Implication and entailment I have several questions about entailment and implication. 
1) Let $A = \{p_1, ..., p_n \}$ be some set of logical statements and $q$ some logical statement. Is it right that "$A \models q$" is a correct logical statement, i.e. it may be true, or false and we may always define its logical meaning?
2) What if $A = \varnothing$?
3) Is it right that $(A \models p) \equiv ((p_1 \land ... \land p_n) \rightarrow q)$ ?
4) How should i treat the following situation: $\{ q, q->p \} \models p \rightarrow r$. Is it a correct entailment? I'm confused because there is $r$ on the right and there is no $r$ on the left.
Thanks in advance.
EDIT
Sorry for my English. I suppose "logical proposition" is a more correct term then "logical statement".
 A: *

*If $q$ is a "logical statement" (assuming you mean is logically valid), then we have $\vDash q$, and so a fortiori $A \vDash q$. (If $q$ is true on all interpretations, it is true in particular on all intepretations which make the members of $A$ true.)

*If $q$ is a "logical statement", then we have $\emptyset \vDash q$. (For $\emptyset \vDash q$ says that $q$ is true on those interpretations which make any $p$ true if $p$ is a member of $\emptyset$ -- which are, vacuously, all interpretations).

*Not quite. You mean $(A \models q)$ iff $\models ((p_1 \land ... \land p_n) \rightarrow q)$ or $(A \models q) \Leftrightarrow\ \models ((p_1 \land ... \land p_n) \rightarrow q)$ [You want metalinguistic claims on both sides of the equivalence.]

*Finally $\{ q, q->p \} \models p \rightarrow r$ is plainly not correct. Suppose $p$ and $q$ are true, and $r$ false.
A: I'm assuming (contrary to Peter Smith), that by "logical statement" you mean "well-formed formula" or possibly even "sentence".
1) It is true that if $p_1$ through $p_n$ and $q$ are formulas, then we can say $p_1,\ldots,p_n\vDash q$, and that $p_1,\ldots,p_n\vDash q$ will be either true or false. But $p_1,\ldots,p_n\vDash q$ is not itself a formula. It is a claim about formulas at the metalevel.
A metalevel claim such as $\vDash q$ has a definite truth value in itself, (though it can be difficult to determine what it is). On the other hand a formula only acquires a truth value when we specify an interpretation for the non-logical symbols in it.
2) All of the above holds for empty sets of assumptions too.
3) No, because $A\vDash p$ is a property at the metalevel and $(p_1\land \cdots \land p_n)\to q$ is a formula itself. They are not even the same kind of things, so they can't be equivalent.
4) $p, p\to q\vDash p\to r$ is a meaningful claim, which just happens to be false. I suspect you're using "correct" to mean "meaningful" rather than "true", in which case you would probably call it "correct".
