A Consequence of Classical Negation

In standard propositional logic, there is a theorem which follows from no premises that can cause some controversy. So-called prelinearity states that, given any three arbitrary propositions, either the first implies the second or the second implies the third, i.e., (P implies Q) or (Q implies R). For example, either "I'm sitting at my computer" implies "the Moon is made of green cheese" or "the Moon is made of green cheese" implies "the Sun rises in the west". I'm sure you, the reader, can come up with many other absurd examples of prelinearity. However, the absurdity of it follows from a seemingly innocuous principle of standard propositional logic: classical negation.

There are four different forms that classical rules of negation take: the excluded middle, double negation elimination, dilemma, and classical reductio. The excluded middle states that there are only two truth-values for any proposition, true or false; that is, (P or ~P). Double negation elimination states that if it is false that some propositions is false then that proposition is true; that is, (~~P implies P). Dilemma states that if a proposition follows from both some other proposition and the second proposition's negation, the original proposition in question is true; that is, (((P implies Q) and (~P implies Q)) implies Q). Classical reductio states that if you can prove a contradiction from the negation of a proposition, that proposition is true; that is, (~P implies an absurdity) implies P. These rules of classical negation follow from the truth-table conception of the meanings of logical operators. An alternative approach, introduction-rule semantics (where the meanings of logical operators are determined by their associated introduction-rule), does not license the classical rules of negation.