Translate the premises and conclusion into the symbols of propositional logic. Construct a truth table in which you analyze the argument for validity. You can construct a truth a table by inserting a table into a Microsoft Word document (from the INSERT option in Word, choose “table.” You will then have an opportunity to choose how many rows and columns you would like your table to be.) Is your argument valid or invalid? If valid, say why it is valid; identify the rows in the truth table that make the argument valid. If the argument is invalid, identify a counterexample; point to a row in your truth table that makes the argument invalid.

1. Humans evolved from lower life forms given that either human life evolved from inanimate matter apart from divine causes or God created human life via evolution. God created life via of evolution. It follows that humans evolved from lower life forms. (H: Human life evolved from lower life forms; M: Human life evolved from inanimate matter apart from divine causes; G: God created human life via evolution)
2. Augustine achieves heaven if Augustine is virtuous. But Augustine is happy provided that he is not virtuous. Augustine does not achieve heaven only if he is not happy. Therefore, Augustine achieves heaven. (A: Augustine achieves heaven; V: Augustine is virtuous; H: Augustine is happy)
3. It is morally permissible for extraterrestrials to eat humans on the condition that it is morally permissible for humans to eat animals. But either it is not morally permissible for extraterrestrials to eat humans or human life lacks intrinsic value. Human life has intrinsic value. Therefore it is not morally permissible for humans to eat animals. (E: It is morally permissible for extraterrestrials to eat humans; H: It is morally permissible for humans to eat animals; V: Human life has intrinsic value)
4. American foreign policy is bankrupt unless it is based on clear moral principles. American foreign policy is not based on clear moral principles just in case it is based primarily on the national interest. Unfortunately American foreign policy is based primarily on the national interest. Therefore, American foreign policy is bankrupt. (B: American foreign policy is bankrupt; M: American foreign policy is based on clear moral principles; N: American foreign policy is based primarily on national interest.)
5. On the condition that landmines are designed to inflict horrible suffering, they ought to be banned unless inflicting horrible suffering is sometimes justified. It is not true that inflicting horrible suffering is sometimes justified, but it is true that landmines are designed to inflict horrible suffering. Therefore, landmines ought to be banned. (L: Landmines are designed to inflict horrible suffering; B: Landmines ought to be banned; S: Inflicting horrible suffering is sometimes justified)
   Then, construct abbreviated truth tables of your argument as described in section 7.4 of the textbook (rules for this are summarized on page 329).

5-1 Journal: Direct Proofs in Natural Deduction
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Instructions
Choose two of the arguments below and write a direct proof using the eight rules of inference introduced in section 8.1 of the textbook. You can do argument 1 or argument 2, but not both, then any of arguments 3–6. Note that commas are used to separate the premises from each other.

1. ~M, (~M • ~N) → (Q → P), P → R, ~N, therefore, Q → R
2. ~F → ~G, P → ~Q, ~F v P, (~G v ~Q) → (L • M), therefore, L
3. ~(Z v Y) → ~W, ~U → ~(Z v Y), (~U → ~W) → (T → S), S → (R v P), [T → (RvP)] → [(~R v K) • ~K], therefore, ~K
4. (S v U) • ~U, S → [T • (F v G)], [T v (J • P)] → (~B • E), therefore, S • ~B
5. ~X → (~Y → ~Z), X v (W → U), ~Y v W, ~X • T, (~Z v U) → ~S, therefore, (R v ~S) • T
6. (C → Q) • (~L → ~R), (S → C) • (~N → ~L), ~Q • J, ~Q → (S v ~N), therefore, ~R

Natural deduction is so called because it is a model for how we naturally reason. This often comes as a surprise to students because all of the symbols seem anything but natural. The symbols, however, allow us to focus on the form of the argument without getting bogged down by content. Recall that each sentence letter represents a simple sentence in English.

After writing your direct proofs, construct a translation key for your argument by assigning each letter a simple sentence, and use that key to fill in the content of the argument.

6-1 Journal: Indirect Proofs in Natural Deduction

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Choose one of the proofs below and use one of the indirect proof techniques (reductio ad absurdum or conditional proof) presented in Chapter 8 to demonstrate the validity of the argument. The proofs below may use any of the rules of inference or replacement rules given in Chapter 8.
1.(G • P) → K, E → Z, ~P → ~ Z, G → (E v L), therefore, (G • ~L) → K
2.(S v T) ↔ ~E, S → (F • ~G), A → W, T → ~W, therefore, (~E • A) → ~G
3.(S v T) v (U v W), therefore, (U v T) v (S v W)
4.~Q → (L → F), Q → ~A, F → B, L, therefore, ~A v B
5.~S → (F → L), F → (L → P), therefore, ~S → (F → P)
In mathematics, it is very common for there to be multiple ways to solve a given a problem; the same can be said of logic. There is often a variety of ways to perform a natural deduction. Now, construct an alternate proof. In other words, if the proof was done using RAA, now use CP; if you used CP, now use RAA. Consider the following questions, as well, in your journal response:
• Will a direct proof work for any of these?
• Can the proof be performed more efficiently by using different equivalence rules?