Translating English sentences into propositional forms
We take an example to explain the procedure:
Example: Convert the following sentence into propositional form:
“If I earn more than 2Lac Rupees then I must file income tax return”.
1. Identify the basic simple sentences and denote them with some variables.
In the above case we can identify two elemental statements:
I earn more than 2Lac rupees. (Let’s denote it by P)
I must file income tax return. (Let’s denote it by Q)
2. Look for the connective keywords.
You can identify a number of keywords very easily in any given sentence. These include ‘OR’, ‘AND’, ‘NOT’, ‘IF… THEN’ and ‘Only… IF’. There are, however, two key points to remember and observe:
a. All keywords except NOT are binary connectives and hence they should be identified only in a manner where they are connecting the two basic statements identified in step 1.
b. NOT is a unary connective and would be taken as part of a whole statement only, rather than connecting two individual statements.
In the current example, the keywords to note are IF…THEN.
3. Use the identified keywords to represent the given statement in a partially propositional form.
In the example, such a partial propositional form can be:
“If P THEN Q”.
4. Use the proper operator symbols corresponding to keywords now present in the partial form obtained in step 3. For “keyword – connective symbol” relationship you may refer to the previous chapter (INTRODUCTION).
In current example we replace “IF… THEN” with the corresponding arrow symbol ‘→’ to get the final propositional form:
P → Q.
NOTE: While looking for the elemental statements within a given problem statement, identify only affirmative sentences as candidates. Any negation (NOT keyword) should not be taken as part of an elemental statement.
More examples:
Translate the following sentences into propositional forms:
(a) If I earn more than 2Lac Rupees and I am honest, then I will file Income Tax return.
(b) I do not earn more than 2 Lac rupees and I will not file income tax return.
(c) I am not honest
(d) I will not file income tax return only if I am not honest.
Solution
Let
P: I earn more than 2 Lac Rupees.
Q: I am honest.
R: I will file income tax return.
Then
(a) (P Ʌ Q) → R
(b) ¬P Ʌ ¬R
(c) ¬Q
(d) ¬R → ¬P