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Chapter: 9 -Problem: 21 >> Translate each of the following into the logical notation of propositional functions and quantifiers, in each case making the formula begin with a quantifier, not with a negation symbol. Nothing is attained in war except by calculation. —Napoleon Bonaparte
Answer Preview: Nothing is attained i…

, Chapter: 9 -Problem: 14 >> For the following statements, if there are forced truth-value assignments, make them. If there are no forced truth-value assignments, determine, using Maxim V, the requisite truth-value combinations on which, and the number of ways that, the statement can be true (or false), as the case requires. A ? (B ? C) T F F
Answer Preview: A T …

, Chapter: 9 -Problem: 1 >> For each of the following arguments, construct an indirect proof of validity. Transcribed Image Text: Pr: A V (B · C) P?: ADC :.c C
Answer Preview: 1 AV B C 2 ADC 3 C …

, Chapter: 9 -Problem: 17 >> For each of the following, either construct a formal proof of validity or prove invalidity by means of the STTT. In each case, use the notation in parentheses. If the butler were present, he would have been seen; and if he had been seen, he would have been questioned. If he had been questioned, he would have replied; and if he had replied, he would have been heard. But the butler was not heard. If
Answer Preview: If the butler were present he would have been seen P S If he had been seen he would have been questioned S Q If he had been questioned he would have r…

, Chapter: 9 -Problem: 1 >> For each numbered statement that is not a premise in each of the formal proofs that follow, state the rule of inference that justifies it. 1. A ? B 2. C ? ~ B ? A ? ~ C3. ~~ B ? ~ C4. B ? ~ C5. A ? ~ C
Answer Preview: Conditional proof Conditional proof Hypothetical syllogism Double negation Modus …

, Chapter: 9 -Problem: 20 >> For each of the following arguments, a formal proof of validity can be constructed without great difficulty, although some of the proofs may require a sequence of eight or nine statements (including premises) for their completion. (P1): (Q ? R) ? (S ? T) (P2): (U ? V) ? (W ? X) (P3): Q ? U ? R ? V
Answer Preview: Q U P3 Given Q R Simplification P1 U V Simplification P2 S T Simplification …

, Chapter: 7 -Problem: 24 >> For each of the following enthymematic arguments: a. Formulate the plausible premise or conclusion, if any, that is missing but understood. b. Write the argument in standard form, including the missing premise or conclusion needed to make the completed argument valid—if possible—using parameters if necessary. c. Name the order of the enthymeme. d. If the argument is not valid even with the underst
Answer Preview: a Unstated conclusion Suicide is forbidden by the law b Stand…

, Chapter: 14 -Problem: 2 >> What is the probability of getting tails every time in three tosses of a coin?
Answer Preview: The probability of getting tails on a single toss of a …

, Chapter: 9 -Problem: 18 >> The following set of arguments involves, in each case, one inference only, in which one of the ten logical equivalences set forth in this section has been employed. Here are two examples, the first two in the exercise set immediately following. (P1): [~ E ? (~~ F ? G)] ? [~ E ? (F ? G)] ? [~ E ? (F ? G)] ? [~ E ? (F ? G)]
Answer Preview: The inference used here is the implication equivalence p …

, Chapter: 8 -Problem: 7 >> Use truth tables to prove the validity or invalidity of each of the argument forms in Section 8.5, Group B, pp. 288–9. (P1): M ? (N · ~ N) (P2): M ? ~(N ? ~ )
Answer Preview: M N N has the p q qM spe…

, Chapter: 9 -Problem: 4 >> Use Conditional Proof to prove the validity of the following arguments. Transcribed Image Text: (P?): JD (AVS) (P?): KD (SVI) (P?): ~S .. (~A~I) (~J. ~K)
Answer Preview: 1 JD AVS 2 KD SVI 3 S 4 5 6 7 8 9 10 11 …

, Chapter: 9 -Problem: 21 >> Use Conditional Proof (C.P.) to prove that the following statements are tautologies. [(W ? X) ? X] ? (W ? X)
Answer Preview: W X X W X Done differently with a nested C …

, Chapter: 9 -Problem: 16 >> The following set of arguments involves, in each case, one inference only, in which one of the ten logical equivalences set forth in this section has been employed. Here are two examples, the first two in the exercise set immediately following. (P1): [V ? ~ (W ? X)] . (Y ? Z) ? {[V ? ~ (W ? X)] ? [V ? ~ (W ? X)]} . (Y ? Z)
Answer Preview: The inference used in this argument is the distributive law of conjunction which states that P …

, Chapter: 9 -Problem: 3 >> For each of the following, either construct a formal proof of validity or prove invalidity by means of the STTT. (P1): I ? (J  ? K) (P2): (J ? K) ? L ? I ? L
Answer Preview: or I…

, Chapter: 9 -Problem: 20 >> Use Conditional Proof (C.P.) to prove that the following statements are tautologies. (W ? X) ? [(W ? Y) ? (X ? Y)]
Answer Preview: W X W Y X Y 1 2 3 4 5 Y W 6 Y …

, Chapter: 9 -Problem: 13 >> Here follows a set of twenty elementary valid arguments. They are valid because each of them is exactly in the form of one of the nine elementary valid argument forms. For each of them, state the rule of inference by which its conclusion follows from its premise or premises. (P1): (C ? D) ? [(J ? K) ? (J ? K)] (P2): ~ [(J ? K) . (J ? K)] ? ~ (C ? D)
Answer Preview: 1 J K J K Premise 2 J K J K Premise 3 J K Simplification from 2 4 J K Simplification from 2 5 C D …

, Chapter: 14 -Problem: 4 >> Some of the following passages contain explanations, some contain arguments, and some may be interpreted as either an argumaent or an explanation. What is your judgment about the chief function of each passage? What would have to be the case for the passage in question to be an argument? To be an explanation? Where you find an argument, identify its premises and conclusion. Where you find an expla
Answer Preview: This is an argument Its premis…

, Chapter: 9 -Problem: 17 >> Translate each of the following into the logical notation of propositional functions and quantifiers, in each case using the abbreviations suggested and making each formula begin with a quantifier, not with a negation symbol. Not every applicant was hired. ( Ax: x is an applicant; Hx: x was hired.)
Answer Preview: The statement Not every applicant was hired can be restated as There exi…

, Chapter: 8 -Problem: 8 >> Use truth tables to prove the validity or invalidity of each of the argument forms in Section 8.5, Group B, pp. 288–9. (P1): (O ? P) ? Q (P2): Q ? (O ? P) ?(O ? P) ? (O ? P)
Answer Preview: O P Q has the p q rQ O P specific form r p q O P O …

, Chapter: 9 -Problem: 3 >> Each of the following exercises presents a flawless formal proof of validity for the indicated argument. For each proof, state the justification for each inferred statement (i.e., each statement that is not a premise). 1. I ? J 2. J ? K 3. L ? M 4. I ? L ? K ? M 5. I ? K 6. (I ? K) ? (L ? M) 7. K ? M
Answer Preview: 5 1 2 …

, Chapter: 9 -Problem: 21 >> For each of the following arguments, a formal proof of validity can be constructed without great difficulty, although some of the proofs may require a sequence of eight or nine statements (including premises) for their completion. (P1): W ? X (P2): (W ? X) ? Y (P3): (W ? Y) ? Z ? W ? Z
Answer Preview: 1 W X 2 W X Y 3 …

, Chapter: 9 -Problem: 11 >> Use Conditional Proof (C.P.) to prove that the following statements are tautologies. [D ? (F ? G)] ?[F ? (D ? G)]
Answer Preview: D FG F DG 1 DO FDG 2 F 3 …

, Chapter: 8 -Problem: 70 >> Using the letters E , I , J , L , and S to abbreviate the simple statements, “Egypt’s food shortage worsens,” “Iran raises the price of oil,” “Jordan requests more U.S. aid,” “Libya raises the price of oil,” and “Saudi Arabia buys fi ve hundred more warplanes,” symbolize these statements.
Answer Preview: I …

, Chapter: 9 -Problem: 19 >> Here follows a set of twenty elementary valid arguments. They are valid because each of them is exactly in the form of one of the nine elementary valid argument forms. For each of them, state the rule of inference by which its conclusion follows from its premise or premises. (P1): [(H ? ~ I) ? C] ? [(I ? ? H) ? D] (P2): (H ? ~ I) ? (I ? ~ H) ? C ? D
Answer Preview: Argument 1 Premises P1 H I C I H D P2 H I I H Conclusion C D Rule of inference The argum…

, Chapter: 9 -Problem: 11 >> For each of the following arguments, inferring just two statements from the premises will produce a formal proof of its validity. Construct a formal proof for each of these arguments. In these formal proofs, and in all the proofs to follow in later sections, note to the right of each inferred statement the rule of inference that is used to infer that statement. It is most convenient if the justifi
Answer Preview: 1 A A Premise 2 A A A Equivalent form of 1 3 A A A Contrapositive of 2 4 …

, Chapter: 8 -Problem: 29 >> Symbolize the following, using capital letters to abbreviate the simple statements involved. If Argentina mobilizes, then Brazil will protest to the UN, and Chile will call for a meeting of all the Latin American states.
Answer Preview: A…

, Chapter: 9 -Problem: 19 >> Use Conditional Proof (C.P.) to prove that the following statements are tautologies. (W ? X) ? [(~ W ? X) ? X]
Answer Preview: W V X W X X 1 2 3 4 5 6 7 8 9 …

, Chapter: 8 -Problem: 27 >> Use truth tables to determine the validity or invalidity of the following arguments: If terrorists’ demands are met, then lawlessness will be rewarded. If terrorists’ demands are not met, then innocent hostages will be murdered. So either lawlessness will be rewarded or innocent hostages will be murdered.
Answer Preview: T L has the p q T I specific form p r …

, Chapter: 9 -Problem: 17 >> For each of the following arguments, a formal proof of validity can be constructed without great difficulty, although some of the proofs may require a sequence of eight or nine statements (including premises) for their completion. (P1): (F ? G) ? (H ? I) (P2): J ? K (P3): (F ? J) ? (H ? L) ? G ? K
Answer Preview: F G H I Premise J K Premise F J Premise H L Premise F G Si…

, Chapter: 9 -Problem: 13 >> The following set of arguments involves, in each case, one inference only, in which one of the ten logical equivalences set forth in this section has been employed. Here are two examples, the first two in the exercise set immediately following. (P1): (~ K ? L) ? (~ M ? ~ N) ? (~ K ? L) ? ~ (M ? N)
Answer Preview: This argument involves the logical equivalence known as De Morgans Law De Morgans La…

, Chapter: 8 -Problem: 69 >> Using the letters E , I , J , L , and S to abbreviate the simple statements, “Egypt’s food shortage worsens,” “Iran raises the price of oil,” “Jordan requests more U.S. aid,” “Libya raises the price of oil,” and “Saudi Arabia buys fi ve hundred more warplanes,” symbolize these statements.
Answer Preview: E …

, Chapter: 8 -Problem: 35 >> Symbolize the following, using capital letters to abbreviate the simple statements involved. It is not the case that if Argentina mobilizes, then both Brazil will protest to the UN, and Chile will call for a meeting of all the Latin American states.
Answer Preview: Let A represent the statement Argentina mobi…

, Chapter: 9 -Problem: 14 >> Use indirect proof to prove that the following statements are tautologies. (A ? B) ? (A ? ~ B)
Answer Preview: A B A B 1 2 3 4 5 AB V A B ADB A B AVB ADB AV B AV B A B AVB …

, Chapter: 14 -Problem: 3 >> If the odds in craps are 8-to-1 against rolling an 8 the “hard way” (that is, with two 4’s), what is the expectation purchased by a $1 bet on a “hard-way” 8?
Answer Preview: In craps rolling an 8 with two 4s is called a hardway …

, Chapter: 9 -Problem: 18 >> Use Conditional Proof (C.P.) to prove that the following statements are tautologies. (W ? X) ? [W ? (W ? X)]
Answer Preview: To prove that W X W W X is a tautology using Conditional Proof we assume W X …

, Chapter: 9 -Problem: 12 >> For each of the following arguments, it is possible to provide a formal proof of validity by validly inferring just three statements. Writing these out, carefully and accurately, will strengthen your command of the rules of inference, a needed preparation for the construction of proofs that are more extended and more complex. (P1): L ? (M . N) (P2): ~ L ? (N . O) (P3): ~ L ? M ? O
Answer Preview: 1 L M N 2 L N O 3 …

, Chapter: 9 -Problem: 2 >> For each numbered statement that is not a premise in each of the formal proofs that follow, state the rule of inference that justifies it. 1. (D ? E) ? F 2. (D ? F) ? G ? E ? G 3. (E ? D) ? F 4. E ? (D ? F) 5. E ? G
Answer Preview: Hypothetical syllogism HS Hypothetical syllogism HS Commutativity of c…

, Chapter: 9 -Problem: 15 >> Use indirect proof to prove that the following statements are tautologies. (A ? B) ? (~ A ? B)
Answer Preview: A B A B AB V A B ADB A B AV B A B AV …

, Chapter: 9 -Problem: 4 >> Translate each of the following into the logical notation of propositional functions and quantifiers, in each case using the abbreviations suggested and making each formula begin with a quantifier, not with a negation symbol. Nurses are always considerate. (Nx: x is a nurse; Cx: x is considerate.)
Answer Preview: x…

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, Chapter: 8 -Problem: 32 >> Symbolize the following, using capital letters to abbreviate the simple statements involved. Either Argentina will mobilize or if Brazil protests to the UN, then Chile will call for a meeting of all the Latin American states.
Answer Preview: A…

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, Chapter: 9 -Problem: 14 >> Use Conditional Proof (C.P.) to prove that the following statements are tautologies. R ? ~~ R
Answer Preview: RD R 1 …

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, Chapter: 9 -Problem: 20 >> Use indirect proof to prove that the following statements are tautologies. R ? ~~ R
Answer Preview: R R 1 R R 2 RR V R R 3 RR V R …

, Chapter: 9 -Problem: 15 >> For each of the following arguments, inferring just two statements from the premises will produce a formal proof of its validity. Construct a formal proof for each of these arguments. In these formal proofs, and in all the proofs to follow in later sections, note to the right of each inferred statement the rule of inference that is used to infer that statement. It is most convenient if the justifi
Answer Preview: To prove the validity of the argument we need to derive the conclusion from the p…

, Chapter: 9 -Problem: 1 >> For each of the following arguments, it is possible to provide a formal proof of validity by validly inferring just three statements. Writing these out, carefully and accurately, will strengthen your command of the rules of inference, a needed preparation for the construction of proofs that are more extended and more complex.(P1): A ? (B ? A) (P2): ~ A ? C ? ~ B
Answer Preview: 1 A B A 2 A …

, Chapter: 9 -Problem: 22 >> For each of the following, either construct a formal proof of validity or prove invalidity by means of the STTT. In each case, use the notation in parentheses. If your prices are low, then your sales will be high, and if you sell quality merchandise, then your customers will be satisfied. So if your prices are low and you sell quality merchandise, then your sales will be high and your customers sa
Answer Preview: 1 LH QS LQ H S 2 LOH 3 LVH 4 LVH V Q 5 LV HV Q LV QVH 6 7 …

, Chapter: 9 -Problem: 12 >> The following set of arguments involves, in each case, one inference only, in which one of the ten logical equivalences set forth in this section has been employed. Here are two examples, the first two in the exercise set immediately following. (P1): (H ? ~ I) ? (~ I ? ~ J) ? (H ? ~ I) ? (J ? I)
Answer Preview: To prove the argument 1 H I I J Premise 2 H I Assumption 3 I J Modus pone…

, Chapter: 9 -Problem: 3 >> For each of the following arguments, construct an indirect proof of validity. Transcribed Image Text: P?: (DVE) (FG) P?: (~GVH) (D.F) .. G
Answer Preview: 1 DVE FDG 2 GVH DF 3 G 4 5 6 7 …

, Chapter: 8 -Problem: 36 >> Use truth tables to decide which of the following biconditionals are tautologies. [p · (q ? r)] = [(p ? q) ·  (p ? r)]
Answer Preview: To create a truth table for this biconditional we ne…

, Chapter: 14 -Problem: 2 >> Identify and explain any fallacies of defective induction or of presumption in the following passages: A national mailing soliciting funds, by People for the Ethical Treatment of Animals (PETA), included a survey in which questions were to be answered “yes” or “no.” Two of the questions asked were these: “Do you realize that the vast majority of painful animal experimentation has no relation at al
Answer Preview: The fallacy in this passage is loaded question or complex question These types of questions are a…

, Chapter: 9 -Problem: 19 >> For each of the following arguments, a formal proof of validity can be constructed without great difficulty, although some of the proofs may require a sequence of eight or nine statements (including premises) for their completion. (P1): (K ? L) ? (M ? N) (P2): (M ? N) ? (O ? P) (P3): K ? O
Answer Preview: We are given the following premises P1 K L M N P2 M N O P P3 K And we need to prove the conclusion …

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, Chapter: 9 -Problem: 1 >> For the following statements, if there are forced truth-value assignments, make them. If there are no forced truth-value assignments, determine, using Maxim V, the requisite truth-value combinations on which, and the number of ways that, the statement can be true (or false), as the case requires. Z ? W F
Answer Preview: Z …

, Chapter: 9 -Problem: 37 >> For each of the following arguments, use the STTT to determine whether the argument is valid or invalid. For some of these arguments, Steps 2 C , 3 C , and 4 (i.e., the C-Sequence) will be most efficient; for some of these arguments Steps 2 P , 3 P , and 4 (i.e., the P-Sequence) will be most efficient. (P1): D ? (E ? F) (P2): G ? (H ? I) (P3): ~ E ? (I ? J) (P4): (I ? G) ? (~ H ? ~ G) (P5): ~ J ?
Answer Preview: P 1 D E F P 2 G H I P 3 E I J P 4 I G H G P 5 J D G I To make the conclus…

, Chapter: 9 -Problem: 21 >> For each of the following, either construct a formal proof of validity or prove invalidity by means of the STTT. In each case, use the notation in parentheses. If we buy a lot, then we will build a house. If we buy a lot, then if we build a house we will buy furniture. If we build a house, then if we buy furniture we will buy dishes. Therefore if we buy a lot, we will buy dishes. ( L, H, F, D )
Answer Preview: 1 LOH 2 LD HDF 3 HD FD LO…

, Chapter: 8 -Problem: 17 >> Use truth tables to determine whether the following statement forms are tautologous, self-contradictory, or contingent. [p ? (q ? r)] ? [( p ? q) ? ( p ? r )]
Answer Preview: Tautolo…

, Chapter: 14 -Problem: 3 >> Identify the premises and conclusions in the following passages. Some premises do support the conclusion, others do not. Note that premises may support conclusions directly or indirectly and that even simple passages may contain more than one argument. Thomas Aquinas argued that human intelligence is a gift from God and therefore “to apply human intelligence to understand the world is not an affro
Answer Preview: Premise Human intelligence is a …

, Chapter: 10 -Problem: 11 >> Prove the invalidity of the following, in each case using the suggested notation: All anarchists are bearded. All communists are bearded. Therefore all anarchists are communists. (Ax, Bx, Cx)
Answer Preview: This argument is invalid Here s a counterexample Let A be the set of all anarchists C be the set of …

, Chapter: 9 -Problem: 18 >> For the following statements, if there are forced truth-value assignments, make them. If there are no forced truth-value assignments, determine, using Maxim V, the requisite truth-value combinations on which, and the number of ways that, the statement can be true (or false), as the case requires. (J ? K) ? (L ? M) T T
Answer Preview: J T T F V T …

, Chapter: 8 -Problem: 2 >> Use truth tables to prove the validity or invalidity of each of the argument forms in Section 8.5, Group B, pp. 288–9. (P1): (C ? D)(C · D) (P2): C · D ?C ? D
Answer Preview: C D C D has the p q p qC …

, Chapter: 9 -Problem: 1 >> Use Conditional Proof to prove the validity of the following arguments. Transcribed Image Text: (P?): ADB (P?): B [(C~~C) D] :. ADD
Answer Preview: 1 A D B 2 B CDCDD 3 4 5 …

, Chapter: 9 -Problem: 28 >> Use Conditional Proof (C.P.) to prove that the following statements are tautologies. (R ? Q) ? [~ (Q ? S) ? ~ (S ? R)]
Answer Preview: RQ 1 2 3 4 5 6 7 8 9 10 Q S S R R…

, Chapter: 9 -Problem: 22 >> For each of the following arguments, inferring just two statements from the premises will produce a formal proof of its validity. Construct a formal proof for each of these arguments. In these formal proofs, and in all the proofs to follow in later sections, note to the right of each inferred statement the rule of inference that is used to infer that statement. It is most convenient if the justifi
Answer Preview: Q S Premise Q S Double negation on line 1 Q R Simplification …

, Chapter: 9 -Problem: 13 >> For each of the following, either construct a formal proof of validity or prove invalidity by means of the STTT. In each case, use the notation in parentheses. If God were willing to prevent evil, but unable to do so, he would be impotent; if he were able to prevent evil, but unwilling to do so, he would be malevolent. Evil can exist only if God is either unwilling or unable to prevent it. There i
Answer Preview: 1 W A I A W M 2 E W A 3 E …

, Chapter: 14 -Problem: 2 >> Hundreds of thousands of recent college graduates today cannot express themselves with the written word. Why? Because universities have shortchanged them, offering strange literary theories, Marxism, feminism, deconstruction, and other oddities in the guise of writing courses. —Stanley Ridgeley, “College Students Can’t Write?” National Review Online , 19 February 2003
Answer Preview: Premise Universities have commonly been offering stran…

, Chapter: 10 -Problem: 1 >> Construct a formal proof of validity for each of the following arguments: Transcribed Image Text: (P?): (x)(Ax-Bx) (P?): (3x)(Cx. Ax) .: (3x) (Cx - Bx)
Answer Preview: To prove the validity of the argument we need to demonstrate that the conclusion follows necessarily …

, Chapter: 9 -Problem: 20 >> The following set of arguments involves, in each case, one inference only, in which one of the ten logical equivalences set forth in this section has been employed. Here are two examples, the first two in the exercise set immediately following. (P1): (~ M ? ~ N) . (O ? ~~ P) ? ~ (M ? N) ? (O ? ~~ P)
Answer Preview: To prove the argument we can use the logical equivale…

, Chapter: 8 -Problem: 20 >> Use truth tables to determine whether the following statement forms are tautologous, self-contradictory, or contingent. {[(p ? q) ? (r ? s)] ? (q ? s)} ? (p ? r)
Answer Preview: Contingent…

, Chapter: 14 -Problem: 4 >> Which of the various functions of language are exemplified by each of the following passages? Moving due south from the center of Detroit, the fi rst foreign country one encounters is not Cuba, nor is it Honduras or Nicaragua or any other Latin American nation;  it is Canada.
Answer Preview: This passage exemplifies the referential function of language The referential function of language i…

, Chapter: 10 -Problem: 5 >> Prove the invalidity of the following: Transcribed Image Text: (P?): (3x)(Mx Nx) (P?): (Ex) (Mx Ox) :: (x) (Ox? Nx)
Answer Preview: To prove the invalidity of the given argument we need to show that the conclusion does not logically …

, Chapter: 9 -Problem: 18 >> Use indirect proof to prove that the following statements are tautologies. (A ? B) ? (~ A ? C)
Answer Preview: To prove this statement by indirect proof we assume the negatio…

, Chapter: 14 -Problem: 4 >> A. Identify three disagreements in current political or social controversy that are of the three types described in this section: one that is genuine, one that is merely verbal, and one that is apparently verbal but really genuine. Explain the disagreements in each case. B. Discuss each of the following disputes. If the dispute is obviously genuine, indicate each of the disputers’ positions with r
Answer Preview: An obviously genuine dispute about whether ear…

, Chapter: 9 -Problem: 39 >> For each of the following, find a normal-form formula that is logically equivalent to the given one: ? (?x) [? (? Qx ? Rx)]
Answer Preview: x Q…

, Chapter: 9 -Problem: 16 >> Use Conditional Proof (C.P.) to prove that the following statements are tautologies. [(W ? X) ? (W ? Y)] ? [W ? (W ? Y)]
Answer Preview: W X WY W WV Y W X WDY …

, Chapter: 14 -Problem: 4 >> Identify and explain the fallacies of relevance in the following passages: However, it matters very little now what the king of England either says or does; he hath wickedly broken through every moral and human obligation, trampled nature and conscience beneath his feet, and by a steady and constitutional spirit of insolence and cruelty procured for himself an universal hatred. —Thomas Paine, Comm
Answer Preview: The fallacy of relevance in this passage is an Ad Hominem which attacks the person making the a…

, Chapter: 9 -Problem: 29 >> If any truth-functional argument is valid, we have the tools to prove it valid; and if it is invalid, we have the tools to prove it invalid. For each of the following arguments determine whether it is valid or invalid using the STTT. If it is valid, prove it valid using the nineteen rules of inference. These proofs will be more difficult to construct than in preceding exercises, but they will offe
Answer Preview: 1 W RV R 2 3 4 RD W 5 6 …

, Chapter: 8 -Problem: 35 >> Use truth tables to decide which of the following biconditionals are tautologies. [p · (q ? r)] = [(p ? q) ? ( p ? r)]
Answer Preview: To determine whether the biconditional is a tautology we need to construct …

, Chapter: 8 -Problem: 29 >> Use truth tables to decide which of the following biconditionals are tautologies. p = [p ? (p ? q)]
Answer Preview: T T F F Not a t…

, Chapter: 9 -Problem: 28 >> If any truth-functional argument is valid, we have the tools to prove it valid; and if it is invalid, we have the tools to prove it invalid. For each of the following arguments determine whether it is valid or invalid using the STTT. If it is valid, prove it valid using the nineteen rules of inference. These proofs will be more difficult to construct than in preceding exercises, but they will offe
Answer Preview: 1 TI BDU 2 K TV B KO IVU 3 TVI BDU 4 TVI BVU 5 TVI 6 TVIVU 7 TV IVU 8 BV…

, Chapter: 9 -Problem: 17 >> The following set of arguments involves, in each case, one inference only, in which one of the ten logical equivalences set forth in this section has been employed. Here are two examples, the first two in the exercise set immediately following. (P1): [(~ A ? B) ? (C ? D)] ? [~ (~ A ? B) ? ~ (C ? D)] ? (~ A ? B) ? (C ? D)
Answer Preview: To prove that the two sides of the argument are logically equivalent we can start by assuming one si…

, Chapter: 14 -Problem: 3 >> An urn contains 27 white balls and 40 black balls. What is the probability of getting four black balls in four successive drawings: A. If each ball drawn is replaced before making the next drawing? B. If the balls are not replaced?
Answer Preview: a 4067 4067 406…

, Chapter: 9 -Problem: 37 >> For each of the following, find a normal-form formula that is logically equivalent to the given one: (?x) [? (Mx ? Nx)]
Answer Preview: x M…

, Chapter: 9 -Problem: 16 >> Use indirect proof to prove that the following statements are tautologies. (A ? B) ? (B ?A)
Answer Preview: To prove that A B B A is a tautology we can use the indirect proof We assume t…

, Chapter: 14 -Problem: 4 >> Orlando Patterson claims that “freedom is a natural part of the human condition.” Nothing could be further from the truth. If it were true, we could expect to find free societies spread throughout human history. We do not. Instead what we fi nd are every sort of tyrannical government from time immemorial. —John Taylor, “Can Freedom Be Exported?” The New York Times , 22 December 2006
Answer Preview: Premise If freedom were a natural part of the human condition we could …

, Chapter: 9 -Problem: 26 >> Translate each of the following into the logical notation of propositional functions and quantifiers, in each case making the formula begin with a quantifier, not with a negation symbol. There isn’t any business we can’t improve. —Advertising slogan, Ernst and Young, Accountants
Answer Preview: There isnt any busine…

, Chapter: 9 -Problem: 16 >> Translate each of the following into the logical notation of propositional functions and quantifiers, in each case using the abbreviations suggested and making each formula begin with a quantifier, not with a negation symbol. Citizens of the United States can vote only in U.S. elections. (Ex: x is an election in which citizens of the United States can vote; Ux: x is a U.S. election.)
Answer Preview: To translate the given statement into logical notation we can …

, Chapter: 9 -Problem: 10 >> For each of the following arguments, construct an indirect proof of validity. Transcribed Image Text: (P?): (NVF) (C.D) (P?): DDV (P?): VOI (P): IDA (P): AD ~C ~
Answer Preview: NVF C D 1 2 DDV 3 VOI 4 IDA 5 AD C 6 F 7 …

, Chapter: 14 -Problem: 2 >> A. Identify three disagreements in current political or social controversy that are of the three types described in this section: one that is genuine, one that is merely verbal, and one that is apparently verbal but really genuine. Explain the disagreements in each case. B. Discuss each of the following disputes. If the dispute is obviously genuine, indicate each of the disputers’ positions with r
Answer Preview: An apparently verbal dispute that is really genuine Th…

, Chapter: 9 -Problem: 42 >> For each of the following arguments, a formal proof may be constructed by making just three inferences. Construct a formal proof of validity for each of them. (P1): ~ B ? (C ? D) ? B ? C
Answer Preview: 1 B C D Premise 2 B B C Conditional proof 3 B A…

, Chapter: 9 -Problem: 16 >> Here follows a set of twenty elementary valid arguments. They are valid because each of them is exactly in the form of one of the nine elementary valid argument forms. For each of them, state the rule of inference by which its conclusion follows from its premise or premises. (P1): Q ? (O ? R) (P2): N ? (O ? P) ? [Q ? (O ? R)] ? [N ? (O ? P)]
Answer Preview: The conclusion of this argument is a conjunction of two conditional statements In order to show that …

, Chapter: 8 -Problem: 33 >> Symbolize the following, using capital letters to abbreviate the simple statements involved. If Argentina does not mobilize, then either Brazil will not protest to the UN or Chile will not call for a meeting of all the Latin American states.
Answer Preview: A…

, Chapter: 10 -Problem: 2 >> Construct a formal proof of validity for each of the following arguments: Transcribed Image Text: - Ex) Ex) (P?): (x)(Dx (P?): (x)(Fx ..(x) (Fx - Dx)
Answer Preview: Argument 1 Premise 1 x Dx Ex Premise 2 x Fx Ex To prove that the argument is valid we need …

, Chapter: 9 -Problem: 21 >> For each of the following arguments, use the STTT to determine whether the  argument is valid or invalid. For some of these arguments, Steps 2 C , 3 C , and 4 (i.e., the C-Sequence) will be most efficient; for some of these arguments Steps 2 P , 3 P , and 4 (i.e., the P-Sequence) will be most efficient. (P1): (E ? F) ? (G ? H) (P2): (G ? H) ? I (P3): E ? I
Answer Preview: P 1 E v F G H P 2 G H I P 3 E I The conclusion is simple statement I so we begin wit…

, Chapter: 8 -Problem: 16 >> Use truth tables to determine whether the following statement forms are tautologous, self-contradictory, or contingent. (p ? p) ? (q · ~q)
Answer Preview: Selfcon…

, Chapter: 9 -Problem: 1 >> Translate each of the following into the logical notation of propositional functions and quantifiers, in each case using the abbreviations suggested and making each formula begin with a quantifier, not with a negation symbol. No beast is without some touch of pity. (Bx: x is a beast; Px: x has some touch of pity.)
Answer Preview: x…

, Chapter: 9 -Problem: 31 >> For each of the following, find a normal-form formula that is logically equivalent to the given one: ? (x) (Ax ? Bx)
Answer Preview: x…

, Chapter: 9 -Problem: 15 >> For each of the following arguments, it is possible to provide a formal proof of validity by validly inferring just three statements. Writing these out, carefully and accurately, will strengthen your command of the rules of inference, a needed preparation for the construction of proofs that are more extended and more complex.(P1): (Z ? A) ? B (P2): B ? A (P3): (B ? A) ? (A ? B) ? (Z ? A) ? (A ? B)
Answer Preview: Here is a formal proof of the argument Z A B Premise B A Premise B A A …

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, Chapter: 10 -Problem: 3 >> Translate the following statements into logical symbolism, in each case using the abbreviations suggested: No car is safe unless it has good brakes. (Cx, Sx, Bx)
Answer Preview: x …

, Chapter: 9 -Problem: 19 >> Use indirect proof to prove that the following statements are tautologies. A ? (A ? B)
Answer Preview: To prove that A A B is a tautology using indirect proof we …

, Chapter: 9 -Problem: 3 >> Translate each of the following into the logical notation of propositional functions and quantifiers, in each case using the abbreviations suggested and making each formula begin with a quantifier, not with a negation symbol. Reporters are present. (Rx: x is a reporter; Px: x is present.)
Answer Preview: x…

, Chapter: 9 -Problem: 35 >> For each of the following arguments, inferring just two statements from the premises will produce a formal proof of its validity. Construct a formal proof for each of these arguments. In these formal proofs, and in all the proofs to follow in later sections, note to the right of each inferred statement the rule of inference that is used to infer that statement. It is most convenient if the justifi
Answer Preview: A B premise C D premise A B repetition line 1 C conjunction elimination line 2 A …

, Chapter: 9 -Problem: 18 >> For each of the following arguments, inferring just two statements from the premises will produce a formal proof of its validity. Construct a formal proof for each of these arguments. In these formal proofs, and in all the proofs to follow in later sections, note to the right of each inferred statement the rule of inference that is used to infer that statement. It is most convenient if the justifi
Answer Preview: To prove that the argument is valid using only two state…

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, Chapter: 9 -Problem: 2 >> The following set of arguments involves, in each case, one inference only, in which one of the ten logical equivalences set forth in this section has been employed. Here are two examples, the first two in the exercise set immediately following. (P1): (E ? F) ? (G ? ~ H) ? (~ E ? F) ? (G ? ~ H)
Answer Preview: Mater…

, Chapter: 9 -Problem: 42 >> For each of the following, find a normal-form formula that is logically equivalent to the given one: ? (?x) [? (? Wx ? Xx)]
Answer Preview: Using De Morgan s law and double negatio…

, Chapter: 9 -Problem: 18 >> Here follows a set of twenty elementary valid arguments. They are valid because each of them is exactly in the form of one of the nine elementary valid argument forms. For each of them, state the rule of inference by which its conclusion follows from its premise or premises. (P1): (W ? ?X) ? (Y ? Z) ? [(W ? ~ X) ? (Y  ? Z)] ? (X ? ~ Z)
Answer Preview: Answer P1 P Q R P2 R S P Q S This argument is an example of the rule of inference called Hypothetica…

, Chapter: 14 -Problem: 4 >> An urn contains 27 white balls and 40 black balls. What is the probability of getting four black balls in four successive drawings: A. If each ball drawn is replaced before making the next drawing? B. If the balls are not replaced?
Answer Preview: A If each ball drawn is replaced before making the next drawing the probability of get…

, Chapter: 9 -Problem: 27 >> Use Conditional Proof (C.P.) to prove that the following statements are tautologies. (R ? Q) ? R
Answer Preview: R Q OR 1 …

, Chapter: 9 -Problem: 22 >> For each of the following arguments, a formal proof of validity can be constructed without great difficulty, although some of the proofs may require a sequence of eight or nine statements (including premises) for their completion. (P1): A ? B (P2): C ? D (P3): A ? C ? (A ? B) ? (C ? D)
Answer Preview: 1 A B 2 C D 3 A C …

, Chapter: 9 -Problem: 15 >> The following set of arguments involves, in each case, one inference only, in which one of the ten logical equivalences set forth in this section has been employed. Here are two examples, the first two in the exercise set immediately following. (P1): [(R ? ~ S) ? ~ T] ? [(R ? ~ S) ? U] ? (R ? ~ S) ? (~ T ? U)
Answer Preview: The given argument is an example of the application of …

, Chapter: 8 -Problem: 29 >> Use truth tables to determine the validity or invalidity of the following arguments: If oil consumption continues to grow, then either oil imports will increase or domestic oil reserves will be depleted. If oil imports increase and domestic oil reserves are depleted, then the nation eventually will go bankrupt. Therefore, if oil consumption continues to grow, then the nation eventually will go ban
Answer Preview: G I D has the p q v r I D B specific form q r s G B p s T 9 T T T T F F F F T T …

, Chapter: 9 -Problem: 27 >> Translate each of the following into the logical notation of propositional functions and quantifiers, in each case making the formula begin with a quantifier, not with a negation symbol. A problem well stated is a problem half solved. —Charles Kettering, former research director for General Motors
Answer Preview: A problem well stated i…

, Chapter: 9 -Problem: 19 >> The following set of arguments involves, in each case, one inference only, in which one of the ten logical equivalences set forth in this section has been employed. Here are two examples, the first two in the exercise set immediately following. (P1): [H ? (I ? J)] ? [H ? (K ? ~ L)] ? H ? [(I ? J) ? (K ? ~ L)]
Answer Preview: Argument 1 This is an example of the associative law of disjunction where the order of grouping of d…

, Chapter: 8 -Problem: 26 >> Symbolize the following, using capital letters to abbreviate the simple statements involved. If Argentina mobilizes, then if Brazil protests to the UN, then Chile will call for a meeting of all the Latin American states.
Answer Preview: A…

, Chapter: 9 -Problem: 20 >> For each of the following arguments, inferring just two statements from the premises will produce a formal proof of its validity. Construct a formal proof for each of these arguments. In these formal proofs, and in all the proofs to follow in later sections, note to the right of each inferred statement the rule of inference that is used to infer that statement. It is most convenient if the justifi
Answer Preview: 1 Z A Premise 2 A B Premise 3 Z B Assumption for Conditional Proof ACP 4 Z B 3 Material Imp…

, Chapter: 14 -Problem: 5 >> Identify the premises and conclusions in the following passages. Some premises do support the conclusion, others do not. Note that premises may support conclusions directly or indirectly and that even simple passages may contain more than one argument. Standardized tests have a disparate racial and ethnic impact; white and Asian students score, on average, markedly higher than their black and Hisp
Answer Preview: Premises 1 Standardized tests have a disparate racial impa…

, Chapter: 9 -Problem: 15 >> Translate each of the following into the logical notation of propositional functions and quantifiers, in each case using the abbreviations suggested and making each formula begin with a quantifier, not with a negation symbol. Only citizens of the United States can vote in U.S. elections. (Cx: x is a citizen of the United States; Vx: x can vote in U.S. elections.)
Answer Preview: The given statement Only citizens of the United States can vote in U S elections can be transla…

, Chapter: 14 -Problem: 4 >> Identify the premises and conclusions in the following passages. Some premises do support the conclusion, others do not. Note that premises may support conclusions directly or indirectly and that even simple passages may contain more than one argument. Sir Edmund Hillary is a hero, not because he was the first to climb Mount Everest, but because he never forgot the Sherpas who helped him achieve t
Answer Preview: Premise Sir Edmund Hilary dedicated …

, Chapter: 9 -Problem: 42 >> For each of the following arguments, use the STTT to determine whether the argument is valid or invalid. For some of these arguments, Steps 2 C , 3 C , and 4 (i.e., the C-Sequence) will be most efficient; for some of these arguments Steps 2 P , 3 P , and 4 (i.e., the P-Sequence) will be most efficient. (P1): T ? (U ? V) (P2): T ? [U ? (W ? X)] (P3): (T ? V) ? ~ (W ? X) ? W ? X
Answer Preview: We can use the Substitution Transposition Tautology Transitivity STTT rule to determine the val…

, Chapter: 9 -Problem: 14 >> For each of the following arguments, inferring just two statements from the premises will produce a formal proof of its validity. Construct a formal proof for each of these arguments. In these formal proofs, and in all the proofs to follow in later sections, note to the right of each inferred statement the rule of inference that is used to infer that statement. It is most convenient if the justifi
Answer Preview: 1 H I J Premise 2 H H Disjunction Introduction 3 H I Di…

, Chapter: 8 -Problem: 28 >> Use truth tables to decide which of the following biconditionals are tautologies. p = [p ? (q ? p)
Answer Preview: Q T T F F Taut…

, Chapter: 9 -Problem: 30 >> For each of the following arguments, use the STTT to determine whether the argument is valid or invalid. For some of these arguments, Steps 2 C , 3 C , and 4 (i.e., the C-Sequence) will be most efficient; for some of these arguments Steps 2 P , 3 P , and 4 (i.e., the P-Sequence) will be most efficient. (P1): A ? G (P2): B ? A (P3): B (P4): H ? (G ? H) ? (I ? J)
Answer Preview: Contrast the following argument with the argument of Appendix A P 1 A G P 2 B A P 3 B P 4 H G H I J As with the similar looking argument in Appendix A this argument would require a complete truth tabl…

, Chapter: 9 -Problem: 11 >> The following set of arguments involves, in each case, one inference only, in which one of the ten logical equivalences set forth in this section has been employed. Here are two examples, the first two in the exercise set immediately following. (P1): (~ F ? G) ? (F ? G) ? (F ? G) ? (F ? G)
Answer Preview: This argument can be simplified by using the logical equivalence of material …

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, Chapter: 10 -Problem: 3 >> Prove the invalidity of the following: Transcribed Image Text: (P?): (x)(GxHx) (P?): (x)(GxIx) :. (x)(Ix? Hx)
Answer Preview: To prove the invalidity of the argument we can use a counterexample Let s consider a domain of objec…

, Chapter: 9 -Problem: 15 >> Here follows a set of twenty elementary valid arguments. They are valid because each of them is exactly in the form of one of the nine elementary valid argument forms. For each of them, state the rule of inference by which its conclusion follows from its premise or premises. (P1): (J ? K) ? (K ? L) (P2): L ? M ? [(J ? K) ? (K ? L)] ? (L ? M)
Answer Preview: This argument is an example of the rule of inference known as Hypothetical Syllogism Hypothetic…

, Chapter: 9 -Problem: 8 >> For each of the following, either construct a formal proof of validity or prove invalidity by means of the STTT. (P1): (O ? P) ? (Q ? R) (P2): S ? ~ R (P3): ~ (P ? ~ S) (P4): ~ (O ? Q) ? ~ O
Answer Preview: O P…

, Chapter: 9 -Problem: 4 >> For each of the following, either construct a formal proof of validity or prove invalidity by means of the STTT. (P1): M ? (N ? O) (P2): (N ? O) ? P ? M ? P
Answer Preview: 1 MD NO 2 NVO P MDP MV NO …

, Chapter: 9 -Problem: 34 >> For each of the following arguments, inferring just two statements from the premises will produce a formal proof of its validity. Construct a formal proof for each of these arguments. In these formal proofs, and in all the proofs to follow in later sections, note to the right of each inferred statement the rule of inference that is used to infer that statement. It is most convenient if the justifi
Answer Preview: Y Z Premise Z Y Premise Y Y Law of excluded mid…

, Chapter: 9 -Problem: 6 >> For each of the following arguments, it is possible to provide a formal proof of validity by validly inferring just three statements. Writing these out, carefully and accurately, will strengthen your command of the rules of inference, a needed preparation for the construction of proofs that are more extended and more complex. (P1): Q ? R (P2): R ? S (P3): ~ S ? ~ Q ? ~ R
Answer Preview: 1 Q R 2 R S 3 S …

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, Chapter: 9 -Problem: 30 >> Each of the following arguments in English may be similarly translated, and for each, a formal proof of validity (using only the nine elementary valid argument forms as rules of inference) may be constructed. These proofs vary in length, some requiring a sequence of thirteen statements (including the premises) to complete the formal proofs. The suggested abbreviations should be used for the sake o
Answer Preview: The argument can be symbolized as follows P1 C R P2 C R B P3 C B S P4 S M C M E To prove the validit…

, Chapter: 9 -Problem: 14 >> For each of the following arguments, it is possible to provide a formal proof of validity by validly inferring just three statements. Writing these out, carefully and accurately, will strengthen your command of the rules of inference, a needed preparation for the construction of proofs that are more extended and more complex. (P1): (T ? U) ? (V ? W) (P2): (U ? X) ? (W ? Y) (P3): T ? X ? Y
Answer Preview: 1 T U V W 2 U X W …

, Chapter: 9 -Problem: 8 >> Use Conditional Proof to prove the validity of the following arguments. Transcribed Image Text: G (P?): (HD) (P?): (CD) (E > F) (P?): G (CVE) (P): (DVF) (HVD) :. G = (DVF)
Answer Preview: 1 CVD EF 2 E EF G 3 GHVH CH 4 C 5 CVD 6 EDF 7 E EF 8 G 9 CDG …

, Chapter: 8 -Problem: 30 >> Use truth tables to determine the validity or invalidity of the following arguments: If oil consumption continues to grow, then oil imports will increase and domestic oil reserves will be depleted. If either oil imports increase or domestic oil reserves are depleted, then the nation will soon be bankrupt. Therefore, if oil consumption continues to grow, then the nation will soon be bankrupt.
Answer Preview: G I D has the p q r I D B specific form q r s G B p s T T T T T T T T F F F F F …

, Chapter: 9 -Problem: 26 >> For each of the following arguments, use the STTT to determine whether the argument is valid or invalid. For some of these arguments, Steps 2 C , 3 C , and 4 (i.e., the C-Sequence) will be most efficient; for some of these arguments Steps 2 P , 3 P , and 4 (i.e., the P-Sequence) will be most efficient. (P1): I ? ~ J (P2): ~ (~ K ? L) (P3): ~ (~ I ? ~ L) ? ~ J ? K
Answer Preview: P 1 I J P 2 K L P 3 I L J K In this case J must be false and K must be …

, Chapter: 8 -Problem: 52 >> Using the letters E , I , J , L , and S to abbreviate the simple statements, “Egypt’s food shortage worsens,” “Iran raises the price of oil,” “Jordan requests more U.S. aid,” “Libya raises the price of oil,” and “Saudi Arabia buys five hundred more warplanes,” symbolize these statements. Either Iran or Libya raises the price of oil.
Answer Preview: Lets use the following abbreviations for the given statements E …

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, Chapter: 9 -Problem: 35 >> For each of the following arguments, use the STTT to determine whether the argument is valid or invalid. For some of these arguments, Steps 2 C , 3 C , and 4 (i.e., the C-Sequence) will be most efficient; for some of these arguments Steps 2 P , 3 P , and 4 (i.e., the P-Sequence) will be most efficient. (P1): (H ? I) ? [J ? (K ? L)] (P2): I ? J ? K
Answer Preview: Using the STTT with the P Sequence Step 1 Write down the premises and conclusion and negate the c…

, Chapter: 9 -Problem: 13 >> Use Conditional Proof (C.P.) to prove that the following statements are tautologies. ~~ R ? R
Answer Preview: RDR 1 2 …

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, Chapter: 8 -Problem: 27 >> Use truth tables to decide which of the following biconditionals are tautologies. p = [p ? (p ? q)]
Answer Preview: T T F F Not a t…

, Chapter: 10 -Problem: 4 >> Translate the following statements into logical symbolism, in each case using the abbreviations suggested: Any tall man is attractive if he is dark and handsome. (Tx, Mx, Ax, Dx, Hx)
Answer Preview: x T…

, Chapter: 9 -Problem: 7 >> For the following statements, if there are forced truth-value assignments, make them. If there are no forced truth-value assignments, determine, using Maxim V, the requisite truth-value combinations on which, and the number of ways that, the statement can be true (or false), as the case requires. A ? ~ M F
Answer Preview: A…

, Chapter: 8 -Problem: 72 >> Using the letters E , I , J , L , and S to abbreviate the simple statements, “Egypt’s food shortage worsens,” “Iran raises the price of oil,” “Jordan requests more U.S. aid,” “Libya raises the price of oil,” and “Saudi Arabia buys five hundred more warplanes,” symbolize these statements.
Answer Preview: S …

, Chapter: 9 -Problem: 31 >> Each of the following arguments in English may be similarly translated, and for each, a formal proof of validity (using only the nine elementary valid argument forms as rules of inference) may be constructed. These proofs vary in length, some requiring a sequence of thirteen statements (including the premises) to complete the formal proofs. The suggested abbreviations should be used for the sake o
Answer Preview: J R J E R I I T E D J R J E R I I T J R J E J T J R J E T J …

, Chapter: 9 -Problem: 15 >> For the following statements, if there are forced truth-value assignments, make them. If there are no forced truth-value assignments, determine, using Maxim V, the requisite truth-value combinations on which, and the number of ways that, the statement can be true (or false), as the case requires. (C ? M) ? G F  F
Answer Preview: C T…

, Chapter: 9 -Problem: 9 >> For each of the following, either construct a formal proof of validity or prove invalidity by means of the STTT. (P1): T ? (U ? V) (P2): U ? (W ? X) (P3): (T ? W) ? (Y ? Z) (P4): (T ? U) ? ~ Y (P5): ~ Y ? (~ Z ? X) ? X
Answer Preview: 1 T UV 2 UD W X 3 TW YZ 4 TU Y 5 YZ2X X 6TV U V 7 TVU TVV 8 T…

, Chapter: 8 -Problem: 71 >> Using the letters E , I , J , L , and S to abbreviate the simple statements, “Egypt’s food shortage worsens,” “Iran raises the price of oil,” “Jordan requests more U.S. aid,” “Libya raises the price of oil,” and “Saudi Arabia buys five hundred more warplanes,” symbolize these statements.
Answer Preview: E …

, Chapter: 10 -Problem: 5 >> Translate the following statements into logical symbolism, in each case using the abbreviations suggested: A gladiator wins if and only if he is lucky. (Gx, Wx, Lx)
Answer Preview: Translating a statement into logical symbolism is a crucial skill in the field of logic It allows us …

, Chapter: 9 -Problem: 10 >> Translate each of the following into the logical notation of propositional functions and quantifiers, in each case using the abbreviations suggested and making each formula begin with a quantifier, not with a negation symbol. The common cold is never fatal. (Cx: x is a common cold; Fx: x is fatal.)
Answer Preview: The given statement can be translated into logical notation as xCx Fx This can b…

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, Chapter: 9 -Problem: 34 >> For each of the following arguments, use the STTT to determine whether the argument is valid or invalid. For some of these arguments, Steps 2 C , 3 C , and 4 (i.e., the C-Sequence) will be most efficient; for some of these arguments Steps 2 P , 3 P , and 4 (i.e., the P-Sequence) will be most efficient. (P1): A ? (B ? C) (P2): B ? (C ? A) (P3): C ? (A ? B) (P4): ~ A ? B ? C
Answer Preview: A B C Premise B C A Premise C A B Premise A Premise B C Negatin…

, Chapter: 8 -Problem: 9 >> Use truth tables to prove the validity or invalidity of each of the argument forms in Section 8.5, Group B, pp. 288–9. (P1): (R ? S) ? T (P2): T ? (R ? S) ?(R ? S) ? (R ? S)
Answer Preview: R S T has the p q rT R S specific form r p q R S …

, Chapter: 9 -Problem: 8 >> For the following statements, if there are forced truth-value assignments, make them. If there are no forced truth-value assignments, determine, using Maxim V, the requisite truth-value combinations on which, and the number of ways that, the statement can be true (or false), as the case requires. ~ L  ?  K    F  T
Answer Preview: T …

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, Chapter: 9 -Problem: 30 >> If any truth-functional argument is valid, we have the tools to prove it valid; and if it is invalid, we have the tools to prove it invalid. For each of the following arguments determine whether it is valid or invalid using the STTT. If it is valid, prove it valid using the nineteen rules of inference. These proofs will be more difficult to construct than in preceding exercises, but they will offe
Answer Preview: This argument can be represented symbolically as P R v R W C W To determin…

, Chapter: 9 -Problem: 15 >> For each of the following, either construct a formal proof of validity or prove invalidity by means of the STTT. In each case, use the notation in parentheses. If Salome is intelligent and studies hard, then she will get good grades and pass her courses. If Salome studies hard but lacks intelligence, then her efforts will be appreciated; and if her efforts are appreciated, then she will pass her c
Answer Preview: I S G PS I A A P…

, Chapter: 9 -Problem: 9 >> The following set of arguments involves, in each case, one inference only, in which one of the ten logical equivalences set forth in this section has been employed. Here are two examples, the first two in the exercise set immediately following. (P1): Z ? (A ? B) ? Z ? (~~ A ? B)
Answer Preview: This argument is an example of the Rule of Double Ne…

, Chapter: 8 -Problem: 31 >> Symbolize the following, using capital letters to abbreviate the simple statements involved. If either Argentina mobilizes or Brazil protests to the UN, then Chile will call for a meeting of all the Latin American states.
Answer Preview: A…

, Chapter: 10 -Problem: 4 >> Construct a formal proof of validity for each of the following arguments: Transcribed Image Text: (P?): (3x) (Jx Kx) (P?): (x)(JxLx) .. (3x)(Lx.Kx)
Answer Preview: ANSWER To prove the validity of the argument we need to show that if the premises P1 and P2 are true then the conclusion C must also be true We begin by assuming that the premises are true Premise 1 x …

, Chapter: 9 -Problem: 16 >> For each of the following arguments, a formal proof of validity can be constructed without great difficulty, although some of the proofs may require a sequence of eight or nine statements (including premises) for their completion. (P1): A ? B (P2): A ? (C ? D) (P3): ~ B ? ~ E ? C
Answer Preview: Premises P1 A B P2 A C D P3 B E Goal C Proof B E Pr…

, Chapter: 9 -Problem: 12 >> For each of the following arguments, inferring just two statements from the premises will produce a formal proof of its validity. Construct a formal proof for each of these arguments. In these formal proofs, and in all the proofs to follow in later sections, note to the right of each inferred statement the rule of inference that is used to infer that statement. It is most convenient if the justifi
Answer Preview: 1 B C D Premise 2 B Simplification 1 3 …

, Chapter: 9 -Problem: 11 >> Here follows a set of twenty elementary valid arguments. They are valid because each of them is exactly in the form of one of the nine elementary valid argument forms. For each of them, state the rule of inference by which its conclusion follows from its premise or premises. (P1): (A ? B) ?  (C ? D) (P2): A ?  B ? C ? D
Answer Preview: The rule of inference used in this argument is hypothetical …

, Chapter: 14 -Problem: 3 >> Which of the various functions of language are exemplified by each of the following passages? What traveler among the ruins of Carthage, of Palmyra, Persepolis, or Rome, has not been stimulated to reflections on the transiency of kingdoms and men, and to sadness at the thought of a vigorous and rich life now departed . . .? —G. W. F. Hegel, Lectures on the Philosophy of History, 1823
Answer Preview: The passage by GWF Hegel exemplifies the function of language known as rhetoric or persuasion The purpose of rhetoric is to persuade an audience to ta…

, Chapter: 9 -Problem: 32 >> For each of the following, find a normal-form formula that is logically equivalent to the given one: ? (x) (Cx ? ? Dx)
Answer Preview: To find the normal form

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