# Introduction to Logic

## 4. Tautological Implications and Tautological Equivalences

### Tautological Implications

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

[(pq)p]q.

In words: If p implies q, and if p is true, then q must be true.

Example
Letting p: "I love math" and q: "I will pass this course," we get

If my loving math implies that I will pass this course, and if I indeed love math, then I will pass this course.

Another way of setting this up is in the following argument form:

 If I love math, then I will pass this course. I love math. Therefore, I will pass this course.

In symbols:

 pq p q

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.

 p q pq (pq) p [(pq) p] q T T T T T T F F F T F T T F T F F T F T

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 we once again take p: "I love math" and q: "I will pass this course," we get

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:

 If I love math, then I will pass this course. I will fail the course. Therefore, I do not love math.

In symbols:

 pq ~q ~p

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!

### Example 1 Practice with Direct and Indirect Reasoning

Use either modus ponens or modus tollens to draw the appropriate conclusion.
 If I go to Mars, I will run for office. I am going to Mars. Therefore, Select one I am going to Mars. I am not going to Mars. I will run for office. I will not run for office. Neither rule applies. If I go to Mars, I will run for office. I am going to run for office. Therefore, Select one I am going to Mars. I am not going to Mars. I will run for office. I will not run for office. Neither rule applies. If I go to Mars, I will run for office. I am not going to Mars. Therefore, Select one I am going to Mars. I am not going to Mars. I will run for office. I will not run for office. Neither rule applies. If I go to Mars, I will run for office. I will not run for office. Therefore, Select one I am going to Mars. I am not going to Mars. I will run for office. I will not run for office. Neither rule applies.

More tautoligical implications:

Simplification

(pq)p
and
(pq)q

In words, the first says: If both p and q are true, then, in particular, p is true.

Example
If the sky is blue and the moon is round, then (in particular) the sky is blue.

Argument Form

 The sky is blue and the moon is round. Therefore, the sky is blue.

In symbols:

 pq p

The other simplification, (pq)q is similar.

p(pq)

In words, the first says: If p is true, then we know that either p or q is true.

Example
If the sky is blue, then either the sky is blue of some ducks are kangaroos.

Argument Form

 The sky is blue. Therefore, the sky is blue or some ducks are kangaroos.

In symbols:

 p pq

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:

(pq)p;

p(pq).

In the exercise set, you will be asked to check that these are not tautologies.

### Example 1P Practice with Simplification and Addition

Use either simplification or addition to draw the appropriate conclusion.
 That's it! I am going to Mars. Therefore, Select one I am going to Mars and I am the King of England. I am going to Mars or I am the King of England. I am the King of England. I am not the King of England. None of the above. I am going to Mars, but not to Jupiter. Therefore, Select one I am not going to Jupiter. I am not going to Mars. I am going to Jupiter. I am going to Mars. None of the above. I am not going to Mars. Therefore, Select one I am going neither to Mars nor Jupiter. I am going to both Mars and Jupiter I am going to Mars but not to Jupiter. I am going to Jupiter but not to Mars. None of the above.

Here are the last two tautoligical implications we shall look at.

Disjunctive Syllogism or One-or-the-Other

[(pq)(~p)]q

[(pq)(~q)]p

Example
If either the cook or the butler did it, but we know that the cook didn't do it, then the butler must have done it.

Argument Form

 Either the cook or the butler did it. The cook didn't do it. Therefore, the butler did it.

In symbols:

 pq ~p q

Transitivity

[(pq)(qr)](pr)

Example
When it rains the ground gets muddy and when the ground is muddy my shoes get dirty. So, when it rains my shoes get dirty.

Argument Form

 When it rains the ground gets muddy. When the ground is muddy my shoes get dirty. Therefore, when it rains my shoes get dirty.

In symbols:

 pq qr pr

We sometimes think of this as allowing us to chain arrows together: The first two implications can be written together as pqr, and if we "follow the arrows" from beginning to end, we get pr.

### Example 2 Practice with Disjunctive Syllogism and Transitivity

Use either disjunctive syllogism or transitivity to draw the appropriate conclusion.
 I cannot go to the moon base. I must either go to Mars or to the moon base. Therefore, Select one I am either going to both Mars and the moon base. I am going to the moon base. I am going to Mars. I am going to Mars or to the moon base. None of the above. If I must go to Mars, I can't go to Jupiter. If I don't go to Jupiter, I will stay home. Therefore, Select one If I must go to Mars, I will stay home. I am not going to Mars and will stay home. I am going to Jupiter. If I stay home, I cannot go to Jupiter None of the above. If I go to Jupiter, I will pass Io. If I don't go to Mars, I will definitely go to Jupiter. Therefore, Select one If I don't go to Io, I am not going to Jupiter. If I do go to Io, I am not going to Mars. If I don't go to Mars I won't go to Io. If I don't go to Mars I will go to Io. None of the above.

### Tautological Equivalences

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

p~(~p)

This is just the Double Negation Law p~(~p). In argument form, we can express this in two ways using argument form:

Argument Form

 p ~(~p)
and      ~(~p) p

Commutativity

(pq)(qp)

This is just the commutativity equivalence pqqp.

Argument Form

 pq qp
and      qp pq

We conclude this section with a summary list of important tautologies.

### Important Tautologies

A. Tautological Implications
Symbolic Form
Argument Form
Name
1. [(pq)p]q
pq
p
q
Modus Ponens
(Direct Reasoning)
2.[(pq)~q]~p
pq
~q
~p
Modus Tollens
(Indirect Reasoning)
3. (pq)p

(pq)q

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

[(pq)(~q)]p

 pq ~p q pq ~q p
Disjunctive Syllogism
(One-or-the-Other)
6. [(pq)(qr)](pr)
pq
qr
pr
Transitivity of

B. Tautological Equivalences
Symbolic Form
Argument Form
Name
1. p~(~p)
 p ~(~p) ~(~p) p
Double Negative
2. pqqp

pqqp

 pq qp pq qp
Commutative Laws
3. (pq)rp(qr)

(pq)rp(qr)

 (pq)r p(qr) p(qr) (pq)r
Associative Laws
4. ~(pq)(~p)(~q)

~(pq)(~p)(~q)

 ~(pq) (~p)(~q) (~p)(~q) ~(pq)
 ~(pq) (~p)(~q) (~p)(~q) ~(pq)
De Morgan's Laws
5. p(qr)
(pq)(pr)

p(qr)
(pq)(pr)

 p(qr) (pq)(pr) (pq)(pr) p(qr)
 p(qr) (pq)(pr) (pq)(pr) p(qr)
Distributive Laws
6. ppp

ppp

 pp p p pp
 pp p p pp
Idempotent Laws
7. (pq)((~p)q)
 pq (~p)q (~p)q pq
Switcheroo
8. (pq)(~q~p)
 pq (~q)(~p) (~q)(~p) pq
Contrapositive
9. (pq)
((pq)(qp))
pq

(pq)(qp)

(pq)(qp)

pq

Meaning of the Biconditional

Last Updated: October, 2001