What is the difference between 'logical symbols' and 'relation symbols' in first order logic?

Question

I am currently looking at some logic notes which give the following definition

However, there is no explanation for what the distinction between 'logical symbols' and 'relation symbols' is.

Can someone please help me to understand this?

Answer 1

Relation symbols can be used to express that certain objects stand in some kind of relation or, in case the relation symbol takes only one argument, that some object has some property. That is, relation symbols, together with a number of 'arguments' or objects, can be used to create claims.

Examples:

$Cube(x)$: this says that x is a cube (arity is 1)

$LeftOf(x,y)$: this says that x is to the left of y (arity is 2)

$Between(x,y,z)$: x is between y and z (arity is 3)

Logical symbols typically perform some kind of operation on those claims. That is, once I have the claims $Cube(x)$ and $LeftOf(x,y)$, I can combine those into something like $Cube(x) \land LeftOf(x,y)$.

In short (and very roughly): Relation symbols take objects as their arguments, and return claims. Logical symbols take claims as their arguments, and return more complicated claims.

Also: The meaning of logical symbols is fixed: $\land$ means the logical and, and always works the way we have defined it. But the meaning of relational symbols is not fixed. As we saw above, I could interpret $Between(x,y,z)$ as 'x is between y and z', but I could also interpret it is 'y is between x and z', or even : 'z is the sum of x and y'!

Answer 2

The distinction is just which belong to logic itself, and which are optionally interchangeable for particular languages. Not every first order language needs "$\in$" or "$\leqslant$"; they all need to contain, or be able to interpret, the logical symbols. (I'm here omitting the more specialized fragments of first order logic like coherent or regular logic; by "first order" I mean "full first order".)

It's worth noting that the logical and extralogical symbols don't have anything like the same purpose. The logical symbols take predicates and spit out new compound predicates; the extralogical symbols combine with terms to form "elementary" predicates. And this brings me to the more serious difference between logical symbols and extralogical symbols: you can take a logical truth in the language of set theory, where the only predicate besides $=$ is $\in$, and you can replace $\in$ in every sentence by $\leqslant$, and it's still logically true. I cannot, however, replace all instances of $\vee$ with $\wedge$ in the set of logical truths without making new false sentences; nor can I just replace all instances of $\exists$ with $\forall$. The logical symbols are essential to the notion of logical truth in a way that extralogical relation symbols are not.