I suspect you've misunderstood what you quoted from the earlier question, since the semantics discussed there was the semantics of sentences, presupposing a notion of sentence. Nevertheless, your present question makes sense, and the (probably rather uninformative) answer is yes. The idea is to mimic semantically the definitions of formulas and their satisfaction, without ever mentioning the strings (or trees, etc.) of symbols that syntax deals with. Fix a vocabulary $L$ and (to avoid set-versus-class problems) consider only $L$-structures included in some fixed large set. "Formulas" will, in this approach, be certain collections of pairs $(A,s)$ where $A$ is a structure and $s$ is a finite tuple of elements of $A$. (The length of the tuple $s$ is to be the same for all pairs in a specific formula, but it may be different for different formulas). To define which collections of pairs count as formulas, one uses an inductive definition whose main clauses look like

If $R$ is an $n$-ary predicate symbol of $L$ then there is a formula consisting of exactly those pairs $(A,s)$ where $s$ is an $n$-tuple satisfying $R$ in $A$.

If $\phi$ and $\psi$ are two formulas in which the $n$-tuples have the same length, then $\phi\cup\psi$ is a formula.

The complement of any formula $\phi$, within the collection of all pairs $(A,s)$ whose tuples $s$ have the same length, is again a formula.

If $\phi$ is a formula whose tuples have non-zero length, say $n+1$, then there is a formula consisting of those pairs $(A,s)$ in which $s$ is an $n$-tuple and $(A,(a)^\frown s)\in\phi$ for some $a\in A$.

If $\phi$ is a formula whose tuples have length $n$ and if $f:\{1,2,\dots,m\}\to\{1,2,\dots,n\}$ then there is a formula consisting of exactly the pairs $(A,s\circ f)$ with $(A,s)\in\phi$.

The first clause says that atomic formulas are formulas; the next three say that the class of formulas is closed under the usual first-order constructors (disjunction, negation, and existential quantification), and the last allows trivial manipulations of the components in tuples (permutation, identifications, and adding dummy components).

syntacticversion of compactness: a theory is consistent iff its every finite subset is consistent. – Gyorgy Sereny Jun 30 '11 at 15:59