Psychology

LESSON1-LOGIC

STATEMENTS AND TRUTH

Statements with Words

A statement (or proposition) is a sentence that can be true or false, but not both simultaneously. Example 1.1:“Mary is awake" is a statement because at any given time either Mary is awake or Mary is not awake (also known as Mary being asleep), and Mary cannot be both awake and asleep at the same time.

Example 1.2: The sentence “Wake up!” is not a statement because it cannot be true or false.

An atomic statement expresses a single idea. The statement “Mary is awake" that we discussed above is an example of an atomic statement. Let's look at a few more examples. Example 1.3: The following sentences are atomic statements:

1. 17 is a prime number.

2. George Washington was the first president of the United States.

3. 5> 6.

4. David is left-handed.

Sentences 1 and 2 above are true, and sentence 3 is false. We can't say for certain whether sentence 4 is true or false without knowing who David is. However, it is either true or false. It follows that each of the four sentences above are atomic statements.

We use logical connectives to form compound statements. The most commonly used logical connectives are "and," “or," “if...then," “if and only if," and “not."

Example 1.4: The following sentences are compound statements:

1. 17 is a prime number and 0=1.

2. Michael is holding a pen or water is a liquid.

3. If Joanna has a cat, then fish have lungs.

4. Albert Einstein is alive today if and only if 5 + 7 = 12.

5. 16 is not a perfect square.

Sentence 1 above uses the logical connective "and." Since the statement “0 = 1" is false, it follows that sentence 1 is false. It does not matter that the statement “17 is a prime number" is true. In fact, “T and F" is always F.

Sentence 2 uses the logical connective “or." Since the statement “water is a liquid” is true, it follows that sentence 2 is true. It does not even matter whether Michael is holding a pen. In fact, “T or T” is always true and “F or T" is always T.

It's worth pausing for a moment to note that in the English language the word “or" has two possible meanings. There is an “inclusive or" and an “exclusive or." The “inclusive or" is true when both statements are true, whereas the “exclusive or” is false when both statements are true. In mathematics, by default, we always use the “inclusive or" unless we are told to do otherwise. To some extent, this is an arbitrary choice that mathematicians have agreed upon. However, it can be argued that it is the better choice since it is used more often and it is easier to work with. Note that we were assuming use of the “inclusive or" in the last paragraph when we said, “In fact, "T or T" is always true." See Problem 4 below for more on the “exclusive or."

Sentence 3 uses the logical connective "if...then." The statement “fish have lungs" is false. We need to know whether Joanna has a cat in order to figure out the truth value of sentence 3. If Joanna does have a cat, then sentence 3 is false (“if T, then F" is always F). lf Joanna does not have a cat, then sentence3 is true ( "if F, then F" is always T).

Sentence 4 uses the logical connective "if and only if." Since the two atomic statements have different truth values, it follows that sentence 4 is false. In fact, “F if and only if T" is always F.

Sentence 5 uses the logical connective "not." Since the statement “16 is a perfect square" is true, it follows that sentence 5 is false. In fact, “not T" is always F.

Notes: (1) The logical connectives “and," “or," “if...then," and “if and only if," are called binary connectives because they join two statements (the prefix “bi" means "two").

(2) The logical connective “not” is called a unary connective because it is applied to just a single statement ("unary” means “acting on a single element"). Example 1.5: The following sentences are not statements:

1. Are you happy?

2. Go away!

3.x-5=7

4. This sentence is false.

5. This sentence is true.

Sentence 1 above is a question and sentence 2 is a command. Sentence 3 has an unknown variable - it can be turned into a statement by assigning a value to the variable. Sentences 4 and 5 are self-referential (they refer to themselves). They can be neither true nor false. Sentence 4 is called the Liar's paradox and sentence 5 is called a vacuous affirmation.