Logic of a higher nature
The logical of a higher nature are formal logics which extend the Calcul of the first order predicates while making it possible to use the variables in the terms as functions, and in the expressions as predicates.
From a semantic point of view, in other words, one regards the functions and predicates as objects with whole share, as well as, for example, an integer. One will be thus authorized, on the one hand, to quantify the predicates and functions and, on the other hand, to give functions or predicates in arguments of other functions and predicates. Nevertheless, one will be able to obtain a system of Typage which will restrict the kind of object which could be given as such or such argument of such or such predicate or such or such function.
A predicate of a higher nature is a predicate which takes as argument one or more other predicates. In a general way, a predicate of order N takes as argument one or more predicates of order n-1, with N > 1. The same thing is valid for the functions of a higher nature.
The Lambda-calculation S typified, like the Calculation of constructions, what is called take as a starting point such logics in the functional paradigm. A strong bond is woven between mathematics and data processing thanks to the Isomorphisme of Curry-Howard which associates a lambda-calculation with a logic. It is of this field that the languages from functional Programmation result.
Logic of the second order
The logic of the second order extends that of the first order by the addition of functional variables, which can thus be quantified. For example, .
Monadic logic of the second order is restricted with the unary functional variables, which are in fact the Ensemble S. It makes it possible to have results of Décidabilité more interesting than to the first order: for example, the monadic theory of a Ordinal Dénombrable is décidable.
Third order logic
It uses objects of third type, like the filters or Ultrafiltre S, but is seldom mentioned as such.
See too
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