Is there a definition of "truth" without interpretations? I know that given a sentence or formula of a formal system, this formula is a logical truth if it is true under all interpretations. 
Is it possible to define this same concept of logical truth without the reference to models and interpretations?
Thanks!
 A: For first-order logic this is essentially the completeness theorem.
The completeness theorem tells us that if $T$ is a first-order theory, then $\varphi$ is provable from $T$ if and only if $\varphi$ is true in every model of $T$.
If a formula is logically true it means that it is true in every interpretation. Every interpretation is a model for the empty theory, and so by the completeness theorem we can say that something is logically true if and only if it is provable from $\varnothing$.
A: The common distinction is between syntactical truth and semantic truth. Given a deduction system (i.e., some rules telling us what which strings are allowed and how to deduce new the syntactical truth of a sentence given the syntactical truth of others) you get a well-defined notion of syntactic truth as those statements that are derivable in the deduction system from a given theory. 
In contact, semantic truth relates to a given model and is means those statements that when interpreted in $M$ are true. 
Goedel's completeness theorem states that (for first order logic) in a deduction system, a statement is syntactically true given a theory $T$ if, and only if, it is semantically true in all models of $T$. So, this answers your question.
The fact that syntactic truth implies semantic truth is quite easy to prove. The other direction is involved and requires a slightly weaker axiom than the axiom of choice.  
