7.E Contingent and Noncontingent Statements:

We can use truth tables to determine whether the truth value of a compound statement depends solely on its form or whether it depends also on the specific truth values of its components. We say that a compound statement is contingent if its truth value varies with the truth values of its components. However, some statements are noncontingent; that is, the truth value under the main operator does not depend on the truth values of its component parts. There are two kinds of noncontingent statements: it istautologous (or logically true) if it is true regardless of the truth values of its components. It is self-contradictory (logically false) if it is false regardless of the truth values of its components..

Samples:

P |P v ~ P

P|P  ~ P

7.F Logical Equivalence

Two statements are said to be logically equivalent when they have identical truth tables. This is done by placing the two statements next to each other so they can share the same guide for simple statements. Compare:

PQ|P  Q|P v Q

SH|~ (S  H)|~ S v ~ H

FC|~ ( F v C )|~ F  ~ C

7.G: Contradictory, Consistent, and Inconsistent Statements

Logically equivalent statements have identical truth tables, while contradictory statements have opposite truth tables. Consider:

L|L|~L

FS|~ F v S|F  ~ S

Consistent statements have at least one line on which their main operators are true. Consider:

R B|R v B|R v ~ B

Inconsistent statements do not have any one line on which they are both true, though they may be both false at the same time.

CM|C  ~ M|C  M

Symbolization and Truth Tables

To symbolize a statement in symbolic language:

1. Identify statement connectives.

2. Identify components containing no connectives, and replace each distinct component with a distinct statement letter, a capital letter of the alphabet. (A record of which statement letter symbolizes which statement component is called a dictionary.)

3. Replace statement connectives with symbolic connectives.

4. Use the structure of the statement to determine grouping.

5. Write out the symbolized argument, placing a single slash between the premises and a double slash between the last premise and the conclusion.

6. Draw a truth table for the symbolized argument as if it were a proposition broken into parts, outlining the columns representing the premises and conclusion.

7. Look for a line in which all of the premises are true and the conclusion is false. If such a line exists, the argument is invalid. If not, it is valid.

The goal of symbolization is to devise a formula that is true exactly when the corresponding statement is true and false exactly when that statement is false. Remember to represent as much of the logical structure of the corresponding statement as possible.

How do we do this?

  1. Identify truth-functional statement connectives:

If Socrates died, he died either when he was living or when he was dead.

But he did not die while living.

Nor when he died.

Socrates did not die.

  1. Identify and replace components. We judge components to be the same if they clearly have the same meaning.

a. Socrates died

b. he died when he was living

c. (he died) when he was dead.

d. he did die while living.

e. (he did die) when he died.

f. Socrates did die.

Dictionary:

S: Socrates died.

L: Socrates died while he was living.

D: Socrates died while he was dead.

So, replacing our dictionary terms for the phrases, we get:

If S, either L or D.

Butnot L.

Nor D.

not S.

  1. Replace English connectives with symbolic connectives.

S, L D

~ L

~ D

~ S

  1. Determine grouping.

S (L D)

~ L

~ D

~ S

5-7.Write it out on a line and construct the truth table.

LDS|S ( L D ) / ~ L / ~ D // ~ S

Another sample:

KL|K  ~ L / ~ ( L  ~ K ) // K  L

Examine the table. If there is no line on which all the premises are true and the conclusion false, then the argument is valid. Conversely, if there is at least one line on which the premises are all true and the conclusion false, then the argument is invalid.

Note two implications of this. First, any argument with inconsistent premises will always be valid, as there is no instance in which it will have all true premises. And any argument with a tautologous conclusion will also be valid, as a tautology can never be false.