Is logical implication examining only the syntatical component?

Question

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

Answer 1

No, that is not exactly what is said, but close.

We're looking only for structures (and valuations) that happen to satisfy everything in $\Gamma$, and then we ask: Does each of those structures (and valuations) also satisfy $\phi$? If they all do, then we say that $\Gamma$ implies $\phi$.

On the other hand, it does not matter at all what happens to structures that don't satisfy everything $\Gamma$. These ones can either satisfy $\phi$ or not; that has no bearing on whether $\Gamma$ implies $\phi$ or not.

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.

I don't really understand what you're trying to say here.

Logical implication means that no matter which meaning you assign, it will either end up that $\Gamma$ is not satisfied (and then we don't care at all), or $\phi$ will also be satisfied.

This is purely about how $\phi$ and the formulas in $\Gamma$ react to having meanings assigned to them -- it does not depend on any syntactic relationship between them.

You're right insofar that logicial implication is a relation between particular pieces of syntax, in the sense that a formula is always a piece of syntax -- but the relevant point is that the definition works in terms of the semantics.