Introduction to Logic
by
Stefan Waner and Steven R. Costenoble

This Section: 4. Tautological Implications and Tautological Equivalences

3. The Conditional and the Biconditional Section 4 Exercises 5. Rules of Inference Main Logic Page "Real World" Page

4. Tautological Implications and Tautological Equivalences

Our aim in this section is to enlarge our list of "standard" tautologies by adding new ones involving the conditional and the biconditional. This section is devoted to discussing the standard tautologies we shall use. A summary list is provided at the end of the section.

From now on, we shall use small letters p, q, ... only to denote atomic statements, and uppercase letters A, B, C, ... to denote statements of all types, compound or atomic.

The first tautologies we look at are tautological implications, tautologies of the form AB.


Modus Ponens or Direct Reasoning

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

For example, letting p: "I love math" and q: "I will pass this course," we get

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.

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? Also see Example 10 in section 6.) 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

In words, if p implies q, and q is false, then so is p.

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

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!


Simplification
(pq)p
and
(pq)q

The first of these in words: If p and q are both true, then, in particular, p must be true.

For example, if p: "the sky is blue" and q: "the moon is round," then this is saying that:

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

In words: If p is true, then either p is true or q is true.

As an example, let us take p: "the sky is blue" and q: "some ducks are kangaroos," then this is saying that:

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.


Caution

The following are common errors:

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


Disjunctive Syllogism (One-or-the-Other)

Let p: "the cook did it," and let q: "the butler did it." Then the first statement is:

In argument form:

pq
~p
q

Transitivity

Let p: "a crew member was responsible for the oil spill," and let q: "the captain must take responsibility for the oil spill," and let r: "the oil company must take responsibility for the oil spill." Then the statement becomes:

In argument form:

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.


The next batch of tautologies are tautological equivalences, which are tautologies of the form AB.

Before we look at examples, we recall that the statement AB is true exactly when A and B have the same truth value. If AB is always true, then A and B must always have the same truth values. In other words, to say that AB is a tautology is the same thing as saying that A and B are logically equivalent. In other words,

Logical Equivalence and Tautological Equivalences

For any (possibly compound) statements A and B,

    AB is a tautology if and only if AB.

It follows that we already know lots of tautological equivalences. Here are a few examples.


Double Negation

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


Commutativity

This is just the commutativity equivalence pqqp. In argument form:


For each logical equivalence we have written down, we can (and will in a moment) write down a tautological equivalence. 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

3. The Conditional and the Biconditional Section 4 Exercises 5. Rules of Inference Main Logic Page "Real World" Page

We would welcome comments and suggestions for improving this resource. Mail us at:
Stefan Waner (matszw@hofstra.edu) Steven R. Costenoble (matsrc@hofstra.edu)
Last Updated: July, 1996
Copyright © 1996 StefanWaner and Steven R. Costenoble