Question

Philosophy

3. Multiple-Line Truth Functions

Compound statements in propositional logic are truth functional, which means that their truth values are determined by the truth values of their statement components. Because of this truth functionality, it is possible to compute the truth value of a compound proposition from a set of initial truth values for the simple statement components that make up the compound statement, combined with the truth table definitions of the five propositional operators.

To compute the truth value of a compound proposition, first you should enter the truth values of the simple components on the line directly beneath the statement letters. Use T for true and F for false. Then bring down the operators and the parentheses. Compute the truth value of the statement component whose connective governs the smallest sentence component, and write this value on the next line beneath the statement component. Drop the parentheses around those operators that have already been reduced. Then bring down the other operators and parentheses to this line. Repeat this process, eliminating an operator and replacing a statement component with its truth value, at each step on a new line, until the main operator has been eliminated. The truth value of the main operator is the truth value of the overall statement.

Complete the following steps to compute the truth value of the given compound statement. In each step, reduce a single operator to its resulting truth value. For this exercise, you should assume that simple statements Z, R, F, B, and P have the following truth values:

Initial Truth Values for Simple Statements

Z	=	false
R	=	true
F	=	false
B	=	true
P	=	false

Given:

[

(

Z

•

R

)

∨

~

(

F

⊃

B

)

]

≡

P

Step 1:

[

(

•

)

∨

~

(

⊃

)

]

≡

Step 2:

[

∨

~

(

⊃

)

]

≡

Step 3:

[

∨

~

]

≡

Step 4:

[

∨

]

≡

Step 5:

≡

Step 6:

Based on this truth function, the given compound statement is   .

Answer 1

We first replace the truth values of to get to step . Then we replace the conjunction by . Next the expression is replaced by since the antecedent is false. Then we replace by . reduces to . Finally, is true, .