Terminology in logic theory

Question

Edit: I should point out that, I tried searching by myself, but without success. Every time I thought I got the terminology correct, I tried to use it, but, I didn't get relevant results..

I'm taking a logic course in my native language, and I'm having trouble finding the same terminology in english. So here are a few questions.

the first expression, can be assigned True or False, is it called an Atom in english too?

$$ p_i $$ what would an expression like this be called? $$ p_i \land (\neg p_j) $$

what is the term for z called? $$ z ⊨\alpha $$ here's a question from my homework:

prove the K "isn't definable". $$ K = Ass \backslash \{z\} $$

what I mean mean by "isn't definable", is that there isn't a set like, $$ \Sigma = \{\alpha,\beta\,...,\theta\} $$ that: $$ M(\Sigma)=K $$

I hope I explained myself good enough. And thanks for anyone that would help :)

Answer 1

1) Yes, $p_i$ is a propositional variable or atom (or atomic formula): in the context of proposiational logic it represent an "undecomposable" sentence.

2) $p_i ∧ (¬p_j)$ is a formula and specifically a conjunction.

3) In an expression like $\Gamma \vDash \alpha$, meaning that formula $\alpha$ is logical consequence of the set $\Gamma$ of formulas, we may call the formulas in $\Gamma$ premises or assumptions.

But we can have also : $v \vDash \alpha$, where $v$ is a valuation (also called truth assignment), i.e. an assignment of truth values to propositional variables.

A valuation can be uniquely extended to an assignment of truth values to all propositional formulas.

Thus, $v \vDash \alpha$ means that valuation $v$ satisfies formula $\alpha$, i.e. that $v$ assigns to $\alpha$ the value TRUE.

I assume that $\text {Ass}$ is the set of all valuations.

Thus, the question about definability of $K = \text {Ass} \setminus \{ z \}$ amounts to :

find a set $\Sigma = \{ p_i \mid \ldots \}$ of propositional variables, such that $K$ is the set of models of $\Sigma$ (call it : $\text {Mod} (\Sigma)$), i.e. the set of valuations that assign to the formulas in $\Sigma$ the value TRUE (i.e. $\text{Mod}(\Sigma) = \{ v \mid v \text { is a valuation and } v \vDash \sigma \text { for every } \sigma \in \Sigma \}$).