1.5. Expressions and definable structures
Terms and formulas
Given the first two layers of a theory (a list of types and a language), an
expression is a finite system of occurrences of symbols, where,
intuitively speaking, an occurrence of a symbol in an expression
is a place where that symbol is written (more details later).
Each expression comes in the context of a given list of available free
variables. Any expression will give (define) a value (either an object
or a Boolean) for each possible data of
- A system interpreting the given types and structure symbols;
- Fixed values of available free variables in this system.
The type of an expression, determined by its syntax as described below,
gives the type of all its possible values. Expressions with Boolean type are called
formulas; others, whose type belongs to the given types list
(their values will be objects), are called terms.
For example, « x+x » is a term with two occurrences
of the variable «x», and one of the addition symbol «+».
The diverse kinds of symbols
In expressions of first-order theories and set theory, symbols of the following kinds may occur.
- Variables of each type:
- Free variables, from the list of available ones ;
- Bound variables, whose occurrences are contained by binders (see 1.8) ;
- Para-operator
symbols:
- Structure symbols from the language (operators and predicates) ;
- One equality symbol per type (predicate with 2 arguments of
the same type) abusively all written = ;
- Logical connectives (1.4, listed in 1.6) ;
-
The conditional operator may
be introduced for abbreviation (2.4).
- Binders (1.8):
- Quantifiers
∀ and ∃ (1.10) are the only primitive binders of first-order logic ;
- More binders will be introduced in set theory.
In first-order logic, let us call logical symbols the quantifiers and symbols of
para-operators outside the language (equality, connectives and conditional operator):
their list and their meaning in each system are determined by the logical framework and
the given types list, which is why they are not listed as components of individual theories.
Root and sub-expressions
Each expression contains a special occurrence of a symbol called its root,
while each other occurrence of a symbol there, is the root of a unique sub-expression
(another expression contained in the given one, and which we may call the sub-expression
of that occurrence).
The type of an expression is given
by the type of result of
its root.
Expressions are built successively, in parallel between different lists of available free variables.
The first and simplest ones are made of just one symbol (as root, having a value by
itself) : constants and variables are the first terms; the
Boolean constants 1 and 0 are the simplest formulas.
The next expressions are then successively built as made of the
following data:
- A choice of root, occurrence of either a para-operator symbol (beyond constants
we already mentioned) or a binder;
- If the root is a binder: a choice of variable symbol, to be bound by it;
- A list of previously built expressions, whose format (number and
types of entries) is determined by the root : for a
para-operator symbol, this format is given by its list of its arguments.
An algebraic term is a term with only free variables and operator symbols ;
these are the only terms in first-order logic without conditional operator.
This notion for only one type, will be formalized as a kind of system in set theory in 4.1.
Display conventions
The display of this list of sub-expressions directly attached to the root requires a
choice of convention. For a para-operator symbol other than constants :
- Most binary para-operator symbols are displayed as one
character between (separating) both arguments, such as in x+y
- Symbols with higher arities can be similarly displayed by
several characters separating the entries, such as in the addition
x+y+z of 3 numbers.
- Function-like displays, such as +(x,y) instead of
x+y, are more usual for arities other than 2 ;
parenthesis may be omitted when arities are known (Polish notation).
-
A few symbols «appear» only implicitly by their special way of putting
their arguments together : multiplication in xy, exponentiation in
xn.
-
Parenthesis can be part of the notation of a
symbol (function evaluator, tuples...).
Parenthesis can also be used to distinguish (separate) the subexpressions, thus
distinguish the root of each expression from other occurring symbols. For example
the root of (x+y)n is the exponentiation operator.
Variable structures
Usually, only few objects are named by the constants in a given language.
Any other objects can be named by a fixed variable, whose status
depends on the choice of theory to see it:
- An ordinary variable symbol, usable by expressions which by a binder
can let it range over some notion;
- A new constant symbol, to be added to the language, forming another
theory with a richer language.
The difference vanishes in generic interpretations which turn
constant symbols into variables (whose values define different models).
By similarity to constants which are particular structures (nullary operators),
the concept of variable can be generalized to that of variable structure.
But those beyond nullary operations (ordinary variables) escape the above list
of allowed symbols in expressions. Still some specific kinds of use of variable
structure symbols can be justified as abbreviations (indirect descriptions)
of the use of legitimate expressions. The main case of this is explained below ;
another use (looking similar but actually a meta-variable)
will be involved in 1.10.
Structures defined by expressions
Starting with any theory, one can introduce a kind of symbol of variable
structure (operator or predicate, though a nullary predicate is normally called a
"Boolean" rather than a "structure"), defined by the following data
it means to abbreviate:
- An expression (terms define operators, while formulas define predicates
and Booleans);
- Among its available free variables, a selection of those which will be bound
by this definition in the role of arguments of the intended structure;
the rest of them, which remain free, are called parameters.
Each of its possible values as a structure or a Boolean comes by fixing the values of all
parameters. So, its variation somehow abbreviates those of all parameters.
Any theory can be extended by the construction of a new notion (abstract type)
given as the range of a variable structure defined by a given (fixed)
expression, while its parameters range over all possible combinations of values.
This is our first case of a construction rule (kind of development of a theory).
The full review of construction rules will be done in 4.11.
In set theory, any function f is synonymous with the functor defined by the term
f(x) with argument x and parameter f (but the domain of this
functor is Dom f instead of a type).
The terms without argument define simple objects (nullary operators) ; the one made
of a variable of a given type, seen as parameter, suffices to give all objects of its type.
Now let us declare the meta-notion of "structure" in one-model theory,
and thus those of "operator" and "predicate", as having to include at least all those
reachable in this way: defined by any expression with any possible values of
parameters. The minimal version of such a meta-notion can be formalized as a
role given to the range of all combinations of an expression with fixed values of its
parameters. As this involves the infinite set of all expressions, this meta-notion usually
escapes (is inaccessible by) the described theory itself : no fixed expression can suffice
to simulate it. Still when interpreting this in set theory, more operations between
interpreted types (undefinable ones) usually exist in the universe. Among the few
exceptions, the full set theoretical range of a variable structure with all
arguments ranging over finite sets (as interpreted types) with given size
limits, can be reached by one expression whose size depends on these limits.
Invariant structures
An invariant structure of a given system (interpreted language), is a
structure defined from its language without parameters (thus a constant one).
This distinction of invariant structures from other structures, generalizes
the distinction between constants and variables, both to cases of
nonzero arity, and to what can be defined by expressions instead of
directly named in the language. Indeed any structure named by a symbol
in the language is directly defined by it without parameter, and thus invariant.
As will be further discussed in 4.10, theories can be developed by definitions,
which consists in naming another invariant structure by a new symbol added to
the language. Among aspects of the preserved meaning of the theory,
are the meta-notions of structure, invariant structure, and the range of theorems
expressible with the previous language.
Other languages: