Understanding Math Logic and Correct Reasoning Principles

by

Dennis McLynn

Math logic is the systematic study of principles of valid inference and correct reasoning. Logic is a form of mathematics that is not only used for mathematical purposes, but is also commonly used in philosophy, semantics and computer science. It examines the common forms of arguments to distinguish between forms that are valid and fallacies. Philosophers use logic to study epistemology, ethics and metaphysics. Mathematicians use logic to study valid inferences within formal language, as well as in the argument theory.

Logic was first established by Aristotle. Aristotle made logic a fundamental part of philosophy by establishing it as one of his disciplines. He made logic part of the classical trivium. Mathematical logic is divided into the fields of set theory, model theory, recursion theory and proof theory. The set theory studies sets, or collections of objects. It examines the binary relation between a set of objects. The model theory examines mathematical structures using mathematical logic. The mathematical structures examined using the model theory are models for formal languages and the structures that give meaning to the sentences of the formal languages. The model theory itself is very similar to algebra in form and function. The recursion theory, also known as the computability theory studies computable functions and Turing degrees. The recursion theory addresses reasoning behind functions and natural numbers. This form of math logic is very similar to computer science and is commonly used in a variety of computer science careers. The proof theory is the study of proofs as formal mathematical objects. Proofs are presented as data structures in the form of plain lists, box lists, or trees. Tree lists are constructed according to the form of axioms and the rules of inference relating back to the logical system. The proof theory, along with the model theory, the axiomatic set theory and the recursion theory, make the four pillars of the foundations of mathematics. All of the fields of mathematical logic share the basic ideas of first-order logic and definability. First-order logic is a formal logic system to deal with simple declarative propositions, predicates and quantification. It is a deductive system frequently used in philosophy. Definability functions in mathematical logic as a definable set. The definable set is an n-ary relation on the domain of a structure whose elements are the elements that satisfy a formula in the language of the designated structure. These sets are specific because they are not limited to parameters. Understanding mathematical logic and correctly recognizing principles allows mathematicians, philosophers, and people serving a variety of careers to deductively reason between valid and false points.

Dennis McLynn is the Vice President of Strategic Marketing & Business Development for High Points Learning. HighPoints Learning (HPL) is a leader in Web-based math education and instruction. HPL offers an

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program that helps raise participants’ math scores an average of 15 points in pre and post testing. HighPoints Learning services the 3-12 grade market.

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Understanding Math Logic and Correct Reasoning Principles