How would i go about using natural deduction to prove this argument is valid?

Question

How would I use natural deduction to prove this argument is correct?

It's always either night or day. There'd only be a full moon if it were night-time. So, since it's daytime, there's no full moon right now.

I have also formalized the argument using truth functional logic.

I'm not sure if it is completely correct though and I would much appreciate help if is is not.

Symbolization Key:
N:  night 
D:  day 
Fm: full moon
Nt: night time 
Dt: day time 

((N V D) , (Fm → Nt) , (Dt → ¬Fm))

Answer 1

1: (D v N) & ~(D & N) (exclusive or)

2: Fm -> N

3 Dt

Show: ~Fm

4: ~(D & N) 1,&out

5: ~D v ~N 4, ~&out

6: ~~Dt 3, ~~in

7: ~N 5,6, vout

8: ~Fm 2,7, ->out

This assumes Dt and D, and Nt and N have the same truth values (intuitive assumption).

Answer 2

Using N = it's night, D = it's day, F = there's a full moon, the argument is:

1. D ↔ ~N
2. F → N
3. Conclusion: D → ~F 

Note that 1 is formulated using ↔ and ~. That is the standard formulation of exclusive-or: when we want to say that the two options are exclusive, i.e., only one of them can be true.

The derivation is:

1. D ↔ ~N    premise
2. F → N     premise
3. D         assumption
4. ~N        MP 1,3
5. ~N → ~F   contraposition 2
6. ~F        MP 4,5
7. D → ~F    →-intro 3-6

Answer 3

For this answer I will follow the authors of forallx listed below. Here is how they describe a symbolization key: (page 23)

We will use atomic sentences to represent or symbolize, certain English sentences.

So the symbolization key associates a symbol with an atomic sentence. These atomic sentences paraphrase the argument to bring out the form. We will then test the form to see if it is valid. Here are the English sentences to symbolize in some way:

It's always either night or day. There'd only be a full moon if it were night-time. So, since it's daytime, there's no full moon right now.

I will try this symbolization key:

N: It is night.
M: There is a full moon.

With this symbolization I can write It's always either night or day as N v ¬N. I will write There'd only be a full moon if it were night-time as M → N. I see the last sentence as giving another premise that it is daytime, that is, not night. So I have as a premise ¬N. The conclusion I hope to derive from this is ¬M, that is, it is not the case that there is a full moon.

Associated with forallx is a natural deduction proof checker that, assuming I have symbolized this properly, shows that the argument is correct or valid:

The MT step is "modus tollens" and it is described on page 125.

There may be other ways to approach this problem. The resources listed below may be useful supplementary materials for learning natural deduction.

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

P. D. Magnus, Tim Button with additions by J. Robert Loftis remixed and revised by Aaron Thomas-Bolduc, Richard Zach, forallx Calgary Remix: An Introduction to Formal Logic, Winter 2018. http://forallx.openlogicproject.org/

Answer 4

It's always either night or day. There'd only be a full moon if it were night-time. So, since it's daytime, there's no full moon right now.

My interpretation:

1. Night or not-night.

2. (Night or not-FM) and not-(night and not-FM).

2a. (night or not-FM) and (not-night or FM).

3. Not-night.

Thus, not-FM.

1. This design uses "not-night" in place of "day". The exclusive OR takes care of itself because of the law of the excluded middle (a thing cannot both be and not be).

2. Then, it is either Night or there is no Full Moon. Also, there cannot exist at the same time Night and no Full Moon. Here, it is necessary to create the exclusive OR, as that is clearly part of the premise ("only...if").

2a. This statement is equivalent to 2.

3. The premise "Day" becomes "not-Night".

Conclusion. The conclusion follows from (2a). In the first part of (2a), "not-night" denies the statement "night", and so "no Full Moon" follows. As to the second part of (2a), "not-night" simply affirms the first term of the OR statement, and the result neither contradicts nor restates the conclusion.

I am not entirely satisfied with this derivation, but there it is.

Thanks to Bertrand Wittgenstein's Ghost, Mauro Allegranza, and rus9384 for the observations about the exclusive OR.