what is the difference between formula and the abbrevation of a formula? there is a problem which is asking me to determine whether a string is a formula or an abbrevation of a formula 
but i don't know the diffrence of formula and the abbrevation of a formula 
i know the definition of a formula , but what about the abbrevation of it ? 
what does it mean ? 
how can i get it from a fromula ? 
so my question is : 
what is the diffrence between formula and the abbrevation of a formula ? 
can you give examples to make the diffrence clear ? 
 A: It will depend entirely on the local conventions of the text you are using. To give a very simple example from propositional logic: 


*

*Some texts take the basic connectives to be $\neg$ and $\to$, and then introduce $\lor$ with the rule that $(A \lor B)$ is an abbreviation of $(\neg A \to B)$. 

*But another text can take  $\neg$ and $\lor$ as basic, and then introduce $(A \to B)$ as an abbreviation of $(\neg A \lor B)$. 

*Other texts again might have all of $\neg$, $\land$, $\lor$, $\to$ as basic connectives introduced on a par.
As I say, you'll just have to see by inspecting the details of your text whether it counts $(A \lor B)$, for example, as a formula in the official strict sense or merely as a useful 'slang' abbreviation for one.
A: Rif to :


*

*Angelo Margaris, First Order Mathematical Logic (1967 - Dover ed.), Ex.4, page 36.


The terms of $N$ are [see page 33] :

the variables : $x,y,\ldots$ , the (individual) constants : $0,1,2,\ldots$ and all the expressions formed by previous defined terms by means of the operation symbols $+$ ("sum") and $\cdot$ ("multiplication").

The formulae of $N$ are [compare with page 31] :

the atomic ones : $(u = v)$ and $(u < v)$, for $u,v$ terms, and all those like $\sim P, (P \to Q), \forall vP$, for $P,Q$ previous defined formulae and $v$ a variable. 

Thus, the two examples :

$\forall xy(x+y=y+x)$

amd :
$\exists 0 \forall x(x+0=x)$
are not formulae : the first one is an abbreviation, while the second one is not well-formed (i.e. it is "un-grammatical").
For the first one, we have that $x+y$ and $y+x$ are terms of $N$; thus : $(x+y=y+x)$ is an atomic formula. But the "formal" expression of its universal closure must be :

$\forall x \forall y(x+y=y+x)$.

For the second one, we have that $x+0$ and $x$ are terms, and thus $(x+0=x)$ is an atomic formula.
Also $\forall x(x+0=x)$ is a formula, but $\exists 0 \forall x(x+0=x)$ is not (nor an abbreviation) because the quantifiers use only variables.
