Mathematical Logic: Statement and its truth value

Statement and its truth value

The simplest part of Mathematical Logic is the Propositional Logic and its building blocks are statements or propositions. Mostly communication needs the use of language through which we impart our ideas. They are in the form of sentences.

There are various types of sentences like

(1) Declarative (Assertive type)

(2) Imperative (A command or a request type)

(3) Exclamatory (Emotions, excitement type)

(4) Interrogative (Question type)

(5) Open type

Definition 12.7

Any declarative sentence is called a statement or a proposition which is either true or false but not both.

Any imperative sentence such as exclamatory, command and any interrogative sentence cannot be a proposition.

The truth value of a statement refers to the truth or the falsity of that particular statement. The truth value of a true statement is true and it is denoted by T or 1. The truth value of a false statement is false and it is denoted by F or 0.

An open sentence is a sentence whose truth can vary according to some conditions, which are not stated in the sentence. For instance, (i) x × 7 = 35 is an open sentence whose truth value depends on value of x . That is, if x = 5 , it is true and if x ≠ 5, it is false. (ii) He is a bad person. This is an open sentence. Opinion varies from individual to individual.

Example 12.11

Identify the valid statements from the following sentences.

Solution:

(1) Mount Everest is the highest mountain of the world.

(2) 3+4=8.

(3) 7+5>10.

(4) Give me that book.

(5) (10 − x) = 7 .

(6) How beautiful this flower is!

(7) Where are you going?

(8) Wish you all success.

(9) This is the beginning of the end.

The truth value of the sentences (1) and (3) are T, while that of (2) is F. Hence they are statements. The sentence (5) is true for x = 3 and false for x ≠ 3 and hence it may be true or false but not both. So it is also a statement.

The sentences (4), (6), (7), (8) are not statements, because (4) is a command, (6) is an exclamatory, (7) is a question while (8) is a sentence expressing one’s wishes and (9) is a paradox.