A partial type (partial n-type) is a consistent set of first-order formulas with parameters in a chosen tuple of variables that describes some properties a prospective tuple may satisfy but is not required to decide every formula in those variables; it admits at least one model-theoretic realization or extension and can be extended to a complete type.

Principle

Non-maximal consistency: a partial type is any consistent, but not necessarily maximal, collection of formulas; its role is to record partial information or constraints that can be extended to full descriptions when needed.

Demonstration

In the theory of dense linear orders without endpoints, the set of formulas { x > a_n for each element a_n of an increasing sequence } is a partial 1-type over the parameters {a_n} describing an upper cut; it does not decide, for every formula, whether some candidates lie on one side or the other and can be extended to different complete types depending on how the cut is filled.

Misapplication

Assuming a partial type uniquely determines a complete type or that any partial type must be realized in every extension; conflating partial types with quantifier-free or finite descriptions that may be inconsistent in some contexts.

Consequence

Partial types serve as building blocks for constructing models, witnessing consistency of collections of properties, and studying definability and forking; they are the syntactic objects whose maximal extensions are complete types and whose realizability informs saturation properties.

Reversal

A complete type is the maximal consistent extension of a partial type; reversing partiality yields a single decisive description that leaves no relevant formula undecided.

Boundary

Applies only to first-order formulas in specified variables and parameters; partial types are silent about undecided formulas and do not by themselves guarantee realizability unless consistency is established and compactness or saturation applied.

Semantic Tension

Tension between flexibility (many extensions possible) and precision (lack of decision about certain formulas); partial types are useful where open possibilities are needed but problematic when uniqueness or isolation is required.

Synthesis

A partial type is a consistent, intentionally incomplete syntactic specification of properties for a prospective tuple: it records constraints and leaves room for extension, and its study connects local constraints to global realizations via extension to complete types.