What is the difference between relational logic and predicate logic? I am studying the Introduction to logic course from Stanford University and I begin learning about relational logic. However when I search on google for the terms there I end up often with results from websites that teach predicate logic. Is there a difference between the two types of logic ?
I am talking about THIS course from Standford :
http://logic.stanford.edu/intrologic/notes/chapter_06.html
 A: 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'.
A: In the Stanford course "relational logic" just means First-order logic (FOL) with Herbrand semantics. In other words, FOL without model theory. You could say that Herbrand semantics introduces a new kind of model theory, but it really is just a way to set model theory aside and focus on logic.
I think this is a good idea because it emphasizes the ancient distinction between logic and grammar, where logic is concerned with form, and grammar is concerned with meaning and interpretation. Taking this view to the extreme, we could say that model theory should not even be seen as part of logic, but rather grammar. They are closely related, but the distinction is important.
