Comparing 2 models of truth trees

Asked Mar 29 '21 at 00:51

Active Mar 29 '21 at 00:51

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Suppose I have the following truth tree:

I understand how it works. I also have the following tree (assume p = p_1 and q = p_2):

I got it from here. I've seen truth trees represented like this in other examples and couldn't understand them. Why are both branches closed?

Can someone explain to me how the 2nd model is equivalent to the 1st one? I just don't understand how it works.

asked Mar 29 '21 at 00:51

Every branch closes because each branch contains a contradiction: $q, \lnot q$ for the left one and $p,\lnot p$ for the right one. – Mauro ALLEGRANZA Mar 29 '21 at 05:53
If every branch closes the root formula is unsatisfiable. – Mauro ALLEGRANZA Mar 29 '21 at 05:53
You can see this post for the basic definitions as well as this one – Mauro ALLEGRANZA Mar 29 '21 at 05:56
The procedure tests for satisfiability; you have checked formula $A$ and found open branches: thus, the formula is satisfiable. The second picture test formula $\lnot A$ and found it unsatisfiable; this means that $A$ is a tautology. The teo cases are not the same but they are consistent: if $A$ is taut, then it is sat. – Mauro ALLEGRANZA Mar 29 '21 at 06:27

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