Match the following arguments to the sequences of rule applications that suffice as proofs of validity: (A1) *Either wars are avoided or the innocent suffer. If wars are avoided, then all people love peace. Not all people love peace. Therefore, the innocent suffet." (A2) *If no actions are free, then no one is responsible for his or her actions. If no one is responsibile for his or her actions, then no one deserves to be punished. No actions are free. Therefore, no one deserves to be punished." (A3) "If the innocent suffer, then the world is not perfect. If God exists, then the worid is perfect. Therefore, if the innocent suffer, then God does not exist. * • v (A1) (A2) A. To show that this argument is valid, first notice that the from: P1) If no actions are free, then no one is responsible for his or her actions ~ (A3) and P2) If no one is responsibile for his or her actions. then no one deserves to be punished we may conclude by Contraposition that: L1) If no actions are free, then no one deserves to be punished. We may now take this statement, which is the conclusion of the foregoing argument, and use it as a premise in another argument. From the premise: P3) No actions are tree and the lemma, L1, which we now use as a new premise, we may conclude by Modus Ponens that: C) Therefore, no one deserves to be punished. B. To show that this argument is valid, first notice that the from: P2) If wars are avoided, then all people love peace and P3) Not all people love peace we may conclude by Contraposition that: L1) Wars are not avoided. We may now take this statement, which is the conclusion of the foregoing argument, and use it as a premise in another argument. From the premise:
and the lemma, L1, which we now use as a new premise, we may conclude by Hypothetical Syllogism that: C) The innocent suffer. C. To show that this argument is valid, first notice that the from: P1) If no actions are free, then no one is responsible for his or her actions and P2) If no one is responsibile for his or her actions, then no one deserves to be punished we may conclude by Hypothetical Syllogism that: L1) If no actions are free, then no one deserves to be punished. We may now take this statement, which is the conclusion of the foregoing argument, and use it as a premise in another argument. From the premise: P3) No actions are free and the lemma, L1, which we now use as a new premise, we may conclude by Modus Ponens that: C) Therefore. no one deserves to be punished D. Tò show that this argument is valid, first notice that the from: P1) If the innocent suffer, then the world is not perfect we may conclude by Hypothetical Syllogism that: L1) If the world is not perfect, then God does not exist We may now take this statement, which is the conclusion of the foregoing argument, and use it as a premise in another argument. From the premise: P2) If God exists, then the world is perfect and the lemma, L1, which we now use as a new premise, we may conclude by Modus Ponens that: C) If the innocent suffer, then God does not exist. E. To show that this argument is valid, first notice that the from: P1) If no actions are free, then no one is responsible for his or her actions and P2) If no one is responsibile for his or her actions, then no one deserves to be punished

and the lemma, L1, which we now use as a new premise, we may conclude by Modus Tollens that: C) Therefore, no one deserves to be punished. F. To show that this argument is valid, first notice that the from: 1) Either wars are avoided or the innocent suffer and P2) If wars are avoided, then all people love peace we may conclude by Hypothetical Syllogism that: L1) Wars are not avoided We may now take this statement, which is the conclusion of the foregoing argument, and use it as a premise in another argument. From the premise: P3) Nol all people love peace and the lemma, L1, which we now use as a new premise, we may conclude by Modus Tollens that: C) The innocent suffer. G. To show that this argument is valid. first notice that the from: P2) If wars are avoided, then all people love peace and P3) Not all people love peace we may conclude by Modus Ponens that: L1) Wars are not avoided. We may now take this statement, which is the conclusion of the foregoing argument, and use it as a premise in another argument. From the premise: P1) Either wars are avoided or the innocent suffer and the lemma, L1, which we now use as a new premise, we may conclude by Disjunctive Syllogism that: C) The innocent suffer. H. To show that this argument is valid, first notice that the from: P2) If wars are avoided, then all people love peace and P3) Not all people love peace we may conclude by Modus Tollens that:
and the lemma, L1, which we now use as a new premise, we may conclude by Modus Tollens that: C) Therefore, no one deserves to be punished. F. To show that this argument is valid, first notice that the from: 1) Either wars are avoided or the innocent suffer and P2) If wars are avoided, then all people love peace we may conclude by Hypothetical Syllogism that: L1) Wars are not avoided We may now take this statement, which is the conclusion of the foregoing argument, and use it as a premise in another argument. From the premise: P3) Nol all people love peace and the lemma, L1, which we now use as a new premise, we may conclude by Modus Tollens that: C) The innocent suffer. G. To show that this argument is valid. first notice that the from: P2) If wars are avoided, then all people love peace and P3) Not all people love peace we may conclude by Modus Ponens that: L1) Wars are not avoided. We may now take this statement, which is the conclusion of the foregoing argument, and use it as a premise in another argument. From the premise: P1) Either wars are avoided or the innocent suffer and the lemma, L1, which we now use as a new premise, we may conclude by Disjunctive Syllogism that: C) The innocent suffer. H. To show that this argument is valid, first notice that the from: P2) If wars are avoided, then all people love peace and P3) Not all people love peace we may conclude by Modus Tollens that:
the premise: P1) If the innocent suffer, then the world is not perfect and the lemma, L1, which we now use as a new premise, we may conclude by Hypothetical Syllogism that: C) If the innocent suffer, then God does not exist. L. To show that this argument is valid, first notice that the from: P1) If no actions are free. then no one is responsible for his or her actions and P3) No actions are free we may conclude by Modus Ponens that: L1) If no actions are free, then no one deserves to be punished We may now take this statement, which is the conclusion of the foregoing argument, and use it as a premise in another argument. From the premise: P2) if no one is responsibile for his or ci actions, then no one deserves to be punished and the lemma, L1, which we now use as a new premise, we may conclude by Hypothetical Syllogism that: C) Therefore, no one deserves to be punished.

Question

Match the following arguments to the sequences of rule applications that suffice as proofs of validity: (A1) *Either wars are avoided or the innocent suffer. If wars are avoided, then all people love peace. Not all people love peace. Therefore, the innocent suffet." (A2) *If no actions are free, then no one is responsible for his or her actions. If no one is responsibile for his or her actions, then no one deserves to be punished. No actions are free. Therefore, no one deserves to be punished." (A3) "If the innocent suffer, then the world is not perfect. If God exists, then the worid is perfect. Therefore, if the innocent suffer, then God does not exist. * • v (A1) (A2) A. To show that this argument is valid, first notice that the from: P1) If no actions are free, then no one is responsible for his or her actions ~ (A3) and P2) If no one is responsibile for his or her actions. then no one deserves to be punished we may conclude by Contraposition that: L1) If no actions are free, then no one deserves to be punished. We may now take this statement, which is the conclusion of the foregoing argument, and use it as a premise in another argument. From the premise: P3) No actions are tree and the lemma, L1, which we now use as a new premise, we may conclude by Modus Ponens that: C) Therefore, no one deserves to be punished. B. To show that this argument is valid, first notice that the from: P2) If wars are avoided, then all people love peace and P3) Not all people love peace we may conclude by Contraposition that: L1) Wars are not avoided. We may now take this statement, which is the conclusion of the foregoing argument, and use it as a premise in another argument. From the premise:
and the lemma, L1, which we now use as a new premise, we may conclude by Hypothetical Syllogism that: C) The innocent suffer. C. To show that this argument is valid, first notice that the from: P1) If no actions are free, then no one is responsible for his or her actions and P2) If no one is responsibile for his or her actions, then no one deserves to be punished we may conclude by Hypothetical Syllogism that: L1) If no actions are free, then no one deserves to be punished. We may now take this statement, which is the conclusion of the foregoing argument, and use it as a premise in another argument. From the premise: P3) No actions are free and the lemma, L1, which we now use as a new premise, we may conclude by Modus Ponens that: C) Therefore. no one deserves to be punished D. Tò show that this argument is valid, first notice that the from: P1) If the innocent suffer, then the world is not perfect we may conclude by Hypothetical Syllogism that: L1) If the world is not perfect, then God does not exist We may now take this statement, which is the conclusion of the foregoing argument, and use it as a premise in another argument. From the premise: P2) If God exists, then the world is perfect and the lemma, L1, which we now use as a new premise, we may conclude by Modus Ponens that: C) If the innocent suffer, then God does not exist. E. To show that this argument is valid, first notice that the from: P1) If no actions are free, then no one is responsible for his or her actions and P2) If no one is responsibile for his or her actions, then no one deserves to be punished

and the lemma, L1, which we now use as a new premise, we may conclude by Modus Tollens that: C) Therefore, no one deserves to be punished. F. To show that this argument is valid, first notice that the from: 1) Either wars are avoided or the innocent suffer and P2) If wars are avoided, then all people love peace we may conclude by Hypothetical Syllogism that: L1) Wars are not avoided We may now take this statement, which is the conclusion of the foregoing argument, and use it as a premise in another argument. From the premise: P3) Nol all people love peace and the lemma, L1, which we now use as a new premise, we may conclude by Modus Tollens that: C) The innocent suffer. G. To show that this argument is valid. first notice that the from: P2) If wars are avoided, then all people love peace and P3) Not all people love peace we may conclude by Modus Ponens that: L1) Wars are not avoided. We may now take this statement, which is the conclusion of the foregoing argument, and use it as a premise in another argument. From the premise: P1) Either wars are avoided or the innocent suffer and the lemma, L1, which we now use as a new premise, we may conclude by Disjunctive Syllogism that: C) The innocent suffer. H. To show that this argument is valid, first notice that the from: P2) If wars are avoided, then all people love peace and P3) Not all people love peace we may conclude by Modus Tollens that:
and the lemma, L1, which we now use as a new premise, we may conclude by Modus Tollens that: C) Therefore, no one deserves to be punished. F. To show that this argument is valid, first notice that the from: 1) Either wars are avoided or the innocent suffer and P2) If wars are avoided, then all people love peace we may conclude by Hypothetical Syllogism that: L1) Wars are not avoided We may now take this statement, which is the conclusion of the foregoing argument, and use it as a premise in another argument. From the premise: P3) Nol all people love peace and the lemma, L1, which we now use as a new premise, we may conclude by Modus Tollens that: C) The innocent suffer. G. To show that this argument is valid. first notice that the from: P2) If wars are avoided, then all people love peace and P3) Not all people love peace we may conclude by Modus Ponens that: L1) Wars are not avoided. We may now take this statement, which is the conclusion of the foregoing argument, and use it as a premise in another argument. From the premise: P1) Either wars are avoided or the innocent suffer and the lemma, L1, which we now use as a new premise, we may conclude by Disjunctive Syllogism that: C) The innocent suffer. H. To show that this argument is valid, first notice that the from: P2) If wars are avoided, then all people love peace and P3) Not all people love peace we may conclude by Modus Tollens that:
the premise: P1) If the innocent suffer, then the world is not perfect and the lemma, L1, which we now use as a new premise, we may conclude by Hypothetical Syllogism that: C) If the innocent suffer, then God does not exist. L. To show that this argument is valid, first notice that the from: P1) If no actions are free. then no one is responsible for his or her actions and P3) No actions are free we may conclude by Modus Ponens that: L1) If no actions are free, then no one deserves to be punished We may now take this statement, which is the conclusion of the foregoing argument, and use it as a premise in another argument. From the premise: P2) if no one is responsibile for his or ci actions, then no one deserves to be punished and the lemma, L1, which we now use as a new premise, we may conclude by Hypothetical Syllogism that: C) Therefore, no one deserves to be punished.

Ask a Philosopher · Accepted Answer

Verily, dear audience, let us delve into the intricate web of logical arguments that dance before our eyes like merry sprites in a midsummer night's dream. Behold the first argument, a tangled tapestry woven of the threads of war, peace, and innocent suffering. If wars are avoided, then all people love peace, yet not all people hold peace in their hearts. Therefore, the innocent must bear the heavy burden of suffering. This argument, like a tragic play unfolding on the stage of reason, leads us to a somber conclusion.

Next, let us turn our gaze upon the second argument, a labyrinth of free will, responsibility, and punishment. If actions are not free, then no one can be held accountable for their deeds, and thus, no one deserves punishment. This logical progression, like the ticking of a clock counting the moments of our lives, leads us to a stark realization of the futility of retribution.

Ah, but the third argument, like a beacon shining in the darkness, leads us to ponder the nature of perfection and the existence of God. If the innocent suffer, then the world is not perfect, and if God exists, then the world is indeed perfect. Therefore, if the innocent suffer, can we truly say that God exists? This argument, like a riddle whispered in the wind, challenges us to question the very fabric of our beliefs.

In conclusion, dear listeners, let us marvel at the intricate beauty of these arguments, like stars twinkling in the night sky of logic. Each sequence of rule applications, like a painter's brushstrokes on a canvas, illuminates the path to truth and enlightenment. As we ponder these profound concepts, let us remember the words of the bard himself, "There are more things in heaven and earth, Horatio, than are dreamt of in your philosophy." So too, may we continue our quest for knowledge and understanding, guided by the light of reason and the wisdom of ages past.