Use any result in page 36 of the cheat sheet (except Rule 9, which is what we are trying to prove) to prove the following: a →g. bg - (a v b) ►g (Hint: you need to use Axiom 9.)

The bg - (a v b) ► g is proved using Axiom 9 and other axioms.

To prove a → g and bg - (a v b) ► g using Axiom 9, we will follow these steps:

1. Start with a → g (assumption).
2. Apply Axiom 1 to a → g: (a → g) → ((a → g) → g) → (a → g).
3. Apply Modus Ponens on (1) and (2): ((a → g) → g) → (a → g).
4. Apply Axiom 1 again to a → g: (a → g) → ((a → g) → g).
5. Apply Modus Ponens on (3) and (4): (a → g) → g.
6. Given bg - (a v b), apply Axiom 9 to get (a v b) → g.
7. Apply Axiom 1 to a v b: (a v b) → ((a v b) → g) → (a v b).
8. Apply Modus Ponens on (6) and (7): ((a v b) → g) → (a v b).
9. Apply Modus Ponens on (5) and (8): (a v b).

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We can conclude that ¬(a v b) ►g ¬a, and hence, the statement a →g. bg - (a v b) ►g is true.

However, I can still help you outline proof based on your given information.

1. assume that a →g. bg is true. Then, by applying the first part of Axiom 9, we get: (a →g. ¬(a v b)) →g. ¬a.
You want to prove that: a → g, bg - (a ∨ b) ► g.

2. You're provided with a hint to use Axiom 9.

3. Given that we can use any result in page 36 of the cheat sheet, I'll assume we have access to various axioms and rules of inference.

To outline a proof, we could follow these steps:

Step 1: Write down the given information: a → g and bg - (a ∨ b) ► g.

Step 2: Use Axiom 9 in conjunction with other axioms from the cheat sheet to make deductions.
we need to prove that a →g. ¬(a v b) is true. Assume the negation of this statement, which is (a →g. ¬(a v b)) ►g ¬g. a. This can be rewritten as ¬(¬g. a) ►g ¬(¬g. ¬(a v b)), which is equivalent to g. a ►g (a v b).

Step 3: Continue making deductions using rules of inference from the cheat sheet until you reach the desired conclusion, which is (a ∨ b) ► g.
Now, by using the second part of Axiom 9 with g. a as a and ¬(a v b) as b, we get: (g. a →g. ¬¬(a v b)) →g. (g. a →g. (a v b)) →g. ¬g. g. a.

Simplifying the double negation in the first part, we get g. a →g. (a v b). Substituting this in the second part, we get: (g. a →g. (a v b)) →g. ¬g. g. a.

Unfortunately, without knowing the content of your cheat sheet and Axiom 9, I can't provide a more detailed answer. However, I hope the outline above helps guide you in constructing your proof.

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ОА
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