**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.

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. WOLI offers immigration law course online - fully accredited. ACE credits online at EES.

**Aristotle****Camus****Descartes****Frege****Heidegger****Hobbes****Hume****Kant****Kierkegaard****Locke****Nietzsche****Plato****Quine****Russell****Sartre****Socrates****Wittgenstein**- All Philosophers…

**Aesthetics****Epistemology****Logic****Ethics****Metaphysics****Language****Mind****Politics****TEST PREP KIT**- All Terms & Topics…

**Proof by contradiction**, or *reductio ad absurdum* is an argument against a proposition or formula in logic which shows that the formula in question results in a contradiction.

In a proof by contradiction, a negation is obtained by proving that its opposite formula creates a contradiction. The method of proof by contradiction is used for both the **negation introduction** (¬I) and **negation elimination** rules of inference, and explains that if "P" results in some contradiction, then ¬P must be true.

Proof by contradiction may be formally presented as follows:

**(α → ⊥) ¬α**

In this example, the symbol ⊥ is used to indicate a contradiction. Alternatively, we may formulate this by showing an actual contradiction of two formulas:

**(α → (β ∧ ¬β)) ¬α**

These examples show the negation introduction (¬I) version of the proof by contradiction, which introduces a negation. The negation elimination (¬E) version is the inverse:

**(¬α ⊥) α**

When beginning with a negation, we have the choice of choosing to conclude either α (by ¬E) or ¬¬α by (¬I), since α and ¬¬α are equivalent, according to the double negation rule.