Let $\varphi$ and $\phi$ be two propositions such that $\vDash_{CL}\varphi\to\phi$, i.e. it's a tautology.
Prove that if $\varphi$ and $\phi$ have atoms in common, say, $p_1,...,p_n$, then there exist a proposition $\rho$ which only contains $p_1,...p_n$ such that $\vDash_{CL}\varphi\to\rho$ and $\vDash_{CL}\rho\to\phi$.
Proof
Let $\rho$ be the disjunction of all eligible conjunctions, such that each eligible conjunction contains all and only statement letters that occur in both $\phi$ and $\varphi$. Then $\vDash_{CL}\varphi\to\rho$.
Now suppose that there is some truth assignment that makes $\rho$ truth but $\phi$ false. Extend this to a truth assignment that covers the statement letters in $\varphi$ that are not in $\phi$ (and so not in $\rho$). Some such truth assignment still makes $\rho$ truth and $\phi$ inadequate, since it doesn’t change anything about the statement letters in either formula. But it also truthifies $\varphi$, per impossible since $\vDash_{CL}\varphi\to\phi$. So there ain’t no such truth assignment. So $\vDash_{CL}\rho\to\phi$.
I do not understand how this is achieved, $\vDash_{CL}\varphi\to\rho$, through the previous hypothesis.
Also, in the part where says ...truth assignment that covers the statement letters in $\varphi$..., this means that a truth assignment might not cover all the statement letters in some formula?, for instance if $\phi=A\to B \land C$, then can I consider v such that only will apply to $(A\to B)$ and not $C$ ??
Why the new extended function will make $\phi$ inadequate? I think the extended function should conserve $\phi$ as false.
- And the last question, why the new extended function will make $\varphi$ true, why is this guaranteed?
Thanks in advance for your time and help.