Wednesday, December 24, 2008

The Vel-Aut Problem

I've found the gift to give myself this Christmas. Well, I already ordered two twelve packs of Cherry RC Cola (not available in my area), but's the season to give. The Genealogy of Disjunction by R.E. Jennings explores the linguistic and logical understanding of "or" in English being essentially disjunctive.

This, out of all my musings thus far, likely has the most immediate bearing on writing a book to aid teachers of logic (as opposed to yet another pre-college logic textbook to enter a crowded market).

I will let Wikipedia explain inclusive and exclusive disjunctions. Pay close attention to the relationship between these concepts and binary!

I will put pictures for color. The first is an inclusive disjunction. This or that, or both - an inclusive disjunction is true as long as one part is true.

This next one represents an exclusive disjunction. This or that, but not both - an exclusive is only true if one part alone is true.

One Ernest W. Adams has a review, part of which can be read online. It seems Jennings takes issue with how he perceives the approach of formal logicians in regards to symbolizing natural language. Much of the book is available on Google Books, and he addresses the idea of the Latin aut being exclusively an exclusive disjunction, whereas vel was inclusive.

In modern formal logic, especially the variety that cleaves to the classical pattern, inclusive disjunction is symbolized with a "v."

One can see giraffes or horses at the zoo.

G v H

Numerous texts on logic, both ones aimed at pre-college students and weighty tomes addressing the history of formal logic have repeated the claim that the use of "v" comes from its connection to vel - but Jennings disputes such a connection!

This of course does not rock my world, but it shows an interesting inductive study of Latin. I will have to ask some of my Latin-teaching colleagues for their take. The connection always seemed too convenient, but understandable given the influence of Latin on Western logic. My heart soared to the Seventh Levels (the ones McCartney reached?) when Jennings brought up Abelard. This book will be a winner, I can tell.

Who cares about the distinction between inclusive and exclusive? Well, it's a matter of how to properly represent language. As the Wikipedia articles linked above note, the use of inclusive vs. exclusive has aided computer language greatly, with its dependence on binary.

I suspect that folks like Jennings take issue with symbolizing disjunction as "this or that," because in natural language, we often connect several terms with "or."

You could take Tim with you to the store in your two-seater. Or Bruce, or Cassandra, or Dick, or Alfred, or Dinah.

Thus, the symbolism of a disjunction as p v q unnerves them. Too unnatural, they might say.

Symbolic logicians have offered other ways to symbolize it. For instance:

Madame, there are many choices on the menu. There is fish, or chicken, or beef. Of course, Madame can have each if she has a suitable appetite.

If this is inclusive, then perhaps it could be symbolized v[A,B,C], meaning one or more may satisfy the requirements.

You may have either coffee, tea, or water. These are your only choices - and you may only have one!

If this is exclusive, then v[A,B,C], meaning only one may be selected.

Now, this symbol of "v" versus "v" is my own invention for the purpose of this blog. One Timothy Williamson addresses disjunction in modal logic quite thoroughly here.

For educators seeking to craft creative, critical thinking, picture a student who can a statement, break it down grammatically, reword it from various angles of tense and voice, and also symbolize it exactly. This is a student who has not only mastered their own language, but can apply logical symbolism to further languages they learn, as well as situations.

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