Proving that a sentence is true in the language of predicate logic

Question

I have recently completed self-studying from a 'transition' type textbook concerning proofs in mathematics and have decided to continue with a philosophical logic book'the logic manual' by Halbach.I tried to input this question in philosophy.stackexchange but for some reason,the math formatting does not work...so i hope i can post this here!

I am having trouble with understanding the following problem and would appreciate some help !

Consider the $L_2-structure$ $A$ with

• Domain of discourse the set of all European cities.
• $|Q^1|_A=\{Florence,Stockholm,Barcelona \}$
• $|R^2|_A=\{<d,e> : \text{d is smaller than e}\}$
• $|a|_A=Florence$
• $|b|_A=London$

Prove : $|\forall x \exists y (R^2xy \lor R^2yx)|_A=T$

Informal Attempt:

Since the formula between $|\dots|$ does not contain any free variables,it is classified as a sentence in $L_2$.

By definition of truth in an $L_2-structure$,$|\forall x \exists y (R^2xy \lor R^2yx)|_A=T$ iff $|\forall x \exists y (R^2xy \lor R^2yx)|_A ^\alpha=T$ for all variable assignments $\alpha$ over $A$.

Therefore:

1. Let $\alpha$ be a variable assignment over $A$.The objective is to prove that $|\forall x \exists y (R^2xy \lor R^2yx)|_A ^\alpha =T$.

2. By successively applying the satisfaction definitions,I ended up with the following proof diagram:

   Let $\alpha$ be any variable assignment over $A$.
   $\quad$ Let $\beta$ be any variable assignment over $A$ differeing from $\alpha$ at most in $x$.
   $\quad \quad$ Let $\gamma$ be the variable assignment found that differs from $\beta$ in $y$ at most.
   $\quad \quad \quad$ Prove $|R^2xy \lor R^2yx|_A ^\gamma = T$

3. Given the way $\alpha , \beta , \gamma$ are related i thought we could eliminate $\beta$ and directly write:

   Let $\alpha$ be any variable assignment over $A$.
   $\quad \quad$ Let $\gamma$ be the variable assignment found that differs from $\alpha$ in $y$ and $x$ at most.
   $\quad \quad \quad$ Prove $|R^2xy|_A ^\gamma=T$ or $|R^2yx|_A ^\gamma = T$

In the light of the above, the objective is to find a variable assignment $\gamma$ such that it differs from $\alpha$ at most in x and y such that $|R^2xy|_A ^\gamma=T$ or $|R^2yx|_A ^\gamma = T$.

Therefore, I thought I should try to 'reverse-engineer' $\gamma$ as follows:

1. By satisfaction : $|R^2xy|_A ^\gamma=T$ iff $<|x|_A ^\gamma,|y|_A ^\gamma> \in |R^2|_A ^\gamma$.
   Then defining $\gamma$ such that it assigns $|x|_A ^\gamma=NotLondon$ and $|y|_A ^\gamma=London$,yields $<|x|_A ^\gamma,|y|_A ^\gamma> \in |R^2|_A ^\gamma$ as required since any european city is smaller than London.

2. By satisfaction : $|R^2yx|_A ^\gamma=T$ iff $<|y|_A ^\gamma,|x|_A ^\gamma> \in |R^2|_A ^\gamma$.
   Then defining $\gamma$ such that it assigns $|x|_A ^\gamma=London$ and $|y|_A ^\gamma=NotLondon$,yields $<|y|_A ^\gamma,|x|_A ^\gamma> \in |R^2|_A ^\gamma$ as required since any european city is smaller than London.

...now if this is correct so far,how do i relate $\gamma$ such that it differs from $\alpha$ at most in x and y?? Since $\alpha$ is arbitrary,presumably we can choose it to be whatever we need it to be??

The proof in the book begins by making a distinction between 2 cases at the outset:

$|x|_A ^\alpha=NotLondon$ and $|x|_A ^\alpha=London$ and goes on from there.

Although i understnad how it proceeds,as usual with textbooks there is no explanation as to why.

Help someone! I have tried my best to format this stuff! Thanks