I am new to Mathematical logic and I am unable to understand the relationship between Predicates and Domain of discourse in First-order logic. Particularly, the following questions are coming to my mind:
Can we define any number of distinct predicates over a single domain of discourse? For example, say domain of discourse is "Set of all Human Beings", then can we make a First-order logic statement based on the following English sentence:
$Some\:people\:are\:Engineers,\:some\:people\:are\:Doctors\:and\:some\:people\: are\:Musicians.$
The above statement seems to have three predicates based on the same domain of discourse. Is that correct?
- Also, can a First-Order Logic Statement have more than one domain of discourse? For Example, The Quantified Version of The statement: "Every King has at least one Minister", seems to have two domain of discourses: Set of all Kings and Set of all Ministers. Is that correct?
The domain $D$ of an interpretation $\mathscr I$ is a collection of objects (e.g. numbers) over which variables range.
A predicate symbol $P$ of the language is interpreted with a subset of the domain, if $P$ is a unary predicate symbol, i.e. $P^{\mathscr I} \subseteq D$, with a binary relation on the domain $D$, if $P$ is a binary predicate symbol, i.e. $P^{\mathscr I} \subseteq D\times D$, and so on.
Having said that, if $H$ is the domain of Human beings, then the three (unary) predicates $E, D$ and $M$ will be interpreted with subsets of $H$, i.e. with the subset of Humans that are Engineers, Doctors and Musicians respectively.
Regarding the second question, we may have domains partitioned into disjoint subdomains (see many-sorted predicate logic), but your example does not need that; it is enough to use two predicates $K$ amd $M$ denoting Kings and Ministers respectively, in the domain of Humans.