As my title asks, is logical implication only examining the syntatical compinent of the formal language?
I am using Enderton's book on Mathematical Logic in my class and after some work I am finally getting the distinction betweem the syntax and semantics portions of the formalization.
So reading the definition of logical implication:
"Let $\Gamma$ be a set of wffs, $\phi$ a wff. Then $\Gamma$ logically implies $\phi$ iff for every structure $\mathfrak A$ for the language and every function $s: V \to |\mathfrak A|$ such that $\mathfrak A$ satisfies every member of $\Gamma$ with $s$, $\mathfrak A$ also satisfies $\phi$ with $s$"
where $\mathfrak A$ is designated for the "structure" and $|\mathfrak A|$ is for the universe.
So is what he is saying is that we have a set of "wffs", at the moment these "wffs" do not have any meaning attached to them so they are currently just strings of symbols. now these strings of symbols when given a structure all produce the same result i.e. being evaluated as valid (true) . This is the set $\Gamma$. Now there is a wff $\phi$ again just a string of symbols initially, now if this $\phi$ is given the same structures that were applied to the set $\Gamma$ and stisfies all of the evaluations with the exact same results as the formulas in $\Gamma$ this means that $\Gamma$ logically implies $\phi$ ?
Now the only way this would be true to me would be that this is occurring at a syntax level because if we were to try to consider the meanings prior to designating a structure there would be as many ways as there are opinions.
Is my interpretation of what is trying to be explained correct? Thanks