# First order language and symbols

1. What is language?

2. What is metalanguage?

3.What are symbols?

Am I right in saying following:

1. Any first order language consists of logical and non logical symbols.
2. Where logical symbols consists of (i) sentential connective symbols(negation, disjunction and conjuction) (ii)auxiliary symbols(brackets) and (iii) sequence of variables(associated with language). And definition for sentential and auxiliary symbols is given by truth table.
3. Non-logical symbols consists of (i) constant symbols (ii) predicate or relation symbols (iii) Function symbols
4. Are Quantifier symbols logical or non logical symbols?
5. Variables are symbols which hold place for constants. (What we mean by holding place) and constant are symbols which don't change withing a language.
• Truth table definition apply only to sentential connectives. Quantifiers are logical symbols. Commented Sep 21, 2014 at 11:45
• @MauroALLEGRANZA I was studying book on logic by Enderton. It says Quantifier as 'Parameters'. What are parameters? Commented Sep 21, 2014 at 11:49
• In some textbook quantifiers are considered logical symbols. Enderton does not do so: he consider them under "parameters" because he says taht in some sense they are "interpreted" (see page 80). He call parameters the symbols that must be interpreted. Commented Sep 21, 2014 at 11:56
• See Enderton, page 73: the formula $∀v_1(Av_1 ∧ Bv_1)$ is a formula of the language of first-order logic. The sentence : " For example, [the above formula] translates 'Everything is an apple and is bad'" is a sentence in the metalanguage i.e. the language used by Enderton's book to define and studi first-order logic. Commented Sep 21, 2014 at 12:12
• A variable $v_1$ belongs to the language of first-order logic. Having chosen an interpretation, the variables "range over" objects in the domain of the int. If we consider the formula $\forall v_1(v_1=0)$ and interpret it in the domain of the natural numbers, $v_1$ refers to numbers. A metavariable like $\alpha$ stands for a formula of first-order logic. Thus, it is used in the metalanguage "speaking of" f-o formulae, but does not belongs to the variables of the f-o language, like $v_1$. Commented Sep 21, 2014 at 12:25

Variables and constants have meaning with an interpretation.

A constant is a name. In an interpretation we assign to every constant a reference, i.e. an object in the domain $D$ of the interpretation for which the constant symbol stands.

A variable is a sort of "temporary name".

To interpret in e.g. the domain od natural numbers the formula $x=0$ ($0$ is a constant of the language for arithmetic) we have to choose a reference for the variable $x$.

We can do it in several ways.

One way is trough an assignment function :

$s : Var \to D$

where $Var$ is the set of the variables of the language.

Consider for example the assignment $s$ such that $s(x)=0$; then :

$(x=0)[s]$ is true, because $0=0$.

Consider instead another assignment $s'$ such that $s(x)=1$; then :

$(x=0)[s']$ is false, because $1 \ne 0$.