Each of the following exercises presents a flawless formal proof of validity for the indicated argument. For each, state the justification for each numbered line that is not a premise.

From the textbook, 14th edition.

9.2, argument 2.

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.

( D v E ) ⊃ ( F v G )

Therefore D v E

9.3 Ex. 3

Each of the following exercises presents a flawless formal proof of validity for the indicated argument. For each, state the justification for each numbered line that is not a premise.

1. I ⊃ J

2. J ⊃ K

3. L ⊃ M

4. I v L

Therefore K v M

5. I ⊃ K

6. (I ⊃ K) AND (L ⊃ M)

7. K v M

9.4 ex. 9

A formal proof for each argument may be constructed by adding just two additional statements. This will be an easy task if the nine elementary valid argument forms are clearly in mind.

1. Y ⊃ Z

2. Y

Therefore, Y · Z

9.5 A3

For each of the following arguments, it is possible to provide a formal proof

of validity by adding just three statements to the premises. 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.

1. (H ⊃ I) · (H ⊃ J)

2. H · (I V J)

Therefore, I V J

9.5 B1

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 lines (including premises) for their completion.

1. A ⊃ B

2. A v (C · D)

3. B · E

Therefore, C

9.6 argument 1

For each of the following one-step arguments, state the one rule of inference by which its conclusion follows from its premise.

(A ⊃ B) ⋅ (C ⊃ D)

Therefore, (A ⊃ B) ⋅ (∼D ⊃ ∼C)

9.8 B9 — appears in page 373.

For each of the following arguments, adding just two statements to the premises will produce a formal proof of its validity. Construct a formal proof for each of these arguments.

1. W ⊃ X

2. Y ⊃ X

Therefore, W ⊃ Y

9.9 exercise 2

Prove the invalidity of the following by the method of assigning truth values.

Not (E and F)

(not-E and not-F) implies (G and H)

H implies G

Therefore, G

9.10 A Exercise 2

For the following, either construct a formal proof of validity or prove invalidity by the method of assigning truth values to the simple statements involved.

(E implies F) and (G implies H)

Therefore, (E or G) implies (F and H)

9.11, argument 3 —- in page 394.

For each of the following arguments, construct an indirect proof of validity.

1. (D v E) ⊃ (F ⊃ G)

2. ( G v H) ⊃ (D · F)

Therefore, G.