See Herbert Enderton, A Mathematical Introduction to Logic (2nd - 2001), page 23 :
take the special case in which $\Sigma$ is the empty set $\emptyset$.
Observe that it is vacuously true that any truth assignment satisfies every member of $\emptyset$. (How could this fail? Only if there was some unsatisfied member of $\emptyset$, which is absurd.) Hence we are left with : $\emptyset \vDash \psi$ iff every truth assignment (for the sentence symbols in $\psi$) satisfies $\psi$.
In this case we say that $\psi$ is a tautology (written $\vDash \psi$).
As highlighted by Enderton, we have a case of vacuous truth.
The condition for an interpretation $I$ to be a model of $\Sigma$ is that all the sentences in $\Sigma$ must be satisfied by the interpretation $I$.
This is :
for all $\sigma$, if $\sigma \in \Sigma$, then $I$ satisfy $\sigma$.
Thus, vacuous truth applies : there are no $\sigma \in \emptyset$.
What we are trying to convince ourselves is that : if $\emptyset \vDash \psi$, then $\psi$ is a tautology.
Consider again the definition of logically implies; there is a double conditional in place.
Saying that $\Sigma \vDash \psi$ means :
for every interpretation $I$, [ if for every sentence $\sigma$, ( if $\sigma \in \Sigma$, then $I$ satisfy $\sigma$ ), then $I$ satisfy $\psi$ ].
In semi-formal way :
$\forall I [ \forall \sigma(\sigma \in \Sigma \rightarrow I \vDash \sigma) \rightarrow I \vDash \psi ]$.
When we put $\emptyset$ in place of $\Sigma$, the antecedent of the "inner" conditional is false; thus, by truth-table for $\rightarrow$, the conditional is true (and this says nothing about the truth-value of the consequent !).
In this way, the antecedent of the "outer" conditional is true. But we are asserting the fact that $\emptyset \vDash \psi$, i.e. that the "outer" conditional is true.
If it is true and if its antecedent is true, there is only one possibility left : the consequent is true.
I.e.
$I \vDash \psi$.
This hold for every $I$, and thus we can conclude that $\psi$ is a tautology.
New addition
We can try with another approach.
According to the definition of logical consequence, $\varphi \vdash \psi$ iff $\varphi \land \lnot \psi$ is always false.
Now, assuming that $\Gamma$ is a finite set of sentences, i.e. $\Gamma = \{ \gamma_1, \ldots, \gamma_n \}$, we have that :
$\Gamma \vdash \psi$ iff $\gamma_1 \land \ldots \land \gamma_n \land \lnot \psi$ is always false.
Thus, if $\Gamma = \emptyset$, the above condition boils down to :
$\emptyset \vDash \psi$ iff $\lnot \psi$ is always false.
Obviously, $\lnot \psi$ is always false iff $\psi$ is a tautology.
First, some terminological issues. $A\vdash B$ usually means $A$ is provable or derivable from $B$. This is a purely syntactic property that is about building formal proofs and does not require knowing whether anything is "true" or not. Validity usually means that a formula is semantically true in all models and is written $\vDash B$ with $A\vDash B$ as shorthand for "$\mathfrak M\vDash A$ implies $\mathfrak M\vDash B$" for all models $\mathfrak M$ with $\mathfrak M\vDash A$ meaning "$A$ is semantically true in model $\mathfrak M$". "Syllogism" has a fairly specific meaning and is relatively archaic at this point. You'll rarely find it used in a modern logic textbook except in a "history of logic" section. You are also using "sound" in the more philosophical sense. This unfortunately conflicts with "sound" in the mathematical logic sense which becomes relevant... now. $\vdash$ and $\vDash$ are (for a given logic) usually sound and complete. Soundness means "$\vdash B$ implies $\vDash B$", i.e. what we can prove is valid. Completeness means "$\vDash B$ implies $\vdash B$", i.e. we can prove everything that is valid. Soundness and completeness together mean that $\vdash$ and $\vDash$ are the same relation on formulas which is why the terminology often gets muddled. However, soundness and completeness are non-trivial (meta-)theorems (particularly completeness), and you need to understand what $\vdash$ and $\vDash$ mean on their own before you can prove them.
To actually start addressing your question, it doesn't make sense in mathematical logic to talk about a formula just being "true". You can talk about it being provable (i.e. a theorem) or being valid. Validity, as I mentioned before, is defined in terms of a notion of semantic truth, and the key thing here is that truth is with respect to a model written $\mathfrak M\vDash B$ which means $B$ is true in the model $\mathfrak M$. Validity can then be written as "for all models $\mathfrak M$, $\mathfrak M\vDash B$". For propositional logic, the models are often called "valuations" or "(truth) assignments" as in Mauro ALLEGRANZA's answer. In this case, they consist entirely of assignments of truth values to atomic propositions which can then be lifted to assignments of truth values to all formulas via the interpretation of the connectives.
The closest thing to what you want is therefore something like $\mathfrak M\vDash B$ for some particular model $\mathfrak M$.
There is nothing in mathematical logic to say that some formula is "true in reality". Whether something is "true in reality" is not a mathematical question but a physical or maybe a philosophical one. Even semantics in mathematical logic interprets things into mathematical structures, typically sets, so semantic truth is just a statement about certain mathematical structures.
If a mathematical logician wanted to say something about a formula being "true in reality" (which would be a very odd thing for them to do), they'd just say it in natural language.
Best Answer
The error is in “Since the empty set doesn't have any premises then there does not exist any valuation where the premises are true and the conclusion false”. In natural language, one might argue that “the premises” cannot be true if there are no premises, but here the required proposition is that all premises are true, and if there are no premises, then by definition all premises are true (and all premises are false). All elements of the empty set are white ravens.