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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?

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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 :

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

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