Transformation rules |
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Propositional calculus |
Rules of inference |
Rules of replacement |
Predicate logic |
Absorption is a valid argument form and rule of inference of propositional logic.^{[1]}^{[2]} The rule states that if implies , then implies and . The rule makes it possible to introduce conjunctions to proofs. It is called the law of absorption because the term is "absorbed" by the term in the consequent.^{[3]} The rule can be stated:
where the rule is that wherever an instance of "" appears on a line of a proof, "" can be placed on a subsequent line.
Formal notation
The absorption rule may be expressed as a sequent:
where is a metalogical symbol meaning that is a syntactic consequence of in some logical system;
and expressed as 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:
where , and are propositions expressed in some formal system.
Examples
If it will rain, then I will wear my coat.
Therefore, if it will rain then it will rain and I will wear my coat.
Proof by truth table
T | T | T | T |
T | F | F | F |
F | T | T | T |
F | F | T | T |
Formal proof
Proposition | Derivation |
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Given | |
Material implication | |
Law of Excluded Middle | |
Conjunction | |
Reverse Distribution | |
Material implication |
See also
References
- ^ Copi, Irving M.; Cohen, Carl (2005). Introduction to Logic. Prentice Hall. p. 362.
- ^ http://www.philosophypages.com/lg/e11a.htm
- ^ Russell and Whitehead, Principia Mathematica