Kenneth H. Rosen - Propositional Logic

Question

Which of these sentences are propositions? What are the truth values of those that are propositions?

a) Boston is the capital of Massachusetts.

b) Miami is the capital of Florida.

c) 2 + 3 = 5.

d) 5 + 7 = 10.

e) x + 2 = 11.

f ) Answer this question.

Answer 1

Proposition is a Declarative statement which can be confined within the domain of TRUE or FALSE

but NOT BOTH.

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a) The Capital of Massachusetts is Boston. Even if you don't know the capital of Massachusetts you 

can still answer it. How? Because either Boston is capital of Massachusetts or not.

So, we can confine it within TRUE or FALSE. 

Hence it is a Proposition.

 

b) Explanation is same as above. It is also a proposition.

 

c) 2 + 3 is equal to 5. This statement is TRUE. Hence it is a Proposition.

 

d) 5 + 7 is 12 $\neq$ 10. This statement is TRUE. Hence it is a Proposition.

 

e) x + 2 = 11. 

Here, x is a free variable & a statement containing free variables are not propositions. Why?

Reason : Here, x can take any value. Without putting any value of x we cannot confine

this statement within the domain of TRUE or FALSE. 

Hence it is NOT a Proposition.

 

f)  Answer this question. 

This is a command statement & such statements are NOT propositions.

Hence it is NOT a Proposition.

 

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Which statements are NOT Propositions?

1) QUESTION STATEMENTS 

EXAPMLE : Where are you going? 

Are you going? 

Are you Fine? 

What is your age?

2) COMMAND STATEMENTS

EXAMPLE : Go clean your room

Leave this house.

Go and study.

You should read this book.

3) STATEMENTS CONTAINING FREE VARIABLES

EXAMPLE : x = 2 + 2x

$4x + 7$ = 37$x^{2}$

4) PARADOX STATEMENTS

EXAMPLE : This statement is False.

Answer 2

Exept e and f all are propositions

e) x + 2 = 11, here x can take any value in a given domain, hence it can be either true or false, hence it is not a proposition

f) Answer this question, this statement is a command ,hence not a proposition