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Mathematical induction can be informally illustrated by reference to the sequential effect of falling dominoes. [1] [2]Mathematical induction is a method for proving that a statement () is true for every natural number, that is, that the infinitely many cases (), (), (), (), … all hold.
Golem is an inductive logic programming algorithm developed by Stephen Muggleton and Cao Feng in 1990. [1] It uses the technique of relative least general generalisation proposed by Gordon Plotkin, leading to a bottom-up search through the subsumption lattice. [2]
Inductive reasoning refers to a variety of methods of reasoning in which broad generalizations or principles are derived from a set of observations. [1] [2] Unlike deductive reasoning (such as mathematical induction), where the conclusion is certain, given the premises are correct, inductive reasoning produces conclusions that are at best probable, given the evidence provided.
In mathematical logic, geometric logic is an infinitary generalisation of coherent logic, a restriction of first-order logic due to Skolem that is proof-theoretically tractable. Geometric logic is capable of expressing many mathematical theories and has close connections to topos theory .
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.
Most recursive definitions have two foundations: a base case (basis) and an inductive clause. The difference between a circular definition and a recursive definition is that a recursive definition must always have base cases, cases that satisfy the definition without being defined in terms of the definition itself, and that all other instances in the inductive clauses must be "smaller" in some ...
Inductive logic started to take a clearer shape in the early 20th century in the work of William Ernest Johnson and John Maynard Keynes, and was further developed by Rudolf Carnap. Carnap introduced the distinction between pure and applied inductive logic, [ 1 ] and the modern Pure Inductive Logic evolves along the lines of the pure ...
Structural induction is a proof method that is used in mathematical logic (e.g., in the proof of Łoś' theorem), computer science, graph theory, and some other mathematical fields. It is a generalization of mathematical induction over natural numbers and can be further generalized to arbitrary Noetherian induction .
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