(For my proof, I'm assuming sentential logic.)
Given n sentence letters in argument A:
- The truth table TbA for A will have 2n rows.
- The truth tree TtA for A has an unbounded number of open branches.
Therefore, the TbA does not necessarily contain more rows than the number of open branches in TtA.
Proof: (1) is trivial but we can prove that the number of open branches are unbounded, i.e. (2), by induction on the disjunction operator like so:
First, consider the truth tree for P ∨ P:
(Open branches: 2)
Next, consider the truth tree for P ∨ (P ∨ P):
(Open branches: 3)
Finally, the truth tree for P ∨ (P ∨ (P ∨ P)):
(Open branches: 4)
We already provided several counterexamples to the claim that a truth table necessarily contains more rows than the number of open branches in the completed tree, but we can generalize this phenomenon (at least in the disjunctive case) by noticing that the truth tree always branches at a disjunction. If there is no contradiction on one of the branches, the branch will remain open.
The following disjunction will yield n+1 open branches in
TtA but only two rows
Where P is a sentence letter.The three above cases in fact show
this disjunction where n = 1, 2, 3 respectively.
Truth trees generated with http://gablem.com/truth-tree-solver/sentential-logic.