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text: '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.\n\nA 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.\n\nLogical systems are made from a series of elements: a **language**, a **syntax**, **axioms** and **rules of inference**.\n\nA **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 $\\Sigma^* = \\{\\neg,\\wedge,\\vee,=,+,\\cdot ,0,a_1,a_2,a_3,...\\}^*$.\n\nA **syntax** is the collection of rules which determine whether a word of our language is a well-formed formula. \n\nThe **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.\n\nThe **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.\n\nAxioms 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.\n\nNote that logical systems are purely syntactical entities - they talk for themselves about nothing. Logical systems are given meaning through [-semantics]. \n\nLogical systems can relate to one another through [-translations].',
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