Why is "If ψ ∈ Γ then the sequent (Γ ⊢ ψ) is correct" true?

Question

In Chiswell and Hodges Mathematical Logic the authors define a sequent as such

"A sequent is an expression (Γ ⊢ ψ) (or Γ ⊢ ψ when there is no ambiguity) where ψ is a statement (the conclusion of the sequent) and Γ is a set of statements (the assumptions of the sequent) ...There is a proof whose conclusion is ψ and whose undischarged assumptions are all in the set Γ.".

They then go on to provide the axiom as follows:

Sequent Rule (Axiom Rule) If ψ ∈ Γ then the sequent (Γ ⊢ ψ) is correct.

Upon further reading about what exactly it means to be a sequent wikipedia I understand that for $\psi$ to be satisfied or "correct", every element of $\Gamma$ must be true as their all linked by AND conjunctions.

What confuses me is that suppose that there is an element in $\Gamma$ which is false then the whole antecedent is false according to the wikipedia definition by conjunction and then even though $\psi$ exists and is true in $\Gamma$ ultimately $\psi$ would be false. Could someone please explain to me and help clarify how the axiom and the definition can both be true. Thank you for all of your help.

Answer 1

The authors are introducing the basic elements of the proof system.

As you said, the definition of correct sequent $(\Gamma \vdash \psi)$ is :

There is a proof [according to the rules of the system to be specified] whose conclusion is $\psi$ and whose undischarged assumptions [premises] are all in the set $Γ$.

When the semantics of the language will be defined [see para 3.5] the authors will intorduce the concept of semantic sequent : $\Gamma \vDash \psi$, defined as :

for every $σ$-structure $A$, if $A$ is a model of $Γ$ then $A$ is a model of $ψ$.

The definition formalizes the informal concept of valid argument.

Then, they will prove the basic result [see page 87 : the Soundness Theorem of Natural Deduction for Propositional Logic] :

$\Gamma \vdash \psi \text { iff } \Gamma \vDash \psi$.

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Having said that, the rules of the proof system are the "rules of the game" that allows us to derive conclusion from premises.

It is obvious that if $\psi \in \Gamma$, we can derive it from $\Gamma$ and this is formalized with the (Axiom Rule) above.

What if $\psi$ is false ? No problem: the move is "formally" correct but the argument is still valid because the case $\psi$ false does not contradict the definition of valid argument :

the conclusion must be true whenever all the premsies are true.

In general, the reasoning applies if some elements of $\Gamma$ is false; the (Axiom Rule) applies (because a premise can always be derived as conclusion) without contradiction.