Difference between these logical expressions of "if P, then Q"

Question

What's the difference between these expressions?

1. $P\Rightarrow Q$
2. $P\vdash Q$
3. $P\vDash Q$
4. $P \over Q$

Answer 1

As pointed out in the comments, there is unfortunately no universal standard on the use of these symbols.

However, it's not as if the usage is completely arbitrary either. Let me go over the most likely interpretations for each of those:

1. $P \Rightarrow Q$. I see two common uses for this:

Sometimes this is used as a truth-functional logic expression, where the $\Rightarrow$ is the truth-functional operator called the material conditional, which, in turn, is typically expressed in English as "If P then Q", although the match between the material conditional and the English conditional seems imperfect (see, e.g. Paradoxes of the Material Implication). Note that there are a good number of other symbols used to denote this material implication, such as $\rightarrow$ (I am surprised that one didn't make it in your list!) and $\supset$

Other texts, however, use the $\Rightarrow$ to express a relationship between logic statements. As such $P \Rightarrow Q$ would be used to express the claim that $Q$ is a logical consequence of $P$. The $\Rightarrow$ is thus as meta-logical symbol, rather than a logic symbol: it says something about logic statements.

Personally, I use of $\Rightarrow$ as a meta-logic symbol, and use the $\rightarrow$ to express the material implication. Indeed, many logicians use the convention to use single horizontal lines as something that is about syntax, whereas the double horizontal line is more about semantics.

2. $P \vdash Q$: This is almost always used to express the claim that logic statement $Q$ can be derived from logic statement $P$. Of course, such derivations must be made using some kind of inference system, and so this really requires reference to such a system. If we do so, you may see something like $P \vdash_S Q$, expressing that using the rules of system $S$, $Q$ can be derived from $P$. The $S$ is often dropped, hpowever, as the context will often make it clear what inference system we are yusing. And, if we are not making any reference to any inference system at all, it is typically understood that we are using a sound symbol, meaning that if $P \vdash Q$, it must also be true that $Q$ logically follows from $P$ as semantically defined. ... but this sometimes leads to confusion, as some will use $\vdash$ to express exactly the latter, even though that is more consistently expressed as $\vDash$:

3. $P \vDash Q$: So yes, this is almost without exception understood as the claim that $Q$ is a logical consequence of $P$ ... as semantically defined. Note that some etxts will insist that the LHS is a set of statements, and so really want to see $\{ P \} \vDash Q$. As such, you can also differentiate between $\Rightarrow$ (if used metalogically) and $\vDash$: both are about logical consequence, but the $\Rightarrow$ expresses that some statement is the consequence of some other statement, whereas the $\vDash$ expresses that some statement is the consequence of a set of other statements. Many other books are just fine with $P \vDash Q$, and either use it interchnageably with $P \Rightarrow Q$, or use the $\Rightarrow$ as the material implication, or don't use the $\Rightarrow$ at all.

4. $\frac{P}{Q}$: This is probably the most informal notation, and indeed, I can think of 3 uses in which you may such a line:

It could be the expression of an argument "P. Therefore, Q". As such, it does not say whether that argument is valid or not, i.e. whether $Q$ actually follows from $P$ or not

It could also be a (fragment of) a formal proof. That is, it expresses that sing one of the inference rules of whatever inference system is implicitly referred to , $Q$ can be derived from $P$

And finally, I can see how some texts would use it to express that $Q$ logically follows from $P$.

OK, so ... yes, lots of room for confusion and inconsistent usage of the symbols between different books and authors

Finally. though, when in your title you ask about the 'different expressions of "If P, then Q", it could be argued that none of these expressions really capture that. But if you have to pick one, it would be the logic expression that uses a material implication as a logic symbol, rather than any meta-logic symbols. And, as we just saw, most of the symbols you expressed were metalogical symbols.