Remember that we use lower case letters to refer to the form of a wff, rather than to express a particular proposition.  So the conditional in the first line of the modus ponens rule, stands for any conditional.  "p" in the second line stands for any wff which is identical to the antecedent of the conditional, and "q" stands for the consequent of the conditional. So the rule says that whenever we have a derivation where one line is  a conditional, and another line (not necessarily the next line) is the antecedent of that conditional, we can infer, or write down as a new line, the consequent of that conditional.

The rules of inference must be applied correctly.  Modus ponens allows us to infer the consequent of a conditional when we have the conditional on one line and the antecedent of that same conditional as a separate wff on a separate line.  If we had a conditional on one line and the consequent of that conditional on the other, we may not infer the antecedent of the conditional by modus ponens. 

Notice that there are two "versions" of simplification.  Given a conjunction and the left conjunct, we can infer the right conjunct, and given that same conjunction and the right conjunct, we can infer the left conjunct.

Another rule of inference for conjunction goes the other way: The rule called "conjoining" allows us to form the conjunction of any two lines of a derivation:

Compare the two rules we've introduced for conjunctions.  Simplification takes conjunctions apart. Conjoining builds conjunctions.  Simplification can be applied to any one line of a derivation where the main operator of the wff on that line is "&".  Conjoining can be applied to any lines of a derivation. We simply conjoin them.

There are two rules for disjunction as well. Like conjunction, our pair of disjunction rules take us from a disjunction to a disjunct, and the other enables us to infer a disjunction. But there the similarities end. Our first disjunction rule, called "disjoining" allows us to infer a disjunction from any wff on any line of a derivation:

There's no restriction on what can be disjoined to a wff. When we infer (p v q) from p, for example, q need not even occur anywhere in prior in the derivation!

We said that our rule of inference for inferring a disjunct from a disjunction will be different from our rule for obtaining a conjunct from a conjunction. It has to be different, otherwise our proof theory would treat conjunctions and disjunctions as equivalent!  Our rule for obtaining a disjunct from a disjunction is this

This rules says: If you have a disjunction on one line of a derivation, and the negation of one of its disjuncts on another line, you can infer the other disjunct.

Threemore rules complete our rules of inference. Like modus ponens, they all involve conditionals.  The rule "modus tollens" infers the negation of the antecedent of a conditional from a conditional together with the negation of the consequent of that conditional:

The next rule is called "Hypothetical Syllogism." The term "syllogism" was used by philosophers and logicians in ancient times to describe a certain class of arguments.  Hypothetical syllogism is so called because it involves conditionals. Here's the rule:

For HS we need two conditionals related in the following way: The consequent of one conditional is the antecedent of the other. When we have two such conditionals, we can infer the conditional whose antecedent is the antecedent of the first conditional and the consequent of the second.

Last but not least is a rule of inference called "dilemma."  This rule may appear the most complicated, but it is also not used very often.  Like all the rules, once you've studied it for a while, it will make more sense.  

Let's look more closely at this rule. Unlike our others, the inference is from three wffs. We need two conditionals, and a disjunction. But not any old disjunction will do. The disjunction is a disjunction of the antecedents of the two conditionals.  From these three wffs we infer a disjunction of the consequents of the two conditionals. 

We've covered a lot of material in this section.  The best way to learn the material is by constructing derivations and proofs.  Once you've begun to do that, you may want to return to this section to refresh your understanding of the rules of inference, or go to the summary of the rules.  In addition, you can do an exercise which will test your recall of the rules.