Tautology (rule of inference)

Short description: Commonly used rules of replacement in propositional logic

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."

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. ISBN 9780534145156.
2. Copi and Cohen
3. Moore and Parker
4. Logic in Computer Science, p. 13