User:Gregbard/Sandbox/formal interpretations

If the class α consists of all of the logical and non-logical signs of a formal language $\mathcal{L}$ and the class ℑ consists of all sentences of $\mathcal{L}$ then a formal language $\mathcal{L}$ can be defined as the ordered pair <α, ℑ>

The class υ of the expressions of $\mathcal{L}$ is defined as the class of all finite sequences whose members are the elements of the class α.
An n-place sequence can be defined as a many-one relation between the n first natural numbers and the members of the sequence. A syntactic axiom may be adopted that states: For any class α and any class ℑ, if <α, ℑ> is a formal language then every element of ℑ is a finite sequence of elements of α, and every element of α occurs as a member of some element of ℑ.

A formal system can be defined as an ordered triple <α,ℑ,$\mathcal{D}$d>, where $\mathcal{D}$d is the relation of direct derivability. This relation is understood in a comprehensive sense such that the primitive sentences of the formal system are taken as directly derivable from the empty set of sentences. Direct derivability is a relation between a sentence and a finite, possibly empty set of sentences. Axioms are laid down in such a way that every first place member of $\mathcal{D}$d is a member of ℑ and every second place member is a finite subset of ℑ.

It is also possible to define a formal system using only the relation $\mathcal{D}$d. In this way we can omit ℑ, and α in the definitions of interpreted formal language, and interpreted formal system. However, this method can be more difficult to understand and work with.

An interpreted formal language can defined as the ordered triple <α,ℑ,$\mathcal{D}$>. The first domain of the relation $\mathcal{D}$ is identical with the class ℑ.

If an extensional metalanguage is used for semantics, then $\mathcal{D}$ is the relation of value assignment for the sentences of the language.
For example, "$\mathcal{D}$(ℑ1,the pope is catholic)" means the same as "The sentence ℑ1 is true if and only if the pope is catholic." For any p and q and any element ℑ1 of the class ℑ, if $\mathcal{D}$(ℑ1,p) and $\mathcal{D}$(ℑ1,q) then p if and only if q.
If on the other hand, an intensional metalanguage, containing a modal operator, such as "it is necessary that", then D is taken as the relation of designation, That is, the relation between an expression and its intension.
For example, "$\mathcal{D}$(ℑ1,the pope is catholic)" means the same as "The sentence ℑ1 designates that the proposition that the pope is catholic." For any p and q and any element ℑ1 of the class ℑ, if $\mathcal{D}$(ℑ1,p) and $\mathcal{D}$(ℑ1,q) then p and q are identical, i.e it is logically necessary that p if and only if q.
In either of these two metalanguages extensional, or intensional, truth with respect to any given interpreted language (α, ℑ,$\mathcal{D}$) can be defined as follows: A sentence ℑ1 is true if and only if for some p, $\mathcal{D}$(ℑ1,p), and p.
There is another method applicable to either of these two metalanguages which takes the relation $\mathcal{D}$ as applying not only to sentences but to a more comprehensive class d of designators. By this method, an interpreted formal language is an ordered quadruple (α,ℑ,d,$\mathcal{D}$).
In these metalanguages, d is the class of finite sequences of elements of the class α, the class of the first place members of $\mathcal{D}$ is the class d, and that ℑ is a subclass of d.
There is also a third method, which is more explicit, which demands that in order to specify an interpreted formal language a class ds of descriptive signs of the language must be indicated. In this method, an interpreted formal language can be defined as the ordered quintuple <α,ds,ℑ,d,$\mathcal{D}$>
Using this method, ds is a subclass of a. This most explicit method is convenient as a basis for definitions of concepts such as "model", "value assignment", "range of a sentence", "logical truth", and other logical concepts.

An interpreted formal system is a formal language for which both syntactical rules for deduction, and semantical rules of interpretation are given. An interpreted formal system can be defined as the ordered quadruple <α,ℑ,$\mathcal{D}$d,$\mathcal{D}$>. Here axioms are stated, some similar to those stated for a formal system, and some like those for an interpreted formal language. Usually, we wish for $\mathcal{D}$d to be truth-preserving (that is, any sentence which is directly derivable from true sentences is itself true), however other modalities can also preserved in such a system. We can formulate an axiom for these purposes with use of the term "true". For any ℑi1,...,ℑin, ℑj, p1,...,pn,q if $\mathcal{D}$d(ℑj,{ℑi1,...,ℑin}), $\mathcal{D}$(ℑi1,p1) and ... and $\mathcal{D}$(ℑin,pn) and p1 and ... and pn, q.

For interpreted formal systems there are also alternative, more explicit definitions which include ds, or both ds and D, analogous to those given for interpreted formal languages.