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?
[Math] the difference between ‘logical symbols’ and ‘relation symbols’ in first order logic
definitionfirst-order-logiclogicpredicate-logic
Related Solutions
Some books use 'relational logic' to emphasize that it goes beyond unary predicates ... (and there are important pedogogical, practical, and theoretical reasons for doing so). Indeed, many books first discuss something they call 'categorical logic', restricted to just unary predicates. For example, Aristotle studied this kind of logic with claims like 'All humans are mortal'. (Then again, some people hold 'categorical logic' to be something different yet, see e.g. the Wikipedia page on 'Categorical Logic'.)
Your book, however, uses 'relational logic' in a way synonymous with 'predicate logic', which is typically understood as the logic where you can have predicates of any arity. (then again, some will insist that only 1-place relationships are 'predicates' (i.e. more like 'properties'), while 2- or more place relationships are 'relations', but not 'predicates' ...)
In other words ... the terminology here is not fixed, so you will find different people have different definitions for these different logics. But, I think most people would agree with the claim that relational logic is a part of predicate logic, i.e. that 'predicate logic' is the more general logic. This is certainly how this community uses the tag 'predicate-logic'
... all of which means ...
You can probably learn plenty about relational logic on the sites that talk about predicate logic! You can also look for 'first-order logic' or 'quantificational logic'.
First-order, second-order and third-order logic are all logical languages with universal and existential quantifiers. The difference lies in what quantifiers speak about.
In first-order logic one can quantify over individuals. An example of a first-order formula is the commutativity axiom for a group $(G,*,e)$:
$$\forall x. \forall y. x * y = y * x$$
In second-order logic one can quantify over sets of individuals. An example of a second-order formula is the axiom defining the existence of least upper bounds for a complete partial order $(D,\sqsubseteq)$, which says that every subset $Y$ has a least upper bound:
$$\forall Y. Y \subseteq D \Rightarrow (\exists z. (\forall y. y \in Y \Rightarrow y \sqsubseteq z) \wedge (\forall x. (\forall y. y \in Y \Rightarrow y \sqsubseteq x) \Rightarrow z \sqsubseteq x))$$
In third-order logic one can quantify over sets of sets of individuals. An example of a third-order formula is the axiom for topological spaces which states that the union of a family of open sets is an open set.
Best Answer
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'!