PHI 165
Introduction to Logic

Garns Fall 1997

6.3 Truth Tables and Propositions

Concepts in this section:

tautology

self-contradiction

contingent

logical equivalence

contradiciton

consistency

NOTE: Here I must use the " * " for the dot, the " > " for the horseshoe, and the " _ " for the triple bar.

Truth Table Design

The size of the truth table depends on the number of different simple statements (=n). L = 2 n

For the first simple proposition (leftmost column) divide the total number of rows in half and make the first half true and the second half false. Every instance of this proposition will receive the same column of values.

For the second simple proposition divide these halves in half and alternative between true and false.

And so on ....

An Example

( A v C ) É > ~ C

" A " and " C " are the two simple propositions and so the formula L = 2 n tells us we need four rows.

Set up the Table and the First Column

( A

v

C )

É >

~

C

T

T

F

F

Fill in Column for the Second Proposition

( A

v

C )

>É

~

C

T

T

T

F

F

T

F

F

Repeat the Column for "C"

( A

v

C )

É >

~

C

T

T

T

T

F

F

F

T

T

F

F

F

Determine the Values for the Disjunction

( A

v

C )

>É

~

C

T

T

T

T

T

T

F

F

F

T

T

T

F

F

F

F

Determine the Value for the Negation

( A

v

C )

>É

~

C

T

T

T

F

T

T

T

F

T

F

F

T

T

F

T

F

F

F

T

F

Determine the Value for the Conditional

( A

v

C )

É >

~

C

T

T

T

F

F

T

T

T

F

T

T

F

F

T

T

F

F

T

F

F

F

T

T

F

Classifying Statements

Tautologous Statements

True under every assignment of truth values

( A

*

~

B )

É>

A

T

F

F

T

T

T

T

T

T

F

T

T

F

F

F

T

T

F

F

F

T

F

T

F

Self-contradictory Statements

False under every assignment of truth values

( A

*

~

B )

*

~

~

B

T

F

F

T

F

T

F

T

T

T

T

F

F

F

T

F

F

F

F

T

F

T

F

T

F

F

T

F

F

F

T

F

Contingent Statements

True under some assignments of truth values, false under others

( A

v

C )

É >

~

C

T

T

T

F

F

T

T

T

F

T

T

F

F

T

T

F

F

T

F

F

F

T

T

F

Comparing Statements

Logically Equivalent

Same truth value under the main operator

R

É >

S

~

R

v

S

T

T

T

F

T

T

T

T

F

F

F

T

F

F

F

T

T

T

F

T

T

F

T

F

T

F

T

F

Contradictory

Opposite truth value under the main operator

~

R

v

S

R

*

~

S

F

T

T

T

T

F

F

T

F

T

F

F

T

T

T

F

T

F

T

T

F

F

F

T

T

F

T

F

F

F

T

F

Consistent

At least one line on which both are true

R

v

S

R

*

S

T

T

T

T

T

T

T

T

F

T

F

F

F

T

T

F

F

T

F

F

F

F

F

F

Inconsistent

No line on which both are true