Chapter: Equivalent Transformations in Propositional Logic

In rhetoric, the word tautology has a negative connotation. It is an argument, which simply rephrases the assertion in order to obfuscate a lack of evidence. In other words, a tautological argument is invalid and does not really support the stated conclusion.

In mathematical logic, a tautology has quite a different meaning than in rhetoric. We have seen that a string of a formal language which is always valid is called a tautology. In this sense, the word "tautology" has a positive connotation in mathematics, because - unlike in rhetoric - it is an always valid logical argument!

Tautologies in logic can be used to show that two logical statements are equivalent. In propositional logic in particular, $\phi$ and $\psi$ are equivalent propositions, if the compound proposition $\phi\Leftrightarrow \psi$ is a tautology. We will see that propositional logic can be embedded in most logical calculi, based on which mathematical theories are built. Therefore, the concepts of tautology and equivalence are important not only for propositional logic in particular but for logical calculi in general. For instance, all axioms of a given logical calculus are tautologies by definition. If in a given logical calculus, we want to prove that two theorems $\Phi$ and $\Psi$ are equivalent, then we usually do it by showing $\Phi\Leftrightarrow\Psi$ is a tautology.

The notions of Equivalence and tautology are very useful since they allow proving that propositions or theorems with different syntax have in fact the same semantics. We will now learn some important examples of equivalent propositions in propositional logic.

  1. Lemma: Every Proposition Implies Itself
  2. Lemma: It is true that something can be (either) true or false
  3. Lemma: Every Contraposition to a Proposition is a Tautology to this Proposition
  4. Lemma: De Morgan's Laws (Logic)
  5. Lemma: Distributivity of Conjunction and Disjunction

Chapters: 1

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  1. Mendelson Elliott: "Theory and Problems of Boolean Algebra and Switching Circuits", McGraw-Hill Book Company, 1982