Clarifying the definition of a logical system on wikipedia 
Logical system is a deductive system together with additional (non-logical) axioms and a semantics. An example of a logical system is Peano arithmetic.  Wikipedia

Can you please explain what is non-logical axioms and a semantics in this context?
 A: As per discussions on previous similar posts we have many terms with similar meaning.
Maybe the first one is Formal language: an alphabet made of an initial (usually finite) stock of basic symbols and a set (usually finite) of rules to produce expressions (usually with finite length) that are called formulas.
With it we build a Formal system (also called a logical calculus, or a logical system) consisting of a formal language together with a deductive apparatus (also called a deductive system).
The deductive apparatus is a set (usually finite) of rules, called rules of inference, that we use to produce formulas from an initial stock of formulas: the axioms.
This is the syntactical part.
The logical system is used to derive logical truths (like e.g. $\vdash p \lor \lnot p$) and to derive consequences from assumptions: $\Gamma \vdash \varphi$.
We may call it the "pure" logical theory.
When the set of assumptions $\Gamma$ is the formalized version of the collection of axioms of some mathematical theory, like e.g. the firts-order arithmetic that formalizes the theory based on Peano axioms, we have an "applied" logical theory.
A formalized mathematical theory can be interpreted: for first-order arithmetic, the "natural" intepretation is the mathematical structure base on the set $\mathbb N$ of natural numbers.
The intuitive notion of interpretation, i.e. the way to assign meaning to symbols and formulas of a formal language, is made rigorous through the mathematical definition of Interpretation of a formal language.
This is the semantics.
