Let us build the tree for the argument Q>-P | P>Q.

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Presentation transcript:

Let us build the tree for the argument Q>-P | P>Q. An Invalid Argument Let us build the tree for the argument Q>-P | P>Q.

Negate the Conclusion To check and argument for validity ... Begin by negating the conclusion. Q>-P -(P>Q) Q>-P | P>Q

Apply the (->) Rule Q>-P -(P>Q) Q>-P | P>Q

Apply the (->) Rule The (->) Rule: -(A>B) Q>-P -(P>Q) Q>-P | P>Q -(A>B) = A&-B

Apply the (->) Rule The (->) Rule: -(A>B) Q>-P -(P>Q) P -Q Q>-P | P>Q 1 -(A>B) = A&-B

Apply the (>) Rule Q>-P -(P>Q) P -Q Q>-P | P>Q 1

Apply the (>) Rule Q>-P Q>-P | P>Q -(P>Q) A>B -A B 1 A>B = -AvB

Apply the (>) Rule 2 Q>-P -(P>Q) P -Q Q>-P | P>Q A>B -A B 1 -Q -P A>B = -AvB

Check for Validity 2 Q>-P -(P>Q) P -Q Q>-P | P>Q 1 -Q -P

contains P and -P, which is Check for Validity 2 Q>-P -(P>Q) P -Q Q>-P | P>Q 1 The right hand branch contains P and -P, which is impossible, so the branch is closed (*). -Q -P *

contains no contradiction. It is open and indicates Check for Validity 2 Q>-P -(P>Q) P -Q Q>-P | P>Q 1 The left hand branch contains no contradiction. It is open and indicates that the argument has a counterexample. -Q -P *

contains no contradiction. It is open and indicates Check for Validity 2 Q>-P -(P>Q) P -Q Q>-P | P>Q 1 The left hand branch contains no contradiction. It is open and indicates that the argument has a counterexample. -Q -P * P is T Q is F

contains no contradiction. It is open and indicates Check for Validity 2 Q>-P -(P>Q) P -Q Q>-P | P>Q 1 The left hand branch contains no contradiction. It is open and indicates that the argument has a counterexample. -Q -P * P T F T F QT T F F Q>-P | P>Q F T T T T T F T P is T Q is F Counterexample

Check for Validity Counterexample 2 Q>-P -(P>Q) P -Q 1 The left hand branch contains no contradiction. It is open and indicates that the argument has a counterexample. -Q -P * P T F T F QT T F F Q>-P | P>Q F T T T T T F T P is T Q is F Counterexample So the Argument is INVALID.

Check for Validity 2 Q>-P -(P>Q) P -Q Q>-P | P>Q 1 The left hand branch contains no contradiction. It is open and indicates that the argument has a counterexample. -Q -P * P is T Q is F To calculate the counterexample for an open branch. Make single letters (like P) = T and negated letters (like Q) = F.

For more click here Checking for Validity THE BOTTOM LINE If the tree is OPEN there is a counterexample. So the argument is INVALID. If the tree is CLOSED (all branches are closed), there is no counterexample. So the argument is VALID. For more click here