I’m reading through Mathematical Logic by Ian Chiswell and Wilfred Hodges. They talk about, and construct the rules for, $LP$ - which they call ‘the language of propositions’. The construction is done through the use of parsing trees.
In constructing $LP$, they say we need to choose a signature $\sigma$, which is a set containing symbols that can be used to stand for statements. I’m confused when they state:
“For each choice of signature $\sigma$ there is a dialect $LP(\sigma)$ of $LP$.”
I don’t understand what they mean by ‘dialect’. I’ve tried looking it up, but haven’t found anything. What is the difference between the language $LP$ and the dialect $LP(\sigma)$? If choosing a signature $\sigma$ gives us $LP(\sigma)$, then it seems we don’t need it to construct $LP$ - just the dialect $LP(\sigma)$.
In a later definition they state:
“The formulas of $LP(\sigma)$ are the formulas associated to parsing trees of $LP(\sigma)$. A formula of $LP$ is a formula of $LP(\sigma)$ for some signature $\sigma$.
They define what it means for something to be a formula for $LP(\sigma)$, but not what it means to be a formula of $LP$. The book defines parsing trees for $LP(\sigma)$, but it doesn’t define what a parsing tree for $LP$ would be. Likewise, the book defines the “lexicon” of $LP(\sigma)$ and what an “expression” of $LP(\sigma)$ is, but neither the lexicon of $LP$, nor what an expression of $LP$ would be, are defined.
If I made a guess, I’d say the language $LP$ is the same as the dialect $LP(\sigma)$, but where all possible symbols are allowed to be used as propositional symbols, while $LP(\sigma)$ restricts the possible propositional symbols to those in the set $\sigma$. Is this correct, and if so is this the only difference between them?