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?

  • $\begingroup$ Correct; you cannot derive the conclusion, because from the premise you can derive $t$, but there is no way to derive also $u$ $\endgroup$ Dec 9, 2020 at 9:01
  • $\begingroup$ I hope too lmao $\endgroup$
    – Arkilo
    Dec 9, 2020 at 9:45

1 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.

  • $\begingroup$ I think it was made by the instructor himself, I haven't found it anywhere else. $\endgroup$
    – Arkilo
    Dec 10, 2020 at 11:58
  • $\begingroup$ @Arkilo OK, good to know. Also, it is quite possible that your instructor is going to try to point to inference rules anyway in an attempt to show that this is invalid ... indeed, if so, the instructor will probably do something similar to what you did, i.e. successfully get to $t$ .... but then point out that you also need $u$ to get to $t \land u$ ... and that you don't have $u$. But, that is not a demonstration at all. Maybe you can get $u$ some other way .... and even if you can't: how do you really know? You don't really know until you use a different method, such as a truth-table. $\endgroup$
    – Bram28
    Dec 10, 2020 at 14:15
  • $\begingroup$ That truth table is gonna have a LOT of rows $\endgroup$
    – Arkilo
    Dec 10, 2020 at 17:18
  • 1
    $\begingroup$ @Arkilo Yes :). Thankfully, there are other methods to check for validity yet. Instead of trying to systematically explore all possible truth-value assignments (like a truth-table does), you can often do a more focused search for a counterexample: a truth-assignment that sets all premises to true but the conclusion to false. And in this case, it is indeed very easy to come up with one: set $p, q$, and $t$ to true, and $r, s$, and $u$ to false. Once you show that with this assignment all premises are true but the conclusion false, you have a proper demonstration of invalidity. $\endgroup$
    – Bram28
    Dec 10, 2020 at 17:44

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