# Solve using rules of inference

If the ghost is in our imagination, then it is immortal. But if it is not in our imagination, then it is mortal and animal. If the ghost is either immortal or animal, then it is carnivorous. A ghost is ferocious if it is carnivorous.

Prove by rules of inference. if a Ghost is an imagination. Is a ghost ferocious? Is it carnivorous?

• 1. Why not post this rather on puzzling.se ? 2. If I read correctly, yes and yes, ghost is ferocious and carnivorous. – user5751924 Aug 5 '19 at 14:01
• Thanks for the suggestion. I am very new in this site XD. But no worries I was able to solve it, I just needed to think another direction. Thanks :) – user40674 Aug 5 '19 at 14:27

Here are the premises:

If ghost is our imagination, then it is immortal. But if it is not imagination, then it is mortal and animal. If the ghost is either immortal or animal, then it is carnivorous. A ghost is ferocious if it is carnivorous.

Let's use the following symbolization key from which we will construct the statements in these premises.

• G: The ghost is imaginary.
• I: The ghost is immortal.
• A: The ghost is animal.
• C: The ghost is carnivorous.
• F: The ghost is ferocious.

Here are the premises:

1. G > I
2. ~G > (~I & A)
3. (I v A) > C
4. C > F

Here are the goals:

Prove by rules of inference. if a Ghost is an imagination. Is a ghost ferocious? Is it carnivorous?

Using the symbolization key one can write these as follows:

• G > F
• G > C

Putting both into a Fitch-style proof checker one gets the following:

The first four lines are the premises. Since it is a conditional that I want to show, G>I, I assume the antecedent of the conditional on line 5. I want to derive the consequent, F. This is indented because it is a subproof starting with an assumption that I will have to discharge later.

By the first premise and my assumption I can use modus ponens or conditional elimination (→E) to derive I on line 6.

By disjunction introduction (vI) I can create an "or" statement from line 6 having just what I need to use the third premise on line 7.

On line 8, I use the third premise and line 7 with the modus ponens inference rule again to derive line 9.

At this point I have a subproof starting with G as an assumption and ending with the derived statement F. I can close that subproof by introducing a conditional on line 10 giving me what I want: G > F.

Note that if I stopped the subproof earlier I could also derive G > C.

Kevin Klement's JavaScript/PHP Fitch-style natural deduction proof editor and checker http://proofs.openlogicproject.org/

• Thanks a lot :') I solved it in the very same way and glad that its correct. – user40674 Aug 5 '19 at 14:51