Is there a subtle difference between "sa" and "sai" in formal situations?

Question

I've been learning more about sapere. I found the following resource but I'm missing a subtle difference between "sa" and "sai". These quotes are from a university so I assume they are correct but want to understand why.

http://www.uky.edu/~allaire/CONSAP.htm in the answer key under "II. Sapere..." you'll see 2 examples. They have:

"Mi scusi, signore, sa per caso che ora è?"
"Scusi, signora, sa se l'autobus numero 27 si ferma qui?"

In both cases above, instead of "sa" why not use "sai" which literally means "you know"? like

"Mi scusi, signore, sai per caso che ora è?"

The only thing I can think of is, in this formal context "sa" might be "Lei sa"?

Thanks.

Answer 1

The respect form in Italian uses the third person singular (with an originally feminine subject “lei”). Some varieties of Italian still use “voi” (second person plural), but such usage is regional and is not concerned here.

So, when addressing a person whom you don't know or, for any reason, you want to show respect for, you use “lei” as an implicit subject and the verb has to be in the third person singular: since

so sai sa sappiamo sapete sanno

is the conjugation of the present tense of sapere, the correct form is

(lei ) sa

As usual the subject is most of the times left implicit in Italian.

Also scusi is in the third person singular (present subjunctive), but this is not recognized by the termination, which is the same for all three persons in the singular:

Scusi (third person singular), signora (vocative), sa (third person singular) …

This is not special of sapere:

Permetta, signore, va per caso a Milano?
Mi dica, buon uomo, è diretto a Viggiù?
Maria, sai se il treno è in orario?

In the third case, calling a person by first name usually wants the second person singular, but it's not uncommon (or at least it used not to be uncommon) to use “lei” also with the first name by acquaintances who don't deem to be friends enough to use tu.

Answer 2

Yes, exactly. The implicit pronoun is "Lei", and the verbs have to match that. Note that there's a third-person subjunctive to replace the second-person imperative as well:

• Scusami, Marco, (tu) sai per caso che ora è?
• Mi scusi, signore, (Lei) sa per caso che ora è?