what is the diffrence between a term , constant and variable in first order logic languages ?

Question

in the text , the author says that the language contains parenthises , sentintial connectives and n-place functions , n-place predicates , equality sign = , terms , constans and variables

i have two question ,

1- what is the definition of a term , constant and variable ?! and what is the diffrence between them ?

2- what is the definition of n-place predicate ?

thanx !

Answer 1

A constant is a symbol. A variable is also a symbol. The difference comes from how they can be used (syntactically). For example, a symbol $v$ can occur with a quantor (as in $\forall v$) only if $v$ is a vraiable, not if it is a constant. Terms are words (symbol sequences) in the language, again obeying certain rules; for example constants and variables are the simplest terms and new terms are obtained from simpler terms by connectives. As a matter of fact, the details of these distinctions should be apparent with what I assume the author writes in the nest few paragraphs after these remarks.

Answer 2

A variable is a symbol like $x$ standing for a variable, so these are the basic bricks. Terms are built up from variables and operation symbols, recursively:

• Each variable itself is also a term,
• and whenever $\tau_1,..,\tau_n$ are terms, and $f$ is an $n$-ary operation symbol ($n$ place function, as you called), then the string $f(\tau_1,..,\tau_n)$ is again a term.

A constant is basically a nullary operation, i.e. it inputs $0$ pieces of elements and outputs one element, so in practice, a constant will be a particular element in any structure ('incarnation') of the given language. Constants themselves are also terms.

An $n$-ary predicate symbol $R$ is then realized in an $n$-ary relation in the structrues. If $a_1,..,a_n$ are concrete elements in a concrete structure, then $R(a_1,..,a_n)$ is either true or false.

For example consider the language with one binary operation symbol $+$ and one binary relation symbol $\le$ and a constant $0$. A structure of this is e.g. $\Bbb N$ (or $\Bbb Z$ or $\Bbb R$) with the usual interpretations of these symbols. A quantifier-free formula is for example $$(x+y)+x\le 0$$