3

If you have a tableau proof for $\Box A$, Show that there is also a tableau proof for A.

Here is my attempt but I'm not sure if it's correct:

If we have a tableau proof for $\Box A$, it means that all possible worlds that are accessible from the current world satisfy $A$. Therefore, if we can show that there is at least one world that satisfies $A$ in the current world, then $A$ must be true.

To show this, we can take the original tableau proof for $\Box A$ and add an additional branch where we assume $\neg A$. Then, we can apply the $\Box$ rule to this branch, which will generate a contradiction, since we know that all accessible worlds satisfy $A$. This contradiction allows us to close the branch and conclude that $A$ must be true in the current world.

Len
  • 51

1 Answers1

0

You don't state which modal logic you are working in. I will assume that you are working in System K, which places no restrictions on the accessibility relation. See here and here for details.

First, let's check that the statement is actually true semantically.

We want to show that $\models \square A \implies \models A$.

Consider the frame with one world $w$ and an accessibility relation $R$ that is never true. Suppose $A$ is false at $w$.

$ (w, R), w \Vdash \square A$ is vacuously true.

However, $(w, R), w \Vdash A$ is false.

Whichever tableau method you are using, it must be sound and complete, so the theorem you want to show is false provided we don't have assumptions on the accessibility relation that rule out this frame.

Your rule, however, is the converse of the necessitation rule $\models A \implies \models \square A$, which is true even in system $K$.

You can prove $\vdash A \implies \vdash \square A$ by proving it semantically and then appealing to the soundness and completeness of the tableau calculus.

Proving $\vdash A \implies \vdash \square A$ without appealing to the semantics is harder, and doing it requires that you pick a specific tableau calculus and understand how it handles the bookkeeping details of possible worlds. I personally like the world-labeled tableaux with pseudoformulas denoting accessibility assumptions (like $1R2$ denoting that world $2$ is accessible from world $1$) being written off to the side.

Greg Nisbet
  • 12,281