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Unformatted text preview: CHAPTER 6 CLASSICAL TAUTOLOGIES AND LOGICAL EQUIVALENCES We present and discuss here a set of most widely use classical tautologies and logical equivalences. We also discuss the definability of classical connectives and as a consequence, the equivalence classical propositional languages. 1 Implication One of the most frequently used classical tautologies are the laws of detachment for implication and equivalence. The implication law was already known to the Stoics (3rd century B.C) and a rule of inference, based on it is called Modus Ponens , so we use the same name here. Modus Ponens  = (( A ∩ ( A ⇒ B )) ⇒ B ) (1) Detachment  = (( A ∩ ( A ⇔ B )) ⇒ B ) (2)  = (( B ∩ ( A ⇔ B )) ⇒ A ) Mathematical and not only mathematical theorems are usually of the form of an implication, so we will discuss some terminology and more properties of implication. Sufficient Given an implication ( A ⇒ B ) , A is called a sufficient condition for B to hold. Necessary Given an implication ( A ⇒ B ) , B is called a necessary condition for A to hold. 1 Simple The implication ( A ⇒ B ) is called a simple implication . Converse Given a simple implication ( A ⇒ B ) , the implication ( B ⇒ A ) is called a converse implication . Opposite Given a simple implication ( A ⇒ B ) , the implication ( ¬ B ⇒ ¬ A ) is called an opposite implication . Contrary Given a simple implication ( A ⇒ B ) , the implication ( ¬ A ⇒ ¬ B ) is called a contrary implication . Each of the following pairs of implications: a simple and an opposite , and a converse and a contrary are equivalent, i.e. the following formulas are tautolo gies: Laws of contraposition (1)  = (( A ⇒ B ) ⇔ ( ¬ B ⇒ ¬ A )) , (3)  = (( B ⇒ A ) ⇔ ( ¬ A ⇒ ¬ B )) . The laws of contraposition make it possible to replace, in any deductive argu ment, a sentence of the form ( A ⇒ B ) by ¬ B ⇒ ¬ A ), and conversely. The relationships between all implications involved in the contraposition laws are usually shown graphically in a following form, which is called the square of opposition . ( A ⇒ B ) ( B ⇒ A ) ( ¬ A ⇒ ¬ B ) ( ¬ B ⇒ ¬ A ) Equivalent implications are situated at the vertices of one and the same diag onal. It follows from the contraposition laws that to prove all of the following implications: ( A ⇒ B ), ( B ⇒ A ), ( ¬ A ⇒ ¬ B ), ( ¬ B ⇒ ¬ A ), it suffices to prove any pairs of those implications which are situated at one and the same side of the square, since the remaining two implications are equivalent to those already proved to be true. Consider now the following tautology:  = (( A ⇔ B )) ⇔ (( A ⇒ B ) ∩ ( B ⇒ A ))) . (4) 2 Necessary and sufficient The above tautology 4 says that in order to prove a theorem ( A ⇔ B ) it suffices to prove two implications: the simple one ( A ⇒ B ) and the converse one ( B ⇒ A ). Conversely, if ( A ⇔ B ) is a theorem, then the implications ( A ⇒ B ) and ( B ⇒ A ) are also theorems.) are also theorems....
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 Spring '08
 Bachmair,L
 Computer Science, Logic, ¬b, ¬a, A1 ⇔ B1

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