SOLUTION: Please use the 18 rules of natural deduction, the 4 instantiation and generalization rules to derive the conclusion of the problem 1. (x)[Ax ⊃ (Bx ∨ Cx)] 2. Ag • ~Bg

Question 1207196: Please use the 18 rules of natural deduction, the 4 instantiation and generalization rules to derive the conclusion of the problem
1. (x)[Ax ⊃ (Bx ∨ Cx)]
2. Ag • ~Bg / Cg
Found 2 solutions by Edwin McCravy, math_tutor2020:
Answer by Edwin McCravy(20055) (Show Source): You can put this solution on YOUR website!

Answer by math_tutor2020(3817) (Show Source): You can put this solution on YOUR website!

I'll discuss how to solve this problem through use of a verbal outline of sorts.
I'll leave it to the student to construct the formal derivation.

The (x) out front of premise 1 indicates "for all x".
Sometimes an upside down A is used instead. So the notation would be

We use the Universal Instantiation rule to go from
Ax ⊃ (Bx ∨ Cx)
to
Ag ⊃ (Bg ∨ Cg)
where g is a specific element.

For example, x could refer to some country in the set of all countries and g refers to Germany.
The Universal Instantiation rule is valid because if the rule applies to all elements, then it certainly applies to one specific element of the set.

Now let's use the simplification rule to break Ag • ~Bg into the separate pieces of Ag and ~Bg

Next, combine Ag ⊃ (Bg ∨ Cg) with Ag to apply the Modus Ponens rule.
This will leave us with Bg ∨ Cg

Lastly combine Bg ∨ Cg with ~Bg when applying the Disjunctive Syllogism rule. This will let you arrive at Cg.

Once again this is a verbal outline and I'll leave it to the student to construct the formal derivation.

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