Does this semantic tree close?

Question

Does this semantic tree close for the two branches that I have not classified? (The ones ending in $r$ and $\neg r$)

I would really appreciate some help on this! Thank you very much!

Answer 1

You can close the left branch since it contains both $p$ and $\neg p$. In fact, you could have closed that branch before branching on $q \vee r$.

The other branch does not close, since there are no more statements left to decompose in that branch and you have no contradictions there. So, the branch is open and finished, meaning that the set of statements in the root of the tree are satisfiable, i.e can all be true. Moreover, the atomic statements (or negations thereof) in the branch show how the statements can be true: set $p$, $q$, $r$, and $t$ to false, and $s$ to true. If you plug in those values, you see that the statements in the root are indeed all true. (Tip: it is always a good idea to check this, for if that doesn't work out, you did something wrong in the tree)

Finally, what does this all mean? Well, that depends on what you were trying to answer, and why you put the statements in the root of th tree in the fisrst place. But sinc none of the statements are a negation, I assume that you were not looking a an argument, but tried to figure out whether some se of sttements is consistent, and so the answer would be yes, they are consistent. If you wer lookin at an argument rhough, you may have forgotten to negate the conclusion.