1.1.6 Building Formulae
Definition of the concept of a first-order formula.
Terms.
Let's suppose we've composed a certain vocabulary. How do we mix these ingredients together? That is, what is the syntax of first-order languages? First of all, we define a first-order term 
to be any constant or any variable. Roughly speaking, terms are the noun phrases of first-order languages: constants can be thought of as first-order counterparts of proper names, and variables as first-order counterparts of pronouns.
Atomic Formulae.
We can then combine our `noun phrases' with our `predicates' (meaning, the various relation symbols in the vocabulary) to form what we call an atomic formula (also: basic formula):
If 
is a relation symbol of arity 
, and 
are terms, then 
is an atomic formula.
Intuitively, an atomic formula is the first-order counterpart of a natural language sentence consisting of a single clause (that is, a simple sentence). So what does a formula like 
actually mean? As a rough translation, we could say that the entities that are named by the terms 
stand in a relationship that is named by the symbol . An example will clarify this point:
What's meant by this formula is that the entity named 
stands in the relation denoted by 
to the entity named 
- or more simply, that Peter loves Anna.
Complex Formulae.
Now that we know how to build atomic formulae, we can define more complex descriptions as well. The following inductive definition tells us exactly what kinds of well-formed formula e (or wffs, or simply formulae) we can form.
All atomic formulae are wffs.
If
and 
are wffs then so are 
, 
, 
, and 
.If
is a wff, and x is a variable, then both 
and 
are wffs. (We call the matrix or scope of such wffs.)Nothing else is a wff.
Roughly speaking, formulae built using 
, 
, 
and 
correspond to the natural language expressions ``it is not the case that ...'', ``if ... then ...'', ``... or ...'', and ``... and ...'', respectively. First-order formulae of the form and 
correspond to natural language expressions of the form ``some...'' or ``all...''. We shall soon use first-order models to give a more precise formulation of these intuitive correspondences.
).