Finite Model Theory/Logics and Structures

FMT studies logics on finite structures. An outline of the most important of these objects of study is given here.


The logics defined here and used throughout the book are always relational, i.e. without function symbols, and finite, i.e. have a finite universe, without further notice.

Fragments of FOEdit

The subsequent restrictions can analogously be found in other logics like SO.

  • MFO ...
  • ESO and USO ...
  • FOn ...

Second Order Logic (SO)Edit

Second-order logic extends first-order logic by adding variables and quantifiers that range over sets of individuals. For example, the second-order sentence   says that for every set S of individuals and every individual x, either x is in S or it is not. That is, the rules are extended by:

  • If X is a n-ary relation variable and t1 ... tn are terms then X t1 ... tn is a formula
  • If φ is a formula and X a relation variable then   is a formula

Fragments of Second Order LogicEdit

  • The fragment monadic second-order logic (MSO) only unary relation variables ("set variables")are allowed.
  • The existential fragment (ESO) is second-order logic without universal second-order quantifiers, and without negative occurrences of existential second-order quantifiers.
  • USO ...

Infinitary LogicsEdit


The intention is to extend FO by an infinite disjunction element over a set ψ of formulas (of infinitary logic)


So the following infinitary logics can be defined

  •   where ψ is an arbitrary set of formulas, e.g. uncountable
  •   where ψ is a countable set of formulas
  •   ...


syntax ...

semantics ...


General Quantifier LogicsEdit

Fixed Point LogicsEdit

Counting LogicsEdit


Finite GraphsEdit

Kinds of Finite GraphsEdit