Chapter 7 A Deductive System we introduced a derivation system for sentential lo

Question

Chapter 7 A Deductive System we introduced a derivation system for sentential logic. Our primary concerm here is to learn to utilize this system, so we shall turn to the rules. In 2.1 we introduced two rules that applied to conjunctions: &E; &1 From: p &q; p&q; To: P From: p,q pq To: p&q; q&p; Exercise Set 2.1 Determine whether the following are correct or erroneous applications of our conjunction rules. 1. P & (Q & (R & S)) 2. R&S; premise 1 &E; 1. P &(Q&(R & S) 2. P&Q; premise 1 &E; 1.P &(Q& (R & S)) premise 1 &E; 1.P 2.Q &R; 3.P&Q; premise premise 1, 2 & 147

Explanation / Answer

(1)
Premise is P & (Q & ( R & S))

P & (Q & ( R & S))
=> P & ((Q & R) & (Q & S) ) ( & is Distributive)
=> P & ( R & S) ( By conjunction rule &E, Q & R => R ; Q & S => S)
=> (P & R) & (P & S) ( & is Distributive)
=> R & S ( By conjunction rule &E, P & R => R ; P & S => S)

Hence, this is a correct applications of the conjunction rule &E.

(2)
Premise is P & (Q & ( R & S))

P & (Q & ( R & S))
=> P & ((Q & R) & (Q & S) ) ( & is Distributive)
=> P & ( Q & Q) ( By conjunction rule &E, Q & R => Q ; Q & S => Q)
=> P & Q

Hence, this is a correct applications of the conjunction rule &E.

(3)
Premise is P & (Q & ( R & S))

P & (Q & ( R & S))
=> (P & Q) & ( R & S) ( & is Associative)
=> Q & ( R & S) ( By conjunction rule &E, P & Q => Q)

Hence, this is a correct applications of the conjunction rule &E.

(4)
Premise is P and Q & R

From : P , Q & R
=> P & (Q & R) ( By conjunction rule &I)
=> P & Q ( By conjunction rule &E, Q & R => Q)

In this, we require applications of both &E and &I to prove the conclusion.

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