# What does individual variables mean ? Can be propositional variables?

We know that any particular first-order language is determined by its symbols. These consist of ;

$$1-) \$$ A denumerable list of symbols called individual variables.

• A denumerable list of symbols $$(\text{not in }1)$$ called individual parameters.
• the connectives ; $$\land ,\lor ,\lnot , \rightarrow$$
• for each natural number $$n$$, a set of $$n$$-ary relation symbols (also called predicate symbols).
• for each natural number $$n$$, a set of $$n$$-ary function symbols.
• the quantifiers ; $$\forall , \exists$$
• parentheses and the comma. $$( \ , \ )$$

My qeustion :

$$1-) \$$ What does individual variables mean ? Can be propositional variables?

$$2-) \$$ In first-order language ; Can be definition symbol?

1) What does individual variable mean ?

It is a term, i.e. a symbol that acts as a name for an object.

Thus, it cannot be a propositional variables, i.e. a symbols that stands for a sentence.

Consider the simple example from first-order language of arithmetic : $$(x=0)$$.

In this formula $$x$$ must be replaced by a number in order to give an arithmetical meaning to the formula.

2) In first-order language, can be definition symbol ?

A definition must either introduce a term, i.e. a symbol acting as a name for an object, or a predicate letter, i.e. a symbol naming a property.

Again, examples from first order arithmetic : we start from the basic symbols of the language : $$0$$ (an individual constant denoting the number $$\text {zero}$$), the unary function $$s(x)$$ (the $$\text {successor}$$ function) and the binary function $$+(x,y)$$ (the $$\text {sum}$$ operation, abbreviated with : $$(x+y)$$).

With them we define the new constant $$1$$ as $$s(0)$$.

And we define the new binary predicate $$<(n,m)$$ (the relation $$\text {less than}$$, abbreviated with $$(n < m)$$) as follows :

$$(n < m) \text { iff } \exists z \ (m=n+s(z))$$.