I have an assignment and it states that while $(p \vee q) \vee \lnot r$ is valid, $(p \vee q) \vee \lnot(\lnot r)$ is invalid in a logical clause, but I don't see why. A clause is described as being either a literal or a disjunction of literals, and a literal is either an atom or a negated atom.
Can anyone explain why the latter formula is invalid? (This question isn't the assignment.)
"First, in this language there are to be some unambiguously constituted sentences, whose internal structure we shall ignore (for our study of the propositional calculus) except for the purpose of identifying the sentences. We call these sentences prime formulas or atoms; and we denote them by capital Roman letters from late in the alphabet... [emphasis in original]" [1]
So, "r" is an atom and thus a literal. ¬r is a negated atom, and thus a literal. But, ¬¬r is a negated negated atom, and thus not literals. Since ¬¬r is not a literal, it thus doesn't appear in any clause.
[1]-Mathematical Logic p. 4-5. 1967 John Wiley and Sons. Stephen Cole Kleene.