His main achievements were his theory of the logical interrelation of affirmative and negative existential and universal statements and, based on this theory, his syllogistic, which can be interpreted as a system of deductive inference.
(cf. the section on the axioms of ZFC in the entry on set theory) Obviously, it is assumed that our formal systems are also equipped with a system of rules of inferences (and possibly some logical axioms), usually some standard system of classical logic (though the incompleteness theorems do not essentially presuppose classical logic, but also apply to systems with, e.g., intuitionistic logic).
The key innovation is a 'logic of orderly pairing': a system of natural-deduction rules of inference governing the formation of the ordered pair π t,u) from existing objects t and u, and the projections λ u) of the left, and ρ u) of the right, member of any ordered pair u.
Ideally, validity and entailment for L can be precisely captured in a system of axioms and rules of inference, in the sense that, for any given formula φ of L, (i) φ is valid if and only if it is a theorem of the system, and (ii) for any set Γ of formulas of L, Γ entails φ if and only if φ can be derived from Γ in the system.
Roughly, a formal system is a system of axioms equipped with rules of inference, which allow one to generate new theorems.
Axiomatic set theorems are the axioms together with statements that can be deduced from the axioms using the rules of inference provided by a system of logic.
In contrast, fuzzy inference system (FIS) is a quite capable modeling tool to overcome the limitations of ANN and ANFIS models as it is based on fuzzy logic and can simply convert the experience of experts to a set of inference system based rule and is focused on modes of reasoning which are approximate rather than exact.
The modular detection systems feed output to a central system, which is a kind of inference engine.
Patterns may represent images, sound data, data describing the input and output of a control system, the inference rules of an expert system and, in general, any ordered set of data.
The system has a comprehensive set of inference rules to reduce the number of facts stored in the knowledge-base because a considerable number of facts, which otherwise would have to be stored explicitly, can be derived by rules with some extra effort.
