Logic: Truth Tables

Constructing a Truth Table

Truth Table

A truth table for a compound statement is a list of the truth or falsity of the statement for every possible combination of truth and falsity of its components.

In other words, a truth table helps to show whether a statement is true or false.

Rows

To find the number of rows used in a truth table, take the number 2 raised to the power of the number of variables.

For example, if there was a p statement and a q statement, there would be 2 variables, 2^2 is 4.

If there were three statements, it would be 2^3, or 8 rows.

Columns

The columns under the connectives /\, and \/, stand for the conjunction, and disjunction of the expression on the two sides of that connective.

Two statement table

p q

THalf the rows should be true

TThe rows should

alternate T, F

T FThe result should

be

FHalf the rows

should be falseT

one of every possibility

F F TT, TF, FT, FF

The three statement tablep q r

T Half of the rows T Alternate TT T Alternate

T should be true, T and FF F T and F

T the other half F T so that there is

T should be false. F F one of every

F T T possibility

F T F TTT, TTF, TFT

F F T ect.

F F F

Negation Truth Table

p ~ p The opposite of p is ~p

T F “Not true” is “false”

F T “Not false” is “true”

Conjunction Truth Table

p q p /\ q p and q

T T TTrue only if

both are true.

T F F

F T F

F F F

Disjunction Truth Table

p q p \/ q p or q

T T TTrue if either

on is true

T F T

F T T

F F FFalse only if both are false

Lets fill out a table

p q p \/(or)

(~p(not)

/\(and)

q)

T T

T F

F T

F F

Negate the p column

p q p \/(or)

(~p(not)

/\(and)

q)

T T F

T F F

F T T

F F T

Copy the q column

p q p \/(or)

(~p(not)

/\(and)

q)

T T F T

T F F F

F T T T

F F T F

Fill the /\ column

p q p \/(or)

(~p(not)

/\(and)

q)

T T F F T

T F F F F

F T T T T

F F T F F

Copy the p column

p q p \/(or)

(~p(not)

/\(and)

T T T F F

T F T F F

F T F T T

F F F T F

Fill in the \/ column using the p and the /\ columns

p q P \/(or)

P and (~p/\p)

(~p(not)

/\(and)

(~p) and (p)

p

T T T T F F T

T F T T F F T

F T F T T T F

F F F F T F F

Use the final column to determine what type of statements it is

\/(or)

P and (~p/\p)

Tautology

Always True

Contradiction

Always False

ContingencySometimes true, sometimes false

T x x

T x x

T x x

F x x

Contingency

Some were true, while one was false. That makes this statement a contingency.

Real life example In case that was not entirely clear, let’s take a look

at an everyday example. Circuits. There are two different kinds of circuits,

a series circuit and a parallel circuit. When the switch is closed the light will be on.

However, with a series circuit, both switches have to be closed and with a parallel circuit only one switch has to be closed for the light to go on.

Series Circuits

Switch p Switch q Light

Closed Closed On Only on if both are closed

Closed Open Off

Open Closed Off

Open Open Off

Parallel Circuits

Switch p Switch q Light

Closed Closed On

Closed Open On

Open Closed On

Open Open Off Only off when both are open

Conclusion

That concludes the Logic: Truth Tables lesson. For more information, consult Finite Mathematics

by Berresford and Rockett. Or learn logic online:

http://www.earlham.edu/~peters/courses/log/terms2.htm