Domain (FOL)
Definition
A domain in the context of first order languages is a 3-tuple, [ilmath](M,\mathcal{F},\mathcal{R})[/ilmath][1], denoted by [ilmath]\mathbb{M} [/ilmath], where:
- [ilmath]M[/ilmath] is a non-empty set Caution:I am studying this for set theory, so something is needed here
- [ilmath]\mathcal{F} [/ilmath] is a non-empty set of functions, [ilmath]f:\underbrace{M\times\cdots\times M}_{n\text{ times} }\rightarrow M[/ilmath] where the arity, [ilmath]n[/ilmath] is specific to each function in [ilmath]\mathcal{F} [/ilmath]
- [ilmath]\mathcal{R} [/ilmath] is a non-empty set of propositions, each represented by a relation, [ilmath]R\subseteq M\times M[/ilmath]
That is to say: [ilmath]\mathbb{M}:=(M,\mathcal{F},\mathcal{R})[/ilmath]
Conventions
- It is common to identify a domain, [ilmath]\mathbb{M} [/ilmath] with the first part of the tuple, [ilmath]M[/ilmath].
Purpose
An interpretation is a map, [ilmath]I:\mathscr{L}\rightarrow\mathbb{M} [/ilmath] (or [ilmath]I:\mathscr{L}\rightarrow M[/ilmath] with the convention above) where [ilmath]\mathscr{L} [/ilmath] is a first order language and [ilmath]\mathbb{M}:=(M,\mathcal{F},\mathcal{R})[/ilmath] is a domain as defined above. This map does the following[1]:
- For each constant symbol, [ilmath]c[/ilmath] in [ilmath]\mathscr{L} [/ilmath], [ilmath]I(c)[/ilmath] is an element of [ilmath]M[/ilmath]
- For each [ilmath]n[/ilmath]-ary function symbol, [ilmath]f[/ilmath] in [ilmath]\mathscr{L} [/ilmath], [ilmath]I(c)[/ilmath] is an element of [ilmath]\mathcal{F} [/ilmath]
- For each [ilmath]n[/ilmath]-ary predicate symbol, [ilmath]P[/ilmath] in [ilmath]\mathscr{L} [/ilmath], [ilmath]I(P)[/ilmath] is an element of [ilmath]\mathcal{R} [/ilmath]
Be sure to see the structure page (as, for example, we rarely write [ilmath]I(c)[/ilmath] and such)
Caveats
- Warning:The following is my own work and should not be referenced, it indicates a problem I have seen and my thoughts (at the time of writing) to solving it
Given a domain, [ilmath]\mathbb{M}:=(M,\mathcal{F},\mathcal{R})[/ilmath], we are supposed to have all elements of this tuple being non-empty sets. However I am studying this to use it in set theory! So it seems absurd that we should be speaking of sets before we have them.
As I see it there are 2 solutions:
- Define some sort of pre-cursor "finite" set theory and "bootstrap" formal logic with it, to then develop "real" set theory
- Use (proper) classes.
It is not outlandish to consider [ilmath]M[/ilmath] in the tuple above as set that comes after we have defined the FOL of set theory. Although that seems very close to being circular I'll go no further on this.
As for the other two, [ilmath]\mathcal{F} [/ilmath] and [ilmath]\mathcal{R} [/ilmath] these (at least to my current knowledge) do not have to be sets we could instead use classes.
It is my understanding that with classes you do not need to "construct" them to show they exist, simply exhibit things that would be a member of them, as such all that is required is to use the interpretation to take a function or predicate symbol to an actual function or relation. We don't need ALL the functions.
Having said that, also note that these sets, [ilmath]\mathcal{F} [/ilmath] and [ilmath]\mathcal{R} [/ilmath] could also be finite, as in (most?) first order languages there are only finitely many predicate or function symbols.
It just means I have to be very careful, when writing function do I mean the set definition (using a relation) constructed after the domain, a function symbol, or a function in the domain?
