Mathematics normally works with a * two-valued
logic*: Every statement is either * True* or
* False*. You can use * truth
tables* to determine the truth or falsity of a complicated
statement based on the truth or falsity of its simple components.

A statement in sentential logic is built from simple statements using the logical connectives , , , , and . I'll construct tables which show how the truth or falsity of a statement built with these connective depends on the truth or falsity of its components.

Here's the table for negation:

This table is easy to understand. If P is *true*, its negation
is *false*. If P is *false*, then is {\it true}.

should be *true* when both P and Q are
*true*, and *false* otherwise:

is *true* if either P is *true* or Q is
*true* (or both). It's only *false* if both P and Q are
*false*.

Here's the table for logical implication:

To understand why this table is the way it is, consider the following example:

"If you get an A, then I'll give you a dollar."

The statement will be *true* if I keep my promise and
*false* if I don't.

Suppose it's *true* that you get an A and it's *true*
that I give you a dollar. Since I kept my promise, the implication is
{\it true}. This corresponds to the first line in the table.

Suppose it's *true* that you get an A but it's *false*
that I give you a dollar. Since I *didn't* keep my promise,
the implication is *false*. This corresponds to the second
line in the table.

What if it's false that you get an A? Whether or not I give you a dollar, I haven't broken my promise. Thus, the implication can't be false, so (since this is a two-valued logic) it must be true. This explains the last two lines of the table.

means that P and Q are *
equivalent*. So the double implication is *true* if P and
Q are both *true* or if P and Q are both *false*;
otherwise, the double implication is false.

You should remember --- or be able to construct --- the truth tables for the logical connectives. You'll use these tables to construct tables for more complicated sentences. It's easier to demonstrate what to do than to describe it in words, so you'll see the procedure worked out in the examples.

* Remarks.* 1. When you're constructing a truth
table, you have to consider all possible assignments of True (T) and
False (F) to the component statements. For example, suppose the
component statements are P, Q, and R. Each of these statements can be
either true or false, so there are possibilities.

When you're listing the possibilities, you should assign truth values
to the component statements in a systematic way to avoid duplication
or omission. The easiest approach is to use *
lexicographic ordering*. Thus, for a compound statement with
three components P, Q, and R, I would list the possibilities this
way:

2. There are different ways of setting up truth tables. You can, for instance, write the truth values "under" the logical connectives of the compound statement, gradually building up to the column for the "primary" connective.

I'll write things out the long way, by constructing columns for each "piece" of the compound statement and gradually building up to the compound statement.

* Example.* Construct a truth table for the
formula .

A * tautology* is a formula which is "always
true" --- that is, it is true for every assignment of truth
values to its simple components. You can think of a tautology as a
*rule of logic*.

The opposite of a tautology is a *
contradiction*, a formula which is "always false". In
other words, a contradiction is false for every assignment of truth
values to its simple components.

* Example.* Show that is a tautology.

I construct the truth table for and show that the formula is always true.

The last column contains only T's. Therefore, the formula is a tautology.

* Example.* Construct a truth table for .

* Example.* Suppose

" " is true.

" " is false.

"Calvin Butterball has purple socks" is true.

Determine the truth value of the statement

For simplicity, let

P = " ".

Q = " ".

R = "Calvin Butterball has purple socks".

I want to determine the truth value of . Since I was given specific truth values for P, Q, and R, I set up a truth table with a single row using the given values for P, Q, and R:

Therefore, the statement is * true*.

Two statements X and Y are * logically
equivalent* if is a tautology. Another way to say
this is: For each assignment of truth values to the *simple
statements* which make up X and Y, the statements X and Y have
identical truth values.

From a practical point of view, you can replace a statement in a proof by any logically equivalent statement.

To test whether X and Y are logically equivalent, you could set up a truth table to test whether is a tautology --- that is, whether "has all T's in its column". However, it's easier to set up a table containing X and Y and then check whether the columns for X and for Y are the same.

* Example.* Show that and are logically equivalent.

Since the columns for and are identical, the two statements are logically
equivalent. This tautology is called * Conditional
Disjunction*. You can use this equivalence to replace a
conditional by a disjunction.

There are an infinite number of tautologies and logical equivalences; I've listed a few below; a more extensive list is given at the end of this section.

When a tautology has the form of a biconditional, the two statements which make up the biconditional are logically equivalent. Hence, you can replace one side with the other without changing the logical meaning.

* Example.* Write down the negation of the
following statements, simplifying so that only simple statements are
negated.

(a)

(b)

I showed that and are logically equivalent in an earlier example.

* Example.* Use DeMorgan's Law to write the
*negation* of the following statement, simplifying so that
only simple statements are negated:

"Calvin is not home or Bonzo is at the movies."

Let C be the statement "Calvin is home" and let B be the statement "Bonzo is at the moves". The given statement is . I'm supposed to negate the statement, then simplify:

The result is "Calvin is home and Bonzo is not at the movies".

* Example.* Use DeMorgan's Law to write the
*negation* of the following statement, simplifying so that
only simple statements are negated:

"If Phoebe buys a pizza, then Calvin buys popcorn."

Let P be the statement "Phoebe buys a pizza" and let C be
the statement "Calvin buys popcorn". The given statement is
. To simplify the negation, I'll use the * Conditional Disjunction* tautology which says

That is, I can replace with (or vice versa).

Here, then, is the negation and simplification:

The result is "Phoebe buys the pizza and Calvin doesn't buy popcorn".

* Example.* Replace the following statement
with its contrapositive:

"If x and y are rational, then is rational."

By the contrapositive equivalence, this statement is the same as "If is not rational, then it is not the case that both x and y are rational".

* Example.* Show that the inverse and the
converse of a conditional are logically equivalent.

Let be the conditional. The inverse is . The converse is .

I could show that the inverse and converse are equivalent by constructing a truth table for . I'll use some known tautologies instead.

Start with :

Remember that I can replace a statement with one that is logically equivalent. For example, in the last step I replaced with Q, because the two statements are equivalent by Double negation.

* Example.* Suppose x is a real number.
Consider the statement

"If , then ."

Construct the converse, the inverse, and the contrapositive. Determine the truth or falsity of the four statements --- the original statement, the converse, the inverse, and the contrapositive --- using your knowledge of algebra.

The converse is "If , then ".

The inverse is "If , then ".

The contrapositive is "If , then ".

The original statement is false: , but . Since the original statement is eqiuivalent to the contrapositive, the contrapositive must be false as well.

The converse is true. The inverse is logically equivalent to the converse, so the inverse is true as well.

Copyright 2009 by Bruce Ikenaga