Natural Deduction 3

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Natural Deduction:

Natural Deduction 7.3

Rules of Replacement I:

Rules of Replacement I DeMorgan’s Rule Commutativity Associativity Distribution

DeMorgan’s Rule:

DeMorgan’s Rule ~(p . q) :: ~p v ~q ~(p v q) :: ~p . ~q

Commutativity :

Commutativity p v q p . q

Associativity :

Associativity p v q v r p . q . r ( ( ) )

Distribution :

Distribution p v q . r ( ) p v ) ( ) p . q v r ( ) p . ) ( )

An Example:

An Example J v (K . L) ~K /J Here we seem unable to use any of the rules of inference immediately. Let’s try a rule of replacement… distribution?

An Example:

An Example J v (K . L) ~K /J (J v K) . (J v L) 1 dist. The advantage here is we can now simplify! Which part should we simplify? We have a ~K in line two which is a clue!

An Example:

An Example J v (K . L) ~K /J (J v K) . (J v L) 1 dist. J v K 3 simp. Now how can we get J out of line 4? Remember you can’t simplify a disjunction! But speaking of disjunctions there’s a DS!

An Example:

An Example J v (K . L) ~K /J (J v K) . (J v L) 1 dist. J v K 3 simp. J 2,4 DS And the code is solved!

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