First Course in Mathematical Logic (Dover Books on Mathematics) by Patrick Suppes & Shirley Hill

Author:Patrick Suppes & Shirley Hill [Suppes, Patrick]
Language: eng
Format: mobi
ISBN: 9780486150949
Publisher: Dover Publications
Published: 2012-04-30T04:00:00+00:00

Prove: P → (¬Q → R)

(1) S & (¬P V M ) P

(2) M → Q V R P

(3) ¬P V M S 1

(4) P P

(5) M TP 3, 4

(6) ¬Q P

(7) Q V R PP 2, 5

(8) R TP 6, 7

(9) ¬Q → R CP 7, 8

(10) P → (¬Q → R ) CP 4, 9

The conclusion is to be a conditional so a conditional proof is tried.

We introduce the antecedent, P, of the desired conditional, and try to derive the consequent, ¬Q → R. In line (6), we try another conditional proof since the ¬Q → R we are trying to derive is itself a conditional. So we add its antecedent ¬Q and indent a second time. This gives a proof subordinate to the first subordinate proof. After deriving R, the rule for conditional proof is used. This allows a return to the proof to which we arc subordinate and gives the consequent we were trying to derive in the first subordinate proof. Applying CP again brings us back to the main proof. One application of CP ends only one indentation, that is, only one subordination. One other thing needs emphasis: at any stage any line may be used which appears earlier in the same proof or earlier in any proof to which we are subordinate. Thus in line (7) we may use line (2) and line (5). But after line (9) we could not use lines (6) to (8) and after line (10) we could not use lines (4) to (9).

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