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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.
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.
The inductive definition can also be rephrased in terms of a closure operation (Enderton 2002). Let V denote a set of propositional variables and let X V denote the set of all strings from an alphabet including symbols in V , left and right parentheses, and all the logical connectives under consideration.
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 .
Pure inductive logic (PIL) is the area of mathematical logic concerned with the philosophical and mathematical foundations of probabilistic inductive reasoning. It combines classical predicate logic and probability theory ( Bayesian inference ).
First-order logic—also called predicate logic, predicate calculus, quantificational logic—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantified variables over non-logical objects, and allows the use of sentences that contain variables.
Mathematical logic, also called 'logistic', 'symbolic logic', the 'algebra of logic', and, more recently, simply 'formal logic', is the set of logical theories elaborated in the course of the nineteenth century with the aid of an artificial notation and a rigorously deductive method. [5]
In intuitionistic type theory (ITT), a discipline within mathematical logic, induction-induction is for simultaneously declaring some inductive type and some inductive predicate over this type. An inductive definition is given by rules for generating elements of some type. One can then define some predicate on that type by providing ...