Understanding sequent calculus

Question

Are the following rules correct?

$(i)$ $\dfrac{\Phi \Rightarrow \Delta}{\Phi, \psi \Rightarrow \Delta}$

$(ii)$ $\dfrac{\Phi, \psi \Rightarrow \Delta}{\Phi \Rightarrow \Delta}$

Intuitively, I would've said that rule $i$ is incorrect and rule $ii$ is correct. We know that a rule is correct when the validity of the upper sequence(s) implies the validity of the lower sequence(s). So for $i$, if $\Phi \Rightarrow \Delta$ is valid, then $\Phi, \psi \Rightarrow \Delta$ must also be valid, but we can pick $\psi$ to be the empty set, making $\Phi, \psi$ unsatisfiable. The same argument can be used in the different direction for $ii$.

However, my textbook gives the exact opposite answer: $i$ is correct and $ii$ is incorrect. Can someone explain to me why this is the case?

Answer 1

The book is correct. (i) is a valid inference rule and (ii) is not valid. Also, the capital Greek letters $\Phi$ and $\Delta$ refer to sets of well-formed formulas, but $\psi$ refers to a single well-formed formula.

Let's consider the two sequents $\Phi \vdash \Delta$ and $\Phi,\psi \vdash \Delta$ and consider when they are true.

Also, let's define $\Phi''$ as $\lnot\Phi_1, \cdots, \lnot\Phi_n$. The double prime is not standard notation. I will use $\bigvee S$ to denote a disjunction over a finite set $S$.

$$ \Phi \vdash \Delta \; \text{holds} \;\; \text{if and only if} \;\;\bigvee \Phi'' \cup \Delta \;\text{holds} $$

$$ \Phi, \psi \vdash \Delta \; \text{holds} \;\; \text{if and only if} \;\;\bigvee \Phi'' \cup \{\lnot\psi\} \cup \Delta \;\text{holds} $$

$\Phi'' \cup \Delta$ is a subset of $\Phi'' \cup \{\lnot \psi\} \cup \Delta$, therefore the truth of the former implies the truth of the latter and thus

$$ \frac{\Phi \vdash \Delta}{\Phi, \psi \vdash \Delta} $$

Similarly, the converse is invalid because of the case where $\bigvee \Phi'' \cup \Delta$ is false, but $\lnot\psi$ is true.