We must start with a language. A language can be defined in two ways. First let us begin with the axioms of pairing, union, and powerset and schema of separation. This gives us a cartesian product of sets, and hence functions.
Definition 1.1. An –ary operation on a set is a map . A structure is a set together with an -ary operation. The signature of a structure is the sequence where is the number of -ary operations.
Definition 1.2. Let and be structures with the same signature such that each -ary operation of is assigned to a -ary operation of (i.e. where is the th -ary operation of ). A homomorphism between structures and is a map such that
Note that a nullary operation on is a map . That is, it is simply an element of . Now let be a set which we will call an alphabet, and its elements will be called letters. A monoid has a nullary operation, called a space, and a binary operation, which will simply be denoted by concatenation. We define the free monoid on as the monoid consisting of all strings of elements in . We now have two definitions of a language, of which the first is traditional and the second is mine:
Definition 1.3. A language is a subset of .
Alt Definition 1.3. Let , be a relational structure (a set together with an -ary relation), and be a structure. The language is defined as where is the free -structure on . In particular elements of are called words, elements of are called terms, and elements of are called formulas.
Definition 1.4. A theory of is a subset . Elements of a theory are called axioms. Elements of are called propositions. A theory of is called a reduced theory if for all , for all -ary operations of and all placements of in evaluation of the operation. (That is, the theory is reduced if no axiom is in the orbit of another).
For example, the theory is called the trivial theory. The theory is called the empty (or agnostic) theory.
Definition 1.5. An –ary logic system on a theory is a homomorphism where and have the same signature and has cardinality We may also say the logic system is normal if for all
In traditional logic is a two element boolean algebra. Traditional logic also has a special kind of function on its language.
Definition 1.6. A quantifier on is a function . We may write:
In particular it is a pseudo operation, and gives the language a pseudo structure. This is similar to modules, where in this case a product of a term and a formula are sent to a formula.
Hence our initial assumption of four axioms (as well as the ability to understand the English language), have in turn given us the ability to create a notion of a language of which a degenerate English can be construed as a special case. This is certainly circular in some sense, but in foundations we must appeal to some cyclic process. One subtlety worth noting is that the secondary language created will always be “strictly bounded above” by the initial language; they aren’t truly equivalent. (In fact this last statement is similar to the antecedent of Godel’s Second Incompleteness theorem).