The enclosure of $\mathit{GIL}$

The new principle was finally named P₀ by Frank Veltman. He called it thus for its similarity with $P: A \rhd B \rightarrow \Box (A \rhd B)$. P₀ as he noticed can be seen as a weakening of P. Strengthening the antecedent of a conditional weakens it. And indeed $\vdash_{\mathit{IL}}A \rhd \Diamond B \rightarrow A \rhd B$. Similarly M can be weakened to obtain M₀. This calls for a general approach. On the one hand we have principles whose corresponding logic is too weak to be $\mathit{GIL}$, and on the other hand we have the logics $\mathit{ILM}$ and $\mathit{ILP}$ which are clearly too strong to be a candidate for $\mathit{GIL}$. The general approach would be to weaken the logics which are too strong and to strengthen the logics which are too weak. We can alter our logics in three different aspects:

• On an arithmetical basis one can appropriately vary the principles. This way of enclosing $\mathit{GIL}$ is the most direct. It is quite difficult to think of candidate principles though and is outside the realm of this thesis.
  

• Another possibility can be to strengthen for example the frame-conditions of some principle which is too weak. $\mathit{ILP_0}$ was found like this. For example one might try to strengthen the frame condition of W to demand $S\circ R$ conversely well-founded.
  

• The other possibility would be syntactical modulation. Many statements in interpretability logic take the form of a valid conditional. Even the $\rhd$ operator can be considered as a conditional. Weakening a conditional can be done by weakening the consequent or strengthening the antecedent. For weakening or strengthening for example a consequent, one can use implications. The antecedent of a conditional is stronger than the consequent. All the axioms of $\mathit{IL}$ can be written as an implicational statement and hence can be used to modify the other principles. J5 can be written in the equivalent form:
  
  $$J5':\ \ \ A\rhd B\rightarrow \Diamond A\rhd B.$$
  
  
  Using this form of J5 to modify other statements gives interesting results. M₀ can be obtained as a right weakening of M and P₀can be seen as a left weakening of P. It has not yet been systematically investigated if new interesting principles can be found proceeding like this.