To install click the Add extension button. That's it.

The source code for the WIKI 2 extension is being checked by specialists of the Mozilla Foundation, Google, and Apple. You could also do it yourself at any point in time.

4,5
Kelly Slayton
Congratulations on this excellent venture… what a great idea!
Alexander Grigorievskiy
I use WIKI 2 every day and almost forgot how the original Wikipedia looks like.
Live Statistics
English Articles
Improved in 24 Hours
Languages
Recent
Show all languages
What we do. Every page goes through several hundred of perfecting techniques; in live mode. Quite the same Wikipedia. Just better.
.
Leo
Newton
Brights
Milds

# Tautology (rule of inference)

In propositional logic, tautology is either of two commonly used rules of replacement.[1][2][3] The rules are used to eliminate redundancy in disjunctions and conjunctions when they occur in logical proofs. They are:

The principle of idempotency of disjunction:

${\displaystyle P\lor P\Leftrightarrow P}$

and the principle of idempotency of conjunction:

${\displaystyle P\land P\Leftrightarrow P}$

Where "${\displaystyle \Leftrightarrow }$" is a metalogical symbol representing "can be replaced in a logical proof with."

• 1/3
Views:
43 448
3 239
11 915
• Logic Lesson 6: Proofs with the Rules of Inference
• Proving a compound proposition is a tautology part 2 of 2
• CS101 - Discrete Mathematics - Rules of Inference (রুলস অব ইনফারেন্স)

## Formal notation

Theorems are those logical formulas ${\displaystyle \phi }$ where ${\displaystyle \vdash \phi }$ is the conclusion of a valid proof,[4] while the equivalent semantic consequence ${\displaystyle \models \phi }$ indicates a tautology.

The tautology rule may be expressed as a sequent:

${\displaystyle P\lor P\vdash P}$

and

${\displaystyle P\land P\vdash P}$

where ${\displaystyle \vdash }$ is a metalogical symbol meaning that ${\displaystyle P}$ is a syntactic consequence of ${\displaystyle P\lor P}$, in the one case, ${\displaystyle P\land P}$ in the other, in some logical system;

or as a rule of inference:

${\displaystyle {\frac {P\lor P}{\therefore P}}}$

and

${\displaystyle {\frac {P\land P}{\therefore P}}}$

where the rule is that wherever an instance of "${\displaystyle P\lor P}$" or "${\displaystyle P\land P}$" appears on a line of a proof, it can be replaced with "${\displaystyle P}$";

or as the statement of a truth-functional tautology or theorem of propositional logic. The principle was stated as a theorem of propositional logic by Russell and Whitehead in Principia Mathematica as:

${\displaystyle (P\lor P)\to P}$

and

${\displaystyle (P\land P)\to P}$

where ${\displaystyle P}$ is a proposition expressed in some formal system.

## References

1. ^ Hurley, Patrick (1991). A Concise Introduction to Logic 4th edition. Wadsworth Publishing. pp. 364–5.
2. ^ Copi and Cohen
3. ^ Moore and Parker
4. ^ Logic in Computer Science, p. 13