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We will define $L_{pc} = \{P_{A} : A \ is \ an \ elementary \ formula \ in \ L \}$

An elementary formula is an atomic formula or a formula of the type $\exists xB$ for some formula B.

This means, for every elementary formula A in the language L, we define an atomic sentence $P_{A}$. Now we will define a function $\phi$ from formulas to sentences in the next manner:

if A is elementary, then $\phi (A) = P_{A}$

if $ A = \neg B$ then $\phi (A) = \neg \phi (B) $

if $ A = B \lor C$ then $\phi (A) = \phi (B) \lor \phi (C) $

if $ A = (\exists xB \lor x \approx x) \lor \neg D $ then $\phi (A) = (P_{\exists xB} \lor P_{x \approx x} \lor \neg \phi (D)) $

Now this is the definition of the language $ L_{pc}$.

Now I need to translate the following formula, in the language $L = (+, \lt, 0)$

$\forall x (x+y \lt 0 \lor x \lt y) $

to the language $L_{pc}$.

To be honest I haven't fully understood how is this suppose to be translated, it might be too intuitive and that might be the reason I can't really put my finger on it.

Any help that could get me going and get me to the point where I understand how to translate formulas like that, would be highly appreciated!

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    Hm.. What do you mean? – 2017-01-19
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    Probably you have to review the definitions of *atomic sentence* and *elementary formula*... – 2017-01-19
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    Ok, and then? I'm sorry but if you could just be more specific .. – 2017-01-19
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    Or perhaps just tell me what's wrong with my question – 2017-01-19
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    What is an elementary formula?? Is it anything like an atomic formula? – 2017-01-19
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    Oh, I got what you mean. Just know that till now, when you stated I don't accept answers, I didn't know that it is a possible thing. And I went on to accept many answers of previous asked questions. In addition, I do vote to helpful answers, I also write a comment to show my thanks. To you as well on the latest post.. so I'm having a difficulty understanding why you just stated that. – 2017-01-19
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    @bof I've edited the post with the meaning of an elementary formula – 2017-01-19
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    What about $\forall$ and $\land$ ? Are they defined as abbreviations ? Please, specify... – 2017-01-19
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    Case 1: $\phi(∀x(x+y<0∨x$\phi$ is not defined for the $\forall$ case. – 2017-01-19
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    Case 2: $\forall$ is an abbreviation of $\lnot \exists \lnot$; in this case : $ϕ(∀x(x+y<0∨x – 2017-01-19
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    Case 2 is the correct case. Thank you very much! – 2017-01-19

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