What is the conjunctive normal form of the boolean constant TRUE?

Question

I have the following problem:

Is TRUE (or 1) a logically equivalent formel in conjuctive normal form to a tautology?

How can I build the conjunctive normal form of TRUE if the output is always TRUE?

Answer 1

Here's a definition of a clause adapted from Merrie Bergmann's An Introduction to Many-Valued and Fuzzy Logic p. 20:

1. A literal (a letter or negation of a letter) is a clause.
2. If P and Q are clauses, then (P ∨ Q) is a clause.

Definition of conjunctive normal form.

1. Every clause is in conjunctive normal form.
2. If P and Q are in conjunctive normal form, then (P∧Q) is in conjunctive normal form.

So, "P" and "$\lnot$P" are both literals. Thus, by condition 2. of the definition of a clause, (P $\lor$ $\lnot$P) is a clause. So, by condition 1. of the definition of a clause, (P $\lor$ $\lnot$P) is in conjunctive normal form.

If ⊤ is a clause in your language, it also follows that ⊤ is in conjunctive normal form.

Answer 2

I don't think Bergmann's definition is quite right. Each variable should only appear once, so (P ∨ P ∨ P ∨ P), for example, is not allowed.

The empty conjunction is true, and the empty disjunction is false. But "" and "()" are clumsy to use, so normally the constants true and false are written explicitly.