Assignment (mathematical logic)

The article is about assignment in mathematical logic; for other uses, see Assignment

Assignment can be regarded as an auxiliary notion, an important step in a specific way for defining the concept of truth formally (e.g. for first-order theories). It enables us to give meanings to terms (truth to sentences) of a language which deals with (free) variables. Technically, in most cases it is a function

$a:{\mathit {Var}}\to {\mathit {A}}$

from a set of variables to the possible values they can take on. (Notion of types can make things more complicated).

Formal definition

Let

• ${\mathfrak {A}}=\left\langle A,I\right\rangle$ denote a structure, with universe A and interpretation I (see the latter concept explained later)
• ${\mathcal {L}}$ a language (here for simplicity's sake, a first-order language) with
  • variable names Var,
  • and also with names structured in a way that each of them should be known
    • whether it will be used to denote an operation or a relation
    • and of what arity it is.

• I gives a name to each operation and relation of ${\mathfrak {A}}$, using predicate symbols of a language ${\mathcal {L}}$ according to the arity and whether it is a relation or an operation

Now, we can define

• the meaning of a term of ${\mathcal {L}}$
• the truth of a sentence of ${\mathcal {L}}$

Each term can be thought of as being built in the shape of pattern $f(t_{0},\dots ,t_{n})$ where f is the name for an n-ary operation, and $t_{0}$,… $t_{n}$ are subterms .

A structural definition (and using I in a straightforward way) enables us to define a meaning for each terms of ${\mathcal {L}}$: we can map all terms to an appropriate element of universe A (of ${\mathfrak {A}}$)

In a similar way, we can define truth for each sentence of ${\mathcal {L}}$, but there are also important differences, e.g. handling logical connectives, quantifiers, and equality in an appropriate way.

In addition, the existence of variables (and quantifiers) force us to deal with (sub)terms and sentences with free variables. Thus, we build the notion for truth and meaning in a more complicated way:

• first, we introduce these notions parametrized by an assignment
• then, we define their corresponding “unparametrized” counterparts as being invariant for assignment i.e. holding for each possible assignment.

See also

• Semantic theory of truth
• First-order logic
• Model theory