23 Understanding Propositional Logic and the second inference (although the statements are not really propositions) on p, q , r both of which are clearly unsound To summarize, the meaning and importance of logically correct inferences is that only such inferences guarantee that if all premises are true, then the conclusions will also be true That is why only logically correct inferences are safe to be employed in our reasoning Let us now look at a few more examples The inference rule q, p ∨ ¬q p is logically sound You can check this in two ways: by applying the definition or by showing that the corresponding formula (q ∧ (p ∨ ¬q )) → p is a tautology Consequently, the inference Olivia is crying or Olivia is not awake Olivia is awake Olivia is crying is logically correct, being an instance of the rule above Now, take the argument If a divides b or a divides c, then a divides bc a divides bc a does not divide b Therefore a divides c where a, b, c are certain integers This argument is an instance of the following inference rule: (p ∨ q ) → r, r, ¬p q Let us see if we can invalidate this rule For that we need an assignment such that ((p ∨ q ) → r) : T, r : T, and ¬p : T, hence p : F and q : F Indeed, the assignment p : F, q : F, and r : T renders all premises true and the conclusion false (Check this.) The rule is therefore not logically sound, hence the argument above is not logically correct I develop the method behind the last argument above in the next chapter 1.2.3 Fallacies of the implication As an application let us analyze some very common forms of correct and incorrect reasoning related to implications Given the implication A→B