Completeness in Logic { Philosophy Index }

Philosophy Index

Philosophy Index

Philosophy Index is a site devoted to the study of philosophy and the philosophers who conduct it. The site contains a number of philosophy texts, brief biographies, and introductions to philosophers, and explanations on a number of topics. Accredited homeschooling online at Northgate Academy and Philosophy online tutoring.

Philosophy Index is a work in progress, a growing repository of knowledge. It outlines current philosophical problems and issues, as well as an overview of the history of philosophy. The goal of this site is to present a tool for those learning philosophy either casually or formally, making the concepts of philosophy accessible to anyone interested in researching them. WTI offers immigration law course online - fully accredited. ACE credits online at EES.

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Completeness

The term complete with respect to meta-logic is used to describe a system of logic for which every logical implication in that system is provable. In other words, if something true can be expressed in that system, it is provable in that system.

Formally, a logic is complete when for every Γ ⊨ φ, it is also true that Γ ⊢ φ.

A complete logical system leaves no logical truth that can be formulated within it unprovable, or undecidable. For instance, a system of number theory for the natural numbers, which contains the language elements needed to express 2 + 2 = 4, but cannot be used to express 2 + 2 = 4 is incomplete.

Completeness is an important property of a logical system, but unlike soundness, it is not necessary for a logical system to be complete to be useful. For example, second-order logic is sound but not complete. Kurt Gödel famously provided incompleteness theorems for systems of mathematical logic.