An addititional set of rules

A very important but less obvious rule is distribution.  For instance, if we say that either logic is fun or it is both dull and useless, it would be the same as saying that logic is either fun or dull and also that it is either fun or useless.

We can also work with conditionals by understanding that denying a consequent ensures denying an antecedent.  This is what we call contraposition.

A very helpful equivalence rule takes the idea of a conditional and rethinks it as a disjunction.  For instance, if we say that either logic is easy or it will not be fun, it is the same as saying that being easy is a  necessary condition for logic being fun--logic is fun only if it is easy.

We also need to think through what we mean when we negate a conjunction or a disjunction.  The English mathematician Augustus DeMorgan pointed out one of the most useful relationships here, and we still use his name to refer to it (De Morgan's law)

Finally, we will bundle together the things that happen when we negate a quantifier.  The idea here is that if we deny everyone is funny, then there must be someone who is not funny.  In the same way if we deny that there is someone smart, then no one is smart.

In the application of these equivalence or substitution rules we will treat double negation as built in.  For instance, if we are working with the expression ~P -> Q we may substitute P v Q instead of first writing ~~P -> Q and then using the rule of double negation.  We will, however, take the extra step in working with the inference rules we met earlier in order to avoid certain very common errors.  (It is also to avoid a beginner's mistake that we do not simply use a rule called modus tollens to go with modus ponens.)
 

PRACTICE EXERCISES