Natural Deduction Proof using rules of implication and rules

Natural Deduction Proof using rules of implication and rules of replacement. Philosophy Symbolic Logic.

Solution

Answer :

The premises are given by

V ~ M

~ V F

~ ( C v F )

The conclusion is ~( M v C )

Now we shall prove that ~( M v C ) is logically follows from the given premises.

Proof :

1. V ~ M                           Premise

2. M ~ V    ( 1 ) contrapositive

3. ~ V F                           Premise

4. M F                             ( 2 ) , ( 3 ) , Chain rule

5. ~ ( C v F )                        Premise

6. ~ C ~ F ( 5 ) demorgan law

7. ~ F ( 6 ) law of conjuction

8. ~ M    ( 4 ) , ( 7 ) Modus Tollen\'s rule

9.  ~ C ( 6 ) law of conjuction

10. ~ C ~ M    ( 8 ) , ( 9 ) law of conjuction

11. ~ ( C v M )      ( 10 ) Demorgan law

Thus , the conclusion ~ ( C v M ) is logically follows from the given premises