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.)
• If Joe is happy, then he is also smart.
• It is not the case that if John lodges a complaint, then Bill will investigate and Mike will not be disqualified.
• Bill will not investigate unless John lodges a complaint.
• If Carbon is a necessary condition for life, and life exists on Mars, then carbon can be found on Mars.
• Virtue is a necessary and sufficient condition for happiness, unless our life is without purpose.
• 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.
• If Joe is happy, then he is also smart.
• It is not the case that if John lodges a complaint, then Bill will investigate and Mike will not be disqualified.
• Bill will not investigate unless John lodges a complaint.
• If Carbon is a necessary condition for life, and life exists on Mars, then carbon can be found on Mars.
• Virtue is a necessary and sufficient condition for happiness, unless our life is without purpose.
• 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.
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".)
• (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.
1. If you drink enough hemlock, then you will die. You have died, so you must have drunk enough hemlock.
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.
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.
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V1
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V2
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V3
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P1
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P2
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P3
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CN
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1
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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
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Valid or Invalid