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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)
${\mathcal {L}}$ ${\mathcal {L}}$ 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 functor 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 ^[1] $f(t_{0},\dots ,t_{n})$ where f is the name for an n-ary operation, and $t_{0}$ ,… $t_{n}$ are subterms ^[2]. See details in formation rules.

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.

Notes

^ A didactic circumlocution for Alfred Tarski's notions: structural descriptive name, meta vs object language, see also metavariable etc.
^ The case of 0-ary operations — constants — can be handled in a straightforward way.

[strdsc-1] A didactic circumlocution for Alfred Tarski's notions: structural descriptive name, meta vs object language, see also metavariable etc.

[cnst-2] The case of 0-ary operations — constants — can be handled in a straightforward way.

[1]

[2]

@@ Line 1: / Line 1: @@
 ''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
+'''''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 logic|first-order]] theories). It enables us to give meanings to terms (truth to [[Sentence (mathematical logic)|sentence]]s) of a language which deals with (free) ''variables''. Technically, in most cases it is a function
 :<math>a : \mathit{Var} \to \mathit A</math>
 from a set of variables to the possible values they can take on.
@@ Line 16: / Line 16: @@
 *** and of what arity it is.
-* ''I'' gives a name to each operation and relation of <math>\mathfrak A</math>, using predicate symbols of a language <math>\mathcal L</math> according to the arity and whether it is a relation or an operation
+* ''I'' gives a name to each operation and relation of <math>\mathfrak A</math>, using functor symbols of a language <math>\mathcal L</math> according to the arity and whether it is a relation or an operation.
 Now, we can define
 * the meaning of a term of <math>\mathcal L</math>
-* the truth of a sentence of <math>\mathcal L</math>
+* the truth of a [[Sentence (mathematical logic)|sentence]] of <math>\mathcal L</math>
-Each term can be thought of as being built in the shape of pattern <ref name=strdsc>A didactic circumlocution for [[Alfred Tarski]]'s notions: structural descriptive name, meta vs object language, see also metavariable etc.</ref> <math>f(t_0,\dots ,t_n)</math> where ''f'' is the name for an ''n''-ary operation, and <math>t_0</math>,… <math>t_n</math> are subterms <ref name=cnst>The case of 0-ary operations — constants — can be handled in a straightforward way.</ref>.
+Each term can be thought of as being built in the shape of pattern <ref name=strdsc>A didactic circumlocution for [[Alfred Tarski]]'s notions: structural descriptive name, meta vs object language, see also metavariable etc.</ref> <math>f(t_0,\dots ,t_n)</math> where ''f'' is the name for an ''n''-ary operation, and <math>t_0</math>,… <math>t_n</math> are subterms <ref name=cnst>The case of 0-ary operations — constants — can be handled in a straightforward way.</ref>. See details in [[First-order logic#Formation rules|formation rules]].
 A structural definition (and using ''I'' in a straightforward way) enables us to define a meaning for each terms of <math>\mathcal L</math>: we can map all terms to an appropriate element of universe ''A'' (of <math>\mathfrak A</math>)

Revision as of 01:02, 5 November 2006

Formal definition

See also

Notes