DHARMPROPOSITIONAL LOGIC 119and construct the truth table for X. From Fig. 5.23 we find that formula X is a tautology,
therefore argument is a valid argument. Hence, given statement is a valid statement.
S 1 (let) S 2 (let)
SPJP → SS → J ~ J(P → S) ∧ (S → J) S1 ∧ ~ J ~ PS2 → ~ P
FFFT T T T T T T
FFTT T F T F T T
FTFT F T F F F T
FTTT T F T F F T
TFFF T T F F T T
TFTT T F T F T T
TTFF F T F F F T
TTTT T F T F F T
Fig. 5.23
As we observe that when number of propositional variables appeared in the formula are
increases then construction of truth table will become lengthy and tedious. To, overcome this
difficulty, we must go through some other possible methods where truth table is no more
needed.5.6.2 Natural Deduction Method................................................................................
Deduction is the derivation process to investigate the validity of an argument. When a conclu-
sion is derived from a set of premises by using rules of inference then, such a process of deriva-
tion is called a deduction or formal proof.
Natural deduction method is based on the rules of Inference that are shown in Fig 5.24.
The process of derivation can be describe by following two steps,
Step 1. From given set of premises, we derive new premises by using rules of inference.
Step 2. The process of derivation will continues until we reaches the required premise
that is the conclusion (every rule used at each stage in the process of derivation, will be
acknowledged at that stage).
I. Rules of Inference
Here we discuss 9 rules of inference, by truth table we can verify that the arguments followed
by these rules are valid arguments. (Assume P, Q, R and S are propositional variables)
Rule 1. Modes Ponens (MP) Rule 2. Modes Tollens (MT)
(i)P → Q(i)P → Q
(ii)P (ii)~ Q
∴ Q ∴ ~ P