Higher-order Predicate Logics
A formal language for higher-order predicate logic looks much the same as a formal language for first-order logic. The difference is that there are now many different types of variables. Some variables correspond to elements of the domain, as in first-order logic. Other variables correspond to objects of higher type: subsets of the domain, functions from the domain, functions that take a subset of the domain and return a function from the domain to subsets of the domain, etc. All of these types of variables can be quantified.
There are two kinds of interpretations commonly employed for higher-order logic. Full semantics require that, once the domain of discourse is satisfied, the higher-order variables range over all possible elements of the correct type (all subsets of the domain, all functions from the domain to itself, etc.). Thus the specification of a full interpretation is the same as the specification of a first-order interpretation. Henkin semantics, which are essentially multi-sorted first-order semantics, require the interpretation to specify a separate domain for each type of higher-order variable to range over. Thus an interpretation in Henkin semantics includes a domain D, a collection of subsets of D, a collection of functions from D to D, etc. The relationship between these two semantics is an important topic in higher order logic.
Read more about this topic: Interpretation (logic)
Famous quotes containing the words predicate and/or logics:
“The predicate of truth-value of a proposition, therefore, is a mere fictive quality; its place is in an ideal world of science only, whereas actual science cannot make use of it. Actual science instead employs throughout the predicate of weight.”
—Hans Reichenbach (1891–1953)
“When logics die,
The secret of the soil grows through the eye,
And blood jumps in the sun;
Above the waste allotments the dawn halts.”
—Dylan Thomas (1914–1953)