In this section we enlarge our list of "standard" tautologies by adding ones involving the conditional and the biconditional. From now on, we use small letters like p and q to denote atomic statements only, and uppercase letters like A and B to denote statements of all types, compound or atomic.
We first look at some tautological implications, tautologies of the form AB. You should check the truth table of each of the statements we give to see that they are, indeed, tautologies.
Modus Ponens or Direct Reasoning
In words: If p implies q, and if p is true, then q must be true. Example
Another way of setting this up is in the following argument form:^{}
In symbols:

Notice that we draw a line in the argument form to separate what we are given from the conclusion that we draw. This tautology represents the most direct form of everyday reasoning, hence its name "direct reasoning." Another bit of terminology: We say that pq and p together logically imply q.
To check that it is a tautology, we use a truth table.
Once more, modus ponens says that, if we know that p implies q, and we know that p is indeed true, then we can conclude that q is also true. This is sometimes known as affirming the hypothesis. You should not confuse this with a fallacious argument like: "If I were an Olympic athlete then I would drink Boors. I do drink Boors, therefore I am an Olympic athlete." (Do you see why this is nonsense?) This is known as the fallacy of affirming the consequent. There is, however, a correct argument in which we deny the consequent:
Modus Tollens or Indirect Reasoning
[(pq)~q]~p In words, if p implies q, and q is false, then so is p. Example If I love math then I will pass this course; but I know that I will fail it. Therefore, I must not love math. In argument form:
In symbols:

As you can see, this argument is not quite so direct as that in the first example; it seems to contain a little twist: "If p were true then q would also be true. However, q is false. Therefore p must also be false (else q would be true.)" That is why we refer to it as indirect reasoning.
We'll leave the truth table for the exercises. Note that there is again a similar, but fallacious argument form to avoid: "If I were an Olympic athlete then I would drink Boors. However, I am not an Olympic athlete. Therefore I can't drink Boors." This is a mistake Boors sincerely hopes you do not make!
More tautoligical implications:
Simplification
In words, the first says: If both p and q are true, then, in particular, p is true. Example Argument Form
In symbols:
The other simplification, (pq)q is similar. 
Addition
In words, the first says: If p is true, then we know that either p or q is true. Example Argument Form
In symbols:
Notice that it doesn't matter what we use as q, nor does it matter whether it is true or false. The reason is that the disjunction pq is true if at least one of p or q is true. Since we start out knowing that p is true, the truth value of q doesn't matter. 
Warning
The following are not tautologies:
p(pq).
In the exercise set, you will be asked to check that these are not tautologies.
Here are the last two tautoligical implications we shall look at.
Disjunctive Syllogism or OneortheOther
[(pq)(~q)]p Example Argument Form
In symbols:

Transitivity
Example Argument Form
In symbols:

The next batch of tautologies are tautological equivalences, which are tautologies of the form AB. Recall that the statement AB is true exactly when A and B have the same truth value. When A and B are compound statements, this must be true for all truth values of the atomic statements used in A and B. This means that A and B are logically equivalent statements.
Logical Equivalence and Tautological Equivalences
A tautological equivalence has the form AB, where A and B are (possibly compounbd) statements that are logically equivalent. In other words, to say that AB is a tautology is the same as saying that A B. 
So, every logical equivalence we already know gives us a tautological equivalence. Here are some examples. We give lots more in the table at the end of the section.
Double Negation
This is just the Double Negation Law p~(~p). In argument form, we can express this in two ways using argument form: Argument Form
Commutativity
This is just the commutativity equivalence pqqp. Argument Form

We conclude this section with a summary list of important tautologies.
1. [(pq)p]q 

(Direct Reasoning) 

2.[(pq)~q]~p 

(Indirect Reasoning) 

3. (pq)p
(pq)q 


4. p(pq) 


5. [(pq)(~p)]q
[(pq)(~q)]p 

(OneortheOther) 

6. [(pq)(qr)](pr) 

B. Tautological Equivalences
1. p~(~p) 


2. pqqp
pqqp 


3.
(pq)rp(qr)
(pq)rp(qr) 


4.
~(pq)(~p)(~q)
~(pq)(~p)(~q) 


5.
p(qr) (pq)(pr) p(qr) 


6. ppp
ppp 


7. (pq)((~p)q) 


8. (pq)(~q~p) 


9. (pq) ((pq)(qp)) 
