Logical systems (a.k.a. formal systems) are mathematical abstractions that aim to capture the notion of reasoning to reach valid conclusions from certain premises.
A logical system can be thought of as a procedure which divides a [-language] in badly-formed and well-formed sentences, and further splits this last group into theorems and not theorems.
Logical systems are made from a series of elements: a language, a syntax, axioms and rules of inference.
A language consists of the [word words] that can be formed from a set of symbols. Typically, we will want our language to be [-enumerable] and [-computable]. For example, a possible language for arithmetic is $~$\Sigma^* = \{\neg,\wedge,\vee,=,+,\cdot ,0,a_1,a_2,a_3,…\}^*$~$.
A syntax is the collection of rules which determine whether a word of our language is a well-formed formula.
The axioms are distinguished formulas of the language that are taken to true a priori. A logical system is [-axiomatizable] if its set of axioms is computable.
The rules of inference are $~$n+1$~$-[-tuples] that represent a function from $~$n$~$ formulas (premises) to a new formula (conclusion). For example, we have modus ponens as a rule of inference, which says that from a formula of the form $~$A\rightarrow B$~$ and another of the form $~$A$~$ you can deduce $~$B$~$. Almost always we will want our rules of inference to be [-computable]. Axioms can be thought of as rules of inference for which no premise is necessary.
Axioms and rules of inference are used to construct proofs. A proof of a sentence $~$S$~$ of the language is a finite sequence of sentences, such that every sentence is either an axiom or can be deduced from the previous sentences using a rule of inference, and the last sentence in the sequence is $~$S$~$. Sentences which have a proof are called theorems of the system.
Note that logical systems are purely syntactical entities - they talk for themselves about nothing. Logical systems are given meaning through [-semantics].
Logical systems can relate to one another through [-translations].