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A theory about a topic, such as set theory, a theory for groups, [3] or a formal theory of arithmetic, is usually a first-order logic together with a specified domain of discourse (over which the quantified variables range), finitely many functions from that domain to itself, finitely many predicates defined on that domain, and a set of axioms ...
Developed in 1990 by Ross Quinlan, [1] FOIL learns function-free Horn clauses, a subset of first-order predicate calculus.Given positive and negative examples of some concept and a set of background-knowledge predicates, FOIL inductively generates a logical concept definition or rule for the concept.
The theory of finite groups is the set of first-order statements in the language of groups that are true in all finite groups (there are plenty of infinite models of this theory). It is not completely trivial to find any such statement that is not true for all groups: one example is "given two elements of order 2, either they are conjugate or ...
A set is called inductively defined if for some monotonic operator : (), () =, where () denotes the least fixed point of .The language of ID 1, , is obtained from that of first-order number theory, , by the addition of a set (or predicate) constant I A for every X-positive formula A(X, x) in L N [X] that only contains X (a new set variable) and x (a number variable) as free variables.
First-order language; First-order logic, a formal logical system used in mathematics, philosophy, linguistics, and computer science; First-order predicate, a predicate that takes only individual(s) constants or variables as argument(s) First-order predicate calculus; First-order theorem provers; First-order theory; Monadic first-order logic
In first-order ZFC set theory, quantification over predicates is not allowed, but one can still express induction by quantification over sets: (((+))) A may be read as a set representing a proposition, and containing natural numbers, for which the proposition holds. This is not an axiom, but a theorem, given that natural numbers are defined in ...
Inductive reasoning is any of various methods of reasoning in which broad generalizations or principles are derived from a body of observations. [1] [2] Inductive reasoning is in contrast to deductive reasoning (such as mathematical induction), where the conclusion of a deductive argument is certain, given the premises are correct; in contrast, the truth of the conclusion of an inductive ...
Inductive logic programming has adopted several different learning settings, the most common of which are learning from entailment and learning from interpretations. [16] In both cases, the input is provided in the form of background knowledge B, a logical theory (commonly in the form of clauses used in logic programming), as well as positive and negative examples, denoted + and respectively.