What do the paths of a truth tree represent?

Question

If you have a truth tree with proposition P at the root, what do the paths of the truth tree represent?

• Do all the paths collectively represent every possible truths assignment for P? If that is the case why can't we say that P is a tautology if all the paths of P are open? (Since all the paths being open represent that every truth assignment makes the proposition true)

• Consequently, if that is not the case and we say that P isn't necessarily a tautology because the paths don't represent all the possible truth assignments of P (meaning that there are some truth assignments not covered, thus leaving us with the possibility that P is a contingency) . Then why is it allowed to say that P is a contradiction if all its paths are closed? Couldn't we by the same argument in this point say that there are still some truth assignments not covered.( hence there is the possibility that P could still be a contingency)

• The third "argument" would be a sort of special case of the second argument, which is that the paths collectively represent not all but a "special" type of truth assignments, at which point I fail to see what is special about them

Any input would be appreciated

Answer 1

To expand on Mauro Allegranza headline answer. We are working with a truth tree headed by the formula $P$.

1. Suppose you have completed open path on the tree -- i.e. one where every complex wff on the tree has had the relevant tree-building rule applied to it. Take the simple wffs -- atoms and negated atoms -- on that path. Consider a valuation which makes those simple wffs true. Then it is easy to see that this valuation makes every wff on the path true, including $P$ at the top.
2. In short, one completed open path corresponds to (at least) one valuation which makes $P$ true. So: the open paths on a tree collectively do represent the various valuations (if any) which make $P$ true.
3. But of course, there may e.g. only be one open path, corresponding to one valuation which make $P$ true. That obviously won't make $P$ a tautology.
4. If there are no open paths, i.e. if the tree is closed, then that means precisely there is no valuation available which makes $P$ true -- it really is a contradiction. For if there such a valuation available, there would be an open path on the tree (corresponding to that valuation) and the tree wouldn't close.

Answer 2

Every open path defines a truth assignment that satisfies the root formula $P$.

Thus, if all paths are closed, the root formula $P$ is unsatisfiable, i.e. the formula $¬P$ is a tautology.

See e.g. Truth Trees Method.