Topics Logic

Question how would i use natural deduction to prove this argument to be correct? Its 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 would much appreciate the help. symbolization key: N: night D: day Fm: full moon Nt: night time Dt: day time ((N V D) , (Fm → Nt) , (Dt → ¬Fm))

Accepted:
January 15, 2019

Comments

There's a problem with your

There's a problem with your symbolization. The word "since" isn't a conditional. It's more like a conjunction, but better yet, we can treat it as simply giving us another premise. So in a slightly modified version of your notation, the argument would be

N v D
F → N
D
∴ ¬F

But from the premises as given, the conclusion won't be derivable. The reason is simple. You are assuming that if it's day it's not night and vice-versa. That may be part of the meaning of the words, but the symbols 'N' and 'D' aren't enough to capture it.

The easiest fix is to treat "day" as "not night." That gives us

N v ¬N
F → N
¬N
∴ ¬F

In this case, the first premise is a tautology and not needed. The argument is just a case of Modus Tollens. If you want something less trivial, you can drop the first premise and add a premise like this:

D ↔ ¬N
F → N
D
∴ ¬F

The first premise amounts to making the "v" exclusive. From there it's easy to complete a proof.

A couple of extra comments. First, in the English version, you add a quantifier, and presumably you are quantifying over times—roughly "For every time, the time is either during night or during day." The argument can be symbolized accordingly, though it will be more complex. Second, you make one of your premises counterfactual with the "would"/"were" construction. That introduces issues you'd be better off avoiding.