Introduction to Logic

by
Stefan Waner and Steven R. Costenoble

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

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.

 

Addition

    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:

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

Example 1P Practice with Simplification and Addition

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

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
Addition
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
Copyright © 1996 StefanWaner and Steven R. Costenoble

Top of Page