In mathematical logic

Mathematical logic

Mathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...

, an abstract logic is a formal system

Formal system

In formal logic, a formal system consists of a formal language and a set of inference rules, used to derive an expression from one or more other premises that are antecedently supposed or derived . The axioms and rules may be called a deductive apparatus...

 consisting of a class of sentence

Sentence (mathematical logic)

In mathematical logic, a sentence of a predicate logic is a boolean-valued well formed formula with no free variables. A sentence can be viewed as expressing a proposition, something that may be true or false...

s and a satisfaction relation with specific properties related to occurrence, expansion, isomorphism, renaming and quantification.

Based on Lindström

Per Lindström (logician)

Per Lindström was a Swedish logician, after whom Lindström's theorem and the Lindström quantifier are named. He was one of the key followers of Lars Svenonius.-External links:*...

's characterization, first order logic is, up to equivalence, the only abstract logic which is countably compact and has Löwenheim number

Löwenheim number

In mathematical logic the Löwenheim number of an abstract logic is the smallest cardinal number for which a weak downward Löwenheim–Skolem theorem holds...

 ω.

See also

• Abstract algebraic logic
  Abstract Algebraic Logic
  In mathematical logic, abstract algebraic logic is the study of the algebraization of deductive systemsarising as an abstraction of the well-known Lindenbaum-Tarski algebra, and how the resulting algebras are related to logical systems.-Overview:...
  
  
• Abstract model theory
  Abstract model theory
  In mathematical logic, abstract model theory is a generalization of model theory which studies the general properties of extensions of first-order logic and their models....
  
  
• Löwenheim number
  Löwenheim number
  In mathematical logic the Löwenheim number of an abstract logic is the smallest cardinal number for which a weak downward Löwenheim–Skolem theorem holds...
  
  
• Lindström's theorem
  Lindström's theorem
  In mathematical logic, Lindström's theorem states that first-order logic is the strongest logic In mathematical logic, Lindström's theorem (named after Swedish logician Per Lindström) states that first-order logic is the strongest logic In mathematical logic, Lindström's theorem (named after...
  
  
• Universal logic
  Universal logic
  Universal logic is the field of logic that is concerned with giving an account of what features are common to all logical structures. Universal logic aims to be to logic what universal algebra is to algebra; currently there is no universally accepted notion of logic...