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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.



Philosophy Topics





The word therefore and its associated symbol ( ∴ ) are used in logic to indicate a relationship of logical consequence, whether that be a syntactic provability, semantic implication or merely intended entailment.

To say Γ ∴ φ is to say that a set of formulae, Γ, logically entails some other formula, φ. It may mean that φ is provable from Γ, that φ follows from Γ, or both. It may also mean that φ is the expected consequence of Γ, or that φ follows from Γ non-deductively.

Provability, or syntactic conseqence

The relationship of provability, Γ ⊢ φ refers to situations where a proof can be constructed in a logical system to demonstrate that, for reasons of syntax, Γ ∴ φ.

Implication, or semantic consequence

The relationship of implication, Γ implies φ refers to situations where we can semantically demonstrate Γ ∴ φ because φ actually follows from the premises of Γ.

Intended entailment

Unlike the fairly strict relations of provability and implication, the symbol ∴ is sometimes used by logicians to present arguments or argument forms that have not yet been proven either syntactically or semantically.

Similarly, the symbol ∴ may be used to note conclusions of non-deductive reasoning, where neither implication nor provability appropriately describe the relationship between the premises and the conclusion.