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In propositional logic, material implication [1] [2] is a valid rule of replacement that allows a conditional statement to be replaced by a disjunction in which the antecedent is negated. The rule states that P implies Q is logically equivalent to not- P {\displaystyle P} or Q {\displaystyle Q} and that either form can replace the other in ...
FOL is now a core formalism of mathematical logic, and is presupposed by contemporary treatments of Peano arithmetic and nearly all treatments of axiomatic set theory. The 1928 edition included a clear statement of the Entscheidungsproblem ( decision problem ) for FOL, and also asked whether that logic was complete (i.e., whether all semantic ...
Algebraic Methods in Philosophical Logic. Oxford University Press. ISBN 978-0-19-853192-0. Good introduction for readers with prior exposure to non-classical logics but without much background in order theory and/or universal algebra; the book covers these prerequisites at length. This book however has been criticized for poor and sometimes ...
This is a list of mathematical logic topics. For traditional syllogistic logic, see the list of topics in logic . See also the list of computability and complexity topics for more theory of algorithms .
Perhaps the most important achievement of abstract algebraic logic has been the classification of propositional logics in a hierarchy, called the abstract algebraic hierarchy or Leibniz hierarchy, whose different levels roughly reflect the strength of the ties between a logic at a particular level and its associated class of algebras. The ...
In predicate logic, universal instantiation [1] [2] [3] (UI; also called universal specification or universal elimination, [citation needed] and sometimes confused with dictum de omni) [citation needed] is a valid rule of inference from a truth about each member of a class of individuals to the truth about a particular individual of that class.
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]
The use of "Hilbert-style" and similar terms to describe axiomatic proof systems in logic is due to the influence of Hilbert and Ackermann's Principles of Mathematical Logic (1928). [ 2 ] Most variants of Hilbert systems take a characteristic tack in the way they balance a trade-off between logical axioms and rules of inference .