Monday, March 31, 2014

Logical foundations of mathematics

The object of mathematics in general is to reason under conditions of certainty. This involves the consideration of logical propositions with truth values such as true and false. With truth values such as true and false we can form predicates that return such truth values and an ontology of such predicates.

With predicates such as sets we can describe every object encountered in mathematics. And with an ontology we can get a clear picture of all these different types of objects. Then we can use logic and our ontology to reason about mathematical objects. By combining certain mathematical knowledge with logical models of uncertainty domains we can use logic to describe all declarative knowledge.

Wednesday, March 26, 2014

Logical models of uncertain domains

Well the set theory and logical predicate calculus provide a good foundation for understanding certain abstract domains we need to extend our logic to be able to deal with uncertainty. Given a system with unknown characteristics then we can create a model of that domain. Models classify uncertain domains by allowing us to apply predicates. Here are two of the simplest kinds of models:
  • Simple models: the simplest possible model of an uncertain domain is a set of values the domain may take. Applying a predicate to such a model yields true if the predicate is a subclass of the set of values of model, false it if it is independent of the set, and unknown otherwise.
  • Probabilistic models: a more advanced type of model is one in which applications of a predicate return probability values. Probabilities should be higher the more general a predicate is so that the probability that an object is an entity is always one.
Given any system with uncertain characteristics we can create a logical model of the system. If our agent is embodied within some environment then it can create a world model of the entire environment including it. If that environment is our physical reality then a world model containing all physical entities and their mereological and causal properties can be produced. In this way logic can be used to organize all declarative knowledge about real or imagined entities.