METEOROLOGY GEL-1370. Chapter Four Chapter Four Humidity, Condensation & Clouds.
Chapter Four
description
Transcript of Chapter Four
![Page 1: Chapter Four](https://reader035.fdocuments.in/reader035/viewer/2022062500/56815b1c550346895dc8d19d/html5/thumbnails/1.jpg)
Chapter Four
Proofs
![Page 2: Chapter Four](https://reader035.fdocuments.in/reader035/viewer/2022062500/56815b1c550346895dc8d19d/html5/thumbnails/2.jpg)
1. Argument Forms
An argument form is a group of sentence forms such that all of its substitution instances are arguments.
![Page 3: Chapter Four](https://reader035.fdocuments.in/reader035/viewer/2022062500/56815b1c550346895dc8d19d/html5/thumbnails/3.jpg)
Argument Forms, continued
• If an argument form has no substitution instances that are invalid, it is said to be a valid argument form.
• An argument form that has even one invalid argument as a substitution instance is called an invalid argument form.
![Page 4: Chapter Four](https://reader035.fdocuments.in/reader035/viewer/2022062500/56815b1c550346895dc8d19d/html5/thumbnails/4.jpg)
2. The Method of Proof: Modus Ponens and Modus Tollens
• Truth tables give us a decision procedure for any sentential argument.
• There is another method available to demonstrate validity of sentential arguments: the method of proof, or natural deduction.
![Page 5: Chapter Four](https://reader035.fdocuments.in/reader035/viewer/2022062500/56815b1c550346895dc8d19d/html5/thumbnails/5.jpg)
The Method of Proof, continued
A proof of an argument is a series of steps that starts with premises; each step beyond the premises is derived from a valid argument form by being a substitution instance of it;
the last step is the conclusion.
![Page 6: Chapter Four](https://reader035.fdocuments.in/reader035/viewer/2022062500/56815b1c550346895dc8d19d/html5/thumbnails/6.jpg)
Method of Proof, continued
Modus Ponens (MP):
p q⊃
p
Therefore, q.
![Page 7: Chapter Four](https://reader035.fdocuments.in/reader035/viewer/2022062500/56815b1c550346895dc8d19d/html5/thumbnails/7.jpg)
Method of Proof, continued
Modus Tollens (MT):
p q⊃
˜q
Therefore, ˜p
![Page 8: Chapter Four](https://reader035.fdocuments.in/reader035/viewer/2022062500/56815b1c550346895dc8d19d/html5/thumbnails/8.jpg)
Method of Proof, continued
Do not confuse either MP or MT with the invalid arguments that resemble them:
Affirming the Consequent:
p q⊃qTherefore, p
![Page 9: Chapter Four](https://reader035.fdocuments.in/reader035/viewer/2022062500/56815b1c550346895dc8d19d/html5/thumbnails/9.jpg)
Method of Proof, continued
Denying the Antecedent:
p q⊃
˜p
Therefore, ˜q.
![Page 10: Chapter Four](https://reader035.fdocuments.in/reader035/viewer/2022062500/56815b1c550346895dc8d19d/html5/thumbnails/10.jpg)
3. Disjunctive Syllogism (DS) and Hypothetical Syllogism (HS)
Another valid argument form is the Disjunctive Syllogism (DS). This has two forms:
p q∨˜pTherefore, q
Andp q∨˜qTherefore, q
![Page 11: Chapter Four](https://reader035.fdocuments.in/reader035/viewer/2022062500/56815b1c550346895dc8d19d/html5/thumbnails/11.jpg)
DS and HS, continued
Another valid argument form is the Hypothetical Syllogism
(HS):
p q⊃q r⊃Therefore, p r⊃
![Page 12: Chapter Four](https://reader035.fdocuments.in/reader035/viewer/2022062500/56815b1c550346895dc8d19d/html5/thumbnails/12.jpg)
4. Simplification and Conjunction
Another valid argument form is Simplification (Simp.), which has two forms:
p.qTherefore, p
And
p.qTherefore, q
![Page 13: Chapter Four](https://reader035.fdocuments.in/reader035/viewer/2022062500/56815b1c550346895dc8d19d/html5/thumbnails/13.jpg)
Simplification and conjunction, continued
Conjunction (Conj.) is another valid argument form:
p
q
Therefore, p.q
![Page 14: Chapter Four](https://reader035.fdocuments.in/reader035/viewer/2022062500/56815b1c550346895dc8d19d/html5/thumbnails/14.jpg)
5. Addition and Constructive Dilemma
Another valid argument form is Addition (Add.):
p
Therefore, p q∨
![Page 15: Chapter Four](https://reader035.fdocuments.in/reader035/viewer/2022062500/56815b1c550346895dc8d19d/html5/thumbnails/15.jpg)
Addition and constructive dilemma, continued
Another valid argument form is Constructive Dilemma (CD):
p q∨p r⊃q r⊃Therefore, r s ∨
![Page 16: Chapter Four](https://reader035.fdocuments.in/reader035/viewer/2022062500/56815b1c550346895dc8d19d/html5/thumbnails/16.jpg)
6. Principles of Strategy
• Look for forms that correspond to valid rules of inference• Remember: small sentences are your friends!• Once you have mastered the rules of inference, you will
find completing many proofs much easier by working backwards from the conclusion.
• Trace the connections between the letters in the argument, starting with those in the conclusion.
• Begin the proof with the letter most distant from those in the conclusion.
![Page 17: Chapter Four](https://reader035.fdocuments.in/reader035/viewer/2022062500/56815b1c550346895dc8d19d/html5/thumbnails/17.jpg)
7. Double Negation (DN) and DeMorgan’s Theorem (DeM)
Double Negation (DN) is an equivalence argument form:
p
Therefore, ˜˜p
And
˜˜p
Therefore, p
![Page 18: Chapter Four](https://reader035.fdocuments.in/reader035/viewer/2022062500/56815b1c550346895dc8d19d/html5/thumbnails/18.jpg)
DN and DeM, continued
DeMorgan’s Theorem (DeM):
˜(p . q) is equivalent to ˜p ˜q∨
And
˜(p q) is equivalent to ˜p . ˜q∨
![Page 19: Chapter Four](https://reader035.fdocuments.in/reader035/viewer/2022062500/56815b1c550346895dc8d19d/html5/thumbnails/19.jpg)
8. Commutation (Comm.), Association (Assoc.), and
Distribution (Dist.)There are three more valid equivalence argument forms:
Commutation (Comm.):
p q is equivalent to q p∨ ∨
p . q is equivalent to q . p
![Page 20: Chapter Four](https://reader035.fdocuments.in/reader035/viewer/2022062500/56815b1c550346895dc8d19d/html5/thumbnails/20.jpg)
Comm., Assoc., and Dist., continued
Association (Assoc.):
p (q r) is equivalent to (p q) r∨ ∨ ∨ ∨p . (q . r) is equivalent to (p . q) . R
![Page 21: Chapter Four](https://reader035.fdocuments.in/reader035/viewer/2022062500/56815b1c550346895dc8d19d/html5/thumbnails/21.jpg)
Comm., Assoc., and Dist., continued
Distribution (Dist):
p . (q r) is equivalent to (p . q) (p . r)∨ ∨
p (q . r) is equivalent to (p q) . (p r)∨ ∨ ∨
![Page 22: Chapter Four](https://reader035.fdocuments.in/reader035/viewer/2022062500/56815b1c550346895dc8d19d/html5/thumbnails/22.jpg)
9. Contraposition, Implication, and Exportation
Contraposition (Contra.):
p q is equivalent to ˜q ˜p⊃ ⊃
Implication (Impl.):
p q is equivalent to ˜p q⊃ ∨
![Page 23: Chapter Four](https://reader035.fdocuments.in/reader035/viewer/2022062500/56815b1c550346895dc8d19d/html5/thumbnails/23.jpg)
Contraposition, Implication, and Exportation, continued
Exportation (Exp.):
(p . q) r is equivalent to p (q r)⊃ ⊃ ⊃
![Page 24: Chapter Four](https://reader035.fdocuments.in/reader035/viewer/2022062500/56815b1c550346895dc8d19d/html5/thumbnails/24.jpg)
10. Tautology and Equivalence
Another valid equivalence form is Tautology
(Taut.):
p is equivalent to p . p
p is equivalent to p p∨
![Page 25: Chapter Four](https://reader035.fdocuments.in/reader035/viewer/2022062500/56815b1c550346895dc8d19d/html5/thumbnails/25.jpg)
Tautology and Equivalence, continued
There are two valid argument forms called
Equivalence (Equiv.):
p ≡ q is equivalent to (p q) . (q p )⊃ ⊃
p ≡ q is equivalent to (p . q) (˜p . ˜q)∨
![Page 26: Chapter Four](https://reader035.fdocuments.in/reader035/viewer/2022062500/56815b1c550346895dc8d19d/html5/thumbnails/26.jpg)
11. More Principles of Strategy
• Break down complex sentences with DeM, Simp, and Equiv.
• Use DeM and Dist to isolate “excess baggage”• Use Impl when you have a mix of conditionals
and disjunctions• Work backward from the conclusion
![Page 27: Chapter Four](https://reader035.fdocuments.in/reader035/viewer/2022062500/56815b1c550346895dc8d19d/html5/thumbnails/27.jpg)
12. Common Errors in Problem Solving
• Using implicational forms on parts of lines• Reluctance to use Addition• Reluctance to use Distribution• Trying to Prove What Cannot be Proved• Failure to Notice the Scope of a Negation Sign
![Page 28: Chapter Four](https://reader035.fdocuments.in/reader035/viewer/2022062500/56815b1c550346895dc8d19d/html5/thumbnails/28.jpg)
Key Terms
• Argument form• Completeness• Decision procedure• Equivalence argument form• Expressive completeness• Implicational argument form• Invalid argument form• Valid argument form