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Graded Exercise Three (Module Three)

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Graded Exercise Three (Module Three)

Name: _Noah Khan__________________________________

This version of GE3 uses ASCII symbols for the logical operators since many word processing systems cannot use the regular symbols. Here are the symbols:Conjunction: “&” (used to translate “and,” “but,” “further,” and others that link simple sentences)Disjunction: “v” (used to translate “either/or” and other ways of disjoining simple sentences)Conditional: “>” (used to translate “if…then,” “on the condition that,” “only if” and other conditional forms)Bi-conditional: “=” (used to translate “if and only if,” “necessary and sufficient conditions for…” and other bi-conditional forms)Negation: “~” (used to translate “not” and other forms of negation).

Part I: Translation for ordinary language to symbolic form.

· Translate the following sentences into variable form and identify the variables. (NOTE: these are complex sentences, but they are only sentences, not arguments.)

Example:Hilary is both a carpenter and a member of congress.Answer: C & MWhere:C = Hilary is a carpenterM = Hilary is a member of congress“&” = “and”***Please note: not every logical operator is translated with the “&” symbol. Please see the textbook or the lecture notes to find the appropriate symbols for the appropriate logical operator!

· If Joe is happy, then he is also smart.

Answer

· It is not the case that if John lodges a complaint, then Bill will investigate and Mike will not be disqualified.

Answer

· Bill will not investigate unless John lodges a complaint.

Answer

· If Carbon is a necessary condition for life, and life exists on Mars, then carbon can be found on Mars.

Answer

· Virtue is a necessary and sufficient condition for happiness, unless our life is without purpose.

Answer

· Translating Arguments. Translate the following arguments into their formal equivalents. You will have to identify the conclusion, the premises, the variables, and the logical operators, and then arrange them accordingly.

ExampleMost people in America are either Republicans or Democrats. If they are Republicans, then they probably voted for Bush and if they are Democrats, then they probably voted for Gore. So, most people in America voted either for Bush or Gore.
Answer:1. R v D.2. R > B.3. D > G.CN: B v G.***Don’t number the conclusion! Just use “CN:” to indicate “conclusion”*** Where:R = Most people in America are RepublicansD = Most people in America are DemocratsB = They voted for BushG = They voted for Gore

· If you believe in Santa Claus, then you must believe in the Easter Bunny. You don’t believe in the Easter Bunny, so you don’t believe in Santa Claus.

Answer

· The administration is either purposefully lying to us or they are mistaken. If the administration is lying to us, then they must have a good reason. If they are mistaken, then they must be a bad administration. But, there is no good reason to lie to us. So, they must be a bad administration.

Answer

· There is no good reason to be a carnivore, since if there were good reason to be a carnivore, then there would be no good reason to be a vegetarian. But, there is good reason to be a vegetarian.

Answer

· John didn’t notice the behavior of his employees unless he approved of it. He noticed the behavior. So, he must have approved it.

Answer

· If Carol still had her purse, then theft could not have been the motive for her assault. But, it was either theft or a random attack. Since Carol still had her purse, it must have been a random attack.

Answer

Part II A: Determining the TRUTH VALUE of sentences using logical operators. Assuming that A, B, and C are ALWAYS TRUE, and X, Y, and Z are ALWAYS FALSE determine whether the following sentences are true or false. (NOTE: These are like the exercises in 7.2 in the textbook.)

Example: (A & Y) v (Z & X)Answer: False.A&YvZ&XTFFFFFFLeft side of “v” is falseThe whole sentence is FalseRight side of “v” is falseNote: v = either or. This is only true when at least ONE side of the “v” is true. So, the first step is to figure out the truth value of either side. Both sides are conjunctions (), and these are only true when BOTH sides of the conjunction are true. Since, both conjunctions contain at least one false statement, both sides of the “v” are false. So, the whole sentence is false.

REMEMBER: Assume that A, B, C are ALWAYS TRUE, and X, Y, Z are ALWAYS FALSE

· (Z v B) > ~C

· {~[(A > B) v X] > ~Z} = [(B & Y) v (~Z > X)]

Part II B: For each of the following sentences, use a truth table to determine if they are (a) tautological; (b) contradictory; or (c) contingent. ( See textbook—section 7.5 for more information .) (NOTE: I am using ASCII for the symbols here to make sure everyone can read them. Here is the notation: “&” means “and”; “v” means “either/or”; “>” means “if-then”; “=” means “if and only if”; and “~” means “not”.)

EXAMPLE:Sentence: [p > (p > q)] > qAnswer: (set up a truth table—you have to make your own)—YOU MUST HAVE A COMPLETE TRUTH TABLE TO GET CREDIT!!!12345 Answerpq(p > q)[p > (p > q)][p > (p > q)] > qTTTTTTFFFTFTTTTFFTTFANSWER: CONTINGENT. Explanation of your answer: (THIS IS ALSO AN IMPORTANT PART OF THE ANSWER!!!) Since the proposition is sometimes true and sometimes false (notice it is false in the last row, but true in all the others), then this sentence is contingent

· (p & q) v (~p > q)

· (p > p) > (q & ~q)

· (q & ~q) > ~(p v ~p)

Part III: Truth Tables: Use truth tables to determine the validity of the following arguments. 1. Determine the number of variables. 2. Rewrite the argument in symbolic form. 3. Use the table provided below to demonstrate validity or invalidity. If the argument is invalid INDICATE the row or rows that show this (see the example below).

V1 stands for Variable One. V2 stands for variable 2 and so on. P1 stands for Premise One, P2 stands for Premise Two, P3 stands for Premise Three and so on. CN stands for Conclusion.

Example:We are either in Chicago or on the Moon. Since we are on the Moon, we must not be in Chicago.C v MV1V2P1P2CNCMC v MM~C1TTTTF2TFTFF3FTTTT4FFFFTM___~CWhere:C = we are in Chicago.M = we are on the Moon.Valid or Invalid? INVALIDNotice that “we are either in Chicago or on the Moon” is TWO simple sentences with ONE logical operator (either/or). So, we split the two parts: we are in Chicago/we are on the Moon.Notice that we have a NEGATION (~) in the conclusion, so C becomes ~C. Also, remember that we know the second clause of the last sentence is the conclusion because the word “since” indicates a premise and it is in front of the first clause of the second sentence.Finally , look at the truth table. First, note that there are four possible combinations of truth value for C & M (our only two variables). Also note that C v M is ONLY FALSE when BOTH variables are FALSE, hence, it is only false in the last row (#4). Also note that the second premise is simply the same sentence as the variable “M”, hence the second premise’s truth value is the SAME as the truth value of M. Finally, notice that the conclusion is the NEGATION of C, hence its truth value is the OPPPOSITE of C’s. So, where C is true, ~C is false. Now, in the very first row we have TWO TRUE PREMISES and a FALSE CONCLUSION. This is impossible to get if the argument is valid. Since we have it, the argument is INVALID.

1. If you drink enough hemlock, then you will die. You have died, so you must have drunk enough hemlock.

V1 V2 P1 P2 CN
1
2
3
4

Valid or Invalid

2. Either Alito is an honest judge or Alito allows his ideology to inform his judicial opinions. If Alito allows his ideology to inform his judicial opinions, then Alito is not a good candidate for the Supreme Court. Alito is an honest judge. So, he is must be a good candidate for the Supreme Court.

V1 V2 V3 P1 P2 P3 CN

Valid or Invalid.

3. If we focus more on securing individual privacy, then we risk an increase in successful terrorist attacks. If we strengthen our defense against terrorist attacks, then we cannot focus on securing individual privacy. We can either focus on securing individual privacy or strengthen our defense against terrorist attacks. Thus, we will either risk an increase in successful terrorist attacks, or cannot secure individual privacy.

V1 V2 V3 P1 P2 P3 CN
1
2
3
4
5
6
7
8

Valid or Invalid

4. If Carol still had her purse, then theft could not have been the motive for her assault. But, it was either theft or a random attack. Since Carol still had her purse, it must have been a random attack

V1 V2 V3 P1 P2 P3 CN
1
2
3
4
5
6
7
8

Valid or Invalid

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