1. How does one signify that a tree that begins with Γ on the left and ϕ on the right closes?
Γ ____________ ϕ
2. In our argument for the contrapositive of the soundness of TABLEAU, we begin by assuming that there is an interpretation on which all Γ is true and ϕ is false. In other words, in terms of the double turnstile, we suppose that:
Γ ____________ ϕ
3. Our aim, then, will be to show that a tree that begins with Γ on the left and ϕ on the right has a consistent path.
4. To do this, we observe that our assumption means that the root of our tree starts out as consistent, which in turn means that there is a path in which there is an interpretation on which all of the formulas on the left are true and all of the formulas on the right are false.
5. And now we argue that systematically extending a path in accordance with the TABLEAU-rules will always result in at least one consistent path.
6. That argument requires us to show how the TABLEAU-rules direct us to decompose compound formulas in a way that aligns with the consequences of their being true if on the left and false if on the right. For instance, if an interpretation makes a disjunction false, then that same interpretation must make both of its disjuncts false. Accordingly, the TABLEAU-rule for disjunction on the right has us place both the disjuncts on the right. And if a disjunction is true, then it follows that at least one of its disjuncts must be true. The TABLEAU-rule for disjunction on the left captures this thought by splitting the path and placing a disjunct on the left of each branch.
7. Next, we observe that since a consistent path never has one and the same formula on both sides, such a path will not close.
8. So, from our assumption that there is an interpretation on which all Γ is true and ϕ is false, we have shown that the tree that begins with Γ on the left and ϕ on the right does not close; this can be symbolized in terms of our turnstiles as: If Γ __________ ϕ, then Γ __________ ϕ.
9. It follows (by contraposition) that if a tree that begins with Γ on the left and ϕ on the right can be closed, we know that there is no interpretation on which all Γ is true and ϕ is false.
10. And that claim asserts the soundness of the game of TABLEAU, which we can signify in terms of our turnstiles as: If Γ __________ ϕ, then Γ __________ ϕ.
