Proving whether an argument is valid or not when the truth value of a proposition just can't be found.

Question

The question I must solve is the following:

Given Problem: Use the rules of inference, show that the given argument is valid or not. The domain is all persons.

“You are listening, and I am speaking. If you are listening, then it is not silence. If music system is off then it is silence. If music system is not off then we are in happy mood, imply the conclusion that we are in happy mood and we are good friends.”. Give a reason for each step of proof.

My attempt at a Solution: I begin solving it by first creating some propositions as follows:

p = "You are listening"

q = "I am speaking"

r = "It is silence"

s = "Music system is off"

t = "We are in a happy mood"

u = "We are friends"

Now I rewrite the given arguments as follows:

• $p \land q$
• $p \rightarrow \lnot r$
• $s \rightarrow r$
• $(\lnot s \rightarrow t) \rightarrow (t \land u)$

Using these premises I do:

$$p\land q \tag{Premise}\\$$ $$p \tag{Simplification}$$ $$p \rightarrow \lnot r \tag{Premise}$$ $$\lnot r \tag{Modus Ponens}$$ $$s\rightarrow r \tag{Premise}$$ $$\lnot s \tag{Modus Tollens}$$ $$\lnot s \rightarrow t \tag{Premise}$$ $$t \tag{Modus Ponens}$$ $$t \land u \tag{Premise}$$

Now I don't know the truth value of u, if I did I could derive it and I think that would prove that the argument is valid.

Question: Is the argument invalid since there is no way for me to derive anything further? Or is there a flaw in my reasoning here?

Answer 1

It is obvious that you cannot derive that 'we are good friends', and so this argument is invalid.

But, you cannot show the argument to be invalud using rules of inference. Using rules of inference you can only show arguments to be valid.

So .. I have no idea how you are supposed to do what you are asked to do.

What a terrible problem! Please let me know what book this is from so I can be sure never to get it.