Logical Languages - Prolog or Lisp/Smalltalk or something else?
At the risk of offending some, I have a hard time reconciling "easily understood by other people that are fond of the BASIC language" with any of the languages you mentioned. That's not intended as a criticism, but as an observation that each of the languages you mention has a style and natural idiom that's quite different from that of BASIC.
Smalltalk - pure OO from the ground up, usually (e.g. Squeak) coupled with an integrated environment that is simultaneously the IDE and the runtime. IOW you enter the Smalltalk VM and work inside it rather than just writing a text that is "source code".
LISP - much closer to functional programming (although with imperative overtones); the prefix notation is the first barrier to most people who "like" other languages, but the concept and use of macros is a much more substantial one.
Clojure - The combination of LISP, OO, and JVM integration makes this one even less BASIC-like.
Python and Ruby - I lump these together (at the risk of further annoying fans of either ;-) because they are both OO language with distinct notations that will take an outsider a bit of learning curve. The use of indentation-only for control nesting in Python and the Perl-like use of special characters in Ruby are often points of the complaint by newcomers. Although both can be written in an imperative style, that would be considered non-standard by seasoned users.
Prolog - This is the most unlike BASIC of all languages mentioned. All of the other languages you mentioned can be (ab)used in a semi-procedural style, but that is essentially impossible in Prolog. It requires a thorough understanding of, and comfort with, recursion to do anything non-trivial.
Code written with a "native accent" in essentially all of these languages (but especially Prolog, IMHO) will make use of idioms and concepts that are outside the norm for conventional BASIC programming. Put another way, if you pick one of these and then write code "with a BASIC accent" you've pretty much wasted the benefits that the language can offer.
I believe that all of them are worth learning for the concepts they can teach (or at least reinforce, depending on your background). But the similarity to Language X (for a wide range of values of X) is not what you'll get.
Is there any particular reason not to use R? It's sort of a build vs. buy (or in this case download) decision. If you're doing a statistical computation, R has many packages off the shelf. These include many libraries and interfaces for various types of data sources. There are also interface libraries for embedding R in other languages such as Python, so you can build a hybrid application with a GUI in Python (for example) and a core computation engine using R.
In this case, you could possibly reduce the effort needed for implementation and wind up with a more flexible application.
If you've got your heart set on learning another language, by all means, do it. There are several good free (some as in speech, some as in beer) implementations of Smalltalk, Prolog and LISP.
If you're putting a user interface on the system, Smalltalk might be the better option. If you want to create large rule sets as a part of your application, Prolog is designed for this sort of thing. Various people have written about the LISP ephiphany that influences the way you think about programming but I can't really vouch for this from experience - I've only really used AutoLISP for writing automation scripts on AutoCAD.