2145-391 Aerospace Engineering Laboratory I Logic and Reasoning Logically draw conclusions Extract experimental results Construct a valid reasoning Construct explanations for experimental results Spot valid and invalid reasoning Implication: if p, then q. (p  q) Equivalence: p if and only if (iff) q (p  q)

Motivation: Logically Draw Conclusions Premises/Given: If it rains or it is humid, then I wear blue shirt. If it is cold, then I do not wear blue shirt. It rains. Questions: Logically, can you draw conclusions regarding the condition of the weather, the color of the shirt I wear?

Logic and Reasoning: What is it for? Logically draw conclusions Extract experimental results Construct a valid reasoning Construct explanations for experimental results Spot valid and invalid reasoning

Statement p = I have two legs p is True p = I have four legs p is False Statement is either True or False

Argument/Reasoning Premises: What we assume to be true / given. Conclusions: What we derive from the premises. Argument/Reasoning is either Valid or Invalid

Valid and Invalid Arguments Premises: If x = 2p, then sin x = 0. x = 2p. Conclusion: Therefore, sin x = 0. Valid Premises: If x = 2p, then sin x = 0. sin x = 0. Conclusion: Therefore, x = 2p. Invalid e.g., x = p

Statement VS Argument/Reasoning Statement: True or False Argument/Reasoning: Valid or Invalid

Valid Argument / Reasoning Truth of premise guarantees truth of conclusion. Truth of conclusion necessarily follows truth of premise. No valid argument can have true premise and false conclusion. If the argument is not valid, it is invalid.

Argument 1: Are these arguments/reasoning valid or invalid? Premises: If you rob a bank, then you go to jail. You go to jail. Conclusion: Therefore, you rob a bank. Premises: If x = 2p, then sin x = 0. sin x = 0. Conclusion: Therefore, x = 2p. Activity: Class Discussion Invalid e.g., x = p

Argument 2: Are these arguments/reasoning valid or invalid? Premises: If you rob a bank, then you go to jail. You rob a bank. Conclusion: Therefore, you go to jail. Premises: If x = 2p, then sin x = 0. x = 2p. Conclusion: Therefore, sin x = 0. Activity: Class Discussion Valid

Argument 3: Are these arguments/reasoning valid or invalid? Premises: If you rob a bank, then you go to jail. You do not rob a bank. Conclusion: Therefore, you do not go to jail. Premises: If x = 2p, then sin x = 0. x 2p. Conclusion: Therefore, sin x 0. Activity: Class Discussion Invalid e.g., x = p, sin x = 0

Argument 4: Are these arguments/reasoning valid or invalid? Premises: If you rob a bank, then you go to jail. You do not go to jail. Conclusion: Therefore, you do not rob a bank. Premises: If x = 2p, then sin x = 0. sin x 0. Conclusion: Therefore, x 2p. Activity: Class Discussion Valid

Truth VS Validity They are not the same. Truth for statements. Validity for argument/reasoning. Premises: Dogs have eight legs. [If x is a dog, then x has eight legs.] Spooky is a dog. Conclusion: Spooky has eight legs. The argument is valid. However, the conclusion is false. For further clarification, see lecture note.

Truth VS Validity from Peter Suber, Philosophy Department, Earlham College: http://www.earlham.edu/~peters/courses/log/tru-val.htm. a. Valid Argument b. Invalid Argument 1. True Premises, False Conclusion Premises: Impossible. No valid argument can have true premises and a false conclusion. Cats are mammals. Dogs are mammals. Conclusion: Therefore, dogs are cats. 2. True Premises, True Conclusion Tigers are cats. Tigers are mammals. Therefore, tigers are mammals. Therefore, tigers are cats.

Truth VS Validity from Peter Suber, Philosophy Department, Earlham College: http://www.earlham.edu/~peters/courses/log/tru-val.htm. a. Valid Argument b. Invalid Argument 3. False Premises, False Conclusion Premises: Dogs are cats. Cats are birds. Dogs are birds. Conclusion: Therefore, dogs are birds. Therefore, dogs are cats. 4. False Premises, True Conclusion Birds are mammals. Tigers are birds. Therefore, cats are mammals. Therefore, tigers are cats.

Rules of Inference Argument 2: q p x Rules of Inference Argument 2: Premises: If (p: you rob a bank), then (q: you go to jail). p: You rob a bank. Conclusion: Therefore, q: you go to jail. is not only implication/conditional, Modus Ponens: but it is a tautology (always true). Check truth table.

Rules of Inference Argument 4: q p x Rules of Inference Argument 4: Premises: If (p: you rob a bank), then (q: you go to jail). not-q: You do not go to jail. Conclusion: Therefore, not-p: you do not go to jail. is not only implication/conditional, Modus Tollens: but it is a tautology (always true). Check truth table.

Invalid Logical Fallacies e.g., x = p Fallacy of The Converse Premises: If x = 2p, then sin x = 0. sin x = 0. Conclusion: Therefore, x = 2p. Invalid e.g., x = p

Invalid Logical Fallacies e.g., x = p Fallacy of The Inverse Premises: If x = 2p, then sin x = 0. x = 2p. Conclusion: Therefore, sin x = 0. Invalid e.g., x = p

Logically Draw Conclusions Premises/Given: If it rains or it is humid, then I wear blue shirt. If it is cold, then I do not wear blue shirt. It rains. Questions: Logically, can you draw conclusions regarding the condition of the weather, the color of the shirt I wear?

Premises: Conclusions: ? R = It rains H = It is humid C = It is cold B = I wear blue shirt Premises: If it rains or it is humid, then I wear blue shirt. If it is cold, then I do not wear blue shirt. It rains. Conclusions: ?

I wear blue shirt. It is not cold.

Implications / Conditional

Some Logic: Necessary and Sufficient Conditions (Deductive Reasoning) Implication (Conditional Statement): p  q Note: There is also “inductive reasoning.” p q p q p q p  q Equivalence (~q)  (~p) If p, then q. If not q, then not p. q if p. q whenever p. p only if q. q is a necessary condition for p. p is a sufficient condition for q. Converse of p  q: q  p Contrapositive of p  q: ~ q  ~ p p  q and its contrapositive ~ q  ~ p are equivalent. That is: If p  q is true, ~q  ~ p is also true. If p  q is false, ~ q  ~ p is also false. On the other hand, p  q does not imply q  p. The truth of p  q does not automatically guarantee the truth of q  p.

p q (p  q) (~q ~p) (q p) T F Note that (p  q) and (~q  ~p) always have the same truth value. Some other equivalent relations:

If-then Example

Conditional Statements: If p, then q: PV = mRT (If/Under-the-condition-of/)For a fixed gas and mass of the gas, (if/under-the-condition-of/)and for a fixed temperature: If volume increases, then pressure decreases. If pressure does not decrease, then volume does not increase. Pressure decreases or volume does not increase.

in The Two Jet Noise Reduction Techniques Example Application of Logic in The Two Jet Noise Reduction Techniques