Prove the following by deductive system:

As an AI language model, I am not able to present a deductive proof without a specific statement or theorem to prove. Please provide me with a statement or theorem that needs to be proven, and I would be happy to assist you with a deductive proof.

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Prove the following by deductive system:1 ⊨ φ ∨ ¬φ (i.e., ∅ ⊨ φ ∨ ¬φ)

To prove 1 ⊨ φ ∨ ¬φ using a deductive system, we need to show that if we assume 1 as true, then we can derive φ ∨ ¬φ as a logical consequence.

We can start with the premise 1, which means that the proposition "true" is true. From this, we can use the rule of addition to derive φ ∨ true, which is logically equivalent to true.

Next, we can use the rule of negation to derive ¬φ ∨ true, which is also logically equivalent to true.

Finally, we can use the rule of disjunction elimination to derive φ ∨ ¬φ from the two previous conclusions. This rule states that if we have a disjunction A ∨ B and we can derive a conclusion C from both A and B separately, then we can conclude C from the disjunction A ∨ B.

In this case, we have the disjunction φ ∨ true and ¬φ ∨ true, and we have derived true from both of them separately. Therefore, we can apply the rule of disjunction elimination to derive φ ∨ ¬φ, which is our desired conclusion.

Therefore, we have shown that if 1 is true, then φ ∨ ¬φ must also be true, which means that 1 ⊨ φ ∨ ¬φ.

Prove the following by deductive system: ⊨ φ ∨ ¬φ (i.e., ∅ ⊨ φ ∨ ¬φ)

Proof:

1. Assume ¬(φ ∨ ¬φ) for the sake of contradiction.
2. By De Morgan's Law, ¬(φ ∨ ¬φ) is equivalent to ¬φ ∧ ¬¬φ.
3. By Double Negation, ¬¬φ is equivalent to φ.
4. Therefore, ¬φ ∧ φ.
5. By the principle of Non-Contradiction, a statement and its negation cannot both be true.
6. Therefore, ¬(φ ∨ ¬φ) leads to a contradiction.
7. By the principle of Explosion, a contradiction implies anything.
8. Therefore, φ ∨ ¬φ must be true, as assuming its negation leads to a contradiction.
9. Therefore, ⊨ φ ∨ ¬φ, as it is true under all interpretations.

Note: This proof uses the principle of Non-Contradiction and the principle of Explosion, which are two fundamental principles of classical logic.