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 ?

• "Is there a difference...?" Clearly no; see page 1: ". A relation constant that can combined with a single argument is said to be unary; one that can be combined with two arguments is said to be binary; one that can be combined with three arguments is said to be ternary; more enerally, a relation constant that can be combined with n arguments is said to be n-ary." Thus, relational constants (of whatever arity) are usually called predicates. Commented Jan 24, 2017 at 19:10
• @MauroALLEGRANZA How is it possible to receive so contradicting answers to such a seemingly simple and basic question ??? I understand your answer and I tend to agree but man... why there are so many people who say so many different things ??? I am getting confused :( Commented Jan 24, 2017 at 19:32
• Terminology is hard to be "regimented"... The founding fathers of math log (see W&R's Prinicpia) developed separate chapters for (unary) predicates and (binary or more) relations Subsequently, it was understood that there is no need of treating them separately and several names were proposed : Functional Calculus, Predicate Calculus. Maybe, the best solution is to call it First Order Logic. Commented Jan 24, 2017 at 20:07
• @MauroALLEGRANZA Now I understand much better where did this 'split' in naming originated. Thank you for the explanation. This comment should be the top answer. If you want, post it as an aswer. Commented Jan 24, 2017 at 23:35

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'.

• @yoyo_fun That's how most people would look at it, yes. But not everyone agrees. And again, the stress on 'relational' in 'relational logic' is typically meant to indicate that we don't (just) consider unary predicates. Also, theoretically there is a big difference between logics that consider only unary predicates, and logics that use any kind of predicates: the former are decidable, whereas the latter are not. So that is another reason to emphasize that in 'relational logic' we are interested in predicates with multiple arguments, as opposed to 'categorical logic'. Commented Jan 24, 2017 at 18:44
• I disagree with your paragraph about categorical logic. Categorical logic concerns the syntax and semantics of all sorts of kinds of logic (algebraic theories in finite product categories, λ-calculus in cartesian closed categories, type theory in locally cartesian closed categories, higher-order logic in toposes, etc...) and is certainly not restricted to unary predicates. Commented Jan 24, 2017 at 18:50
• @CliveNewstead Ah yes, I would call that 'category theory', but like I said, the terminology is not fixed. I see many uses of 'categorical logic' to mean the kind of logic Aristotle was involved with, i.e. 'All humans are mortal', etc. Think Venn diagrams. But you proved my claim that different people use these terms to mean different things! Commented Jan 24, 2017 at 18:52
• @Bram28 - categorical syllogism, from categorical proposition and not categorical logic :-) Commented Jan 24, 2017 at 19:25
• @MauroALLEGRANZA Google 'Categorical Logic' and you'll see what I mean: lots of pages on categorical syllogism and categorical propositions with an explicit header of 'Categorical Logic'. Again, different people, different uses of the term. :) Commented Jan 24, 2017 at 19:37

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.

• I don't think this opinion-based answer (however interesting) is relevant for the question in the OP. Commented Apr 7, 2018 at 0:02
• Well, it is a fact-based answer with opinion-based explanation and commentary. To clarify the fact, Dr. Genesereth, the author of the Stanford course, gave the reason I explained for distinguishing between relational logic and FOL, i.e., Herbrand semantics, which redefines the notion of what constitutes a model Commented Apr 7, 2018 at 5:29