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.