.