text: **Formal systems**
  
  A formal system of logic consists of a formal language together with a set of axioms and a proof system 
  used to draw inferences from these axioms.[95] In logic, axioms are statements that are accepted without 
  proof. They are used to justify other statements.[96] Some theorists also include a semantics that specifies 
  how the expressions of the formal language relate to real objects.[97] Starting in the late 19th century, 
  many new formal systems have been proposed.[98] 
  
  A*formal language*consists of an*alphabet*and syntactic rules. The alphabet is the set of basic symbols 
  used in expressions. The syntactic rules determine how these symbols may be arranged to result in well- 
  formed formulas.[99] For instance, the syntactic rules of propositional logic determine that " 
  " is a 
  well-formed formula but "   requires terms on both sides.[100]  " is not since the logical conjunction 
  
  A*proof system*is a collection of rules to construct formal proofs. It is a tool to arrive at conclusions from 
  a set of axioms. Rules in a proof system are defined in terms of the syntactic form of formulas 
  independent of their specific content. For instance, the classical rule of conjunction introduction states 
  that 
  . Such rules can be applied sequentially, giving a 
  mechanical procedure for generating conclusions from premises. There are different types of proof 
  systems including natural deduction and sequent calculi.[101] 
  
   follows from the premises   and 