[FOM] Papers on Free Logic

[Dana Scott]

Surprisingly, more than two dozen colleagues asked for
access to the old papers of mine on Free Logic.  I should
also have included a reference to a more recent book:

> Karl Lambert. "Free Logic: Selected Essays." Cambridge
> University Press, 2003, xii + 191 pp.

There are nine essays with an introduction explaining the
historical geneses of the writings.  The essays themselves
have extensive historical footnotes and references.  There
is no book index, however.  (Note to publishers: Never allow
such a lapse.)

Of the two papers of mine, the older in classical logic
found its main motivation in Quine's use of "virtual classes."
He introduces them in a formal way by "contextual definitions."
There is nothing wrong with this, but I felt that having
to prove metatheorems showing that the the usual laws of 
logic applied to notations given contextual definitions 
was a burden.  I preferred to give a semantical explanation
using non-designating terms.  With the semantics in hand,
one can explore which rules of proof are justified. And, it 
was clear that virtual classes could be added to any consistent 
set theory satisfying extensionality.  As to partial functions,
though, Quine did not particularly like my approach.

The second paper in intuitionistic logic developed after
my earlier papers:

> Dana Scott. "Extending the topological interpretation to 
> intuitionistic analysis." Compositio Math., vol. 20 (1968), 
> pp. 194-210.
> 
> Dana Scott. "Extending the topological interpretation to 
> intuitionistic analysis, II." In: Proceedings of Conference 
> on Proof Theory and Intuitionism, edited by J. Myhill. 
> North-Holland Publishing Company, 1970, pp. 235-255.

At the time I wrote those I did not understand why partial
elements were needed.  And in those particular models they
were not needed.  In Boolean-valued models partial elements
are never needed, because one can say, "if the element
exists, take it; if not, take 0 (or some other nullity)."
And that depends on the Law of the Excluded Middle, clearly.

Terms that always denote indicate "global elements".  In
models over complete Heyting algebras (cHa's) without Excluded
Middle, the move to get sufficient global elements is not 
possible.  However, in some models the global elements "cover".  
Care is needed, though, as there can be models (sheaf models) 
without any global elements at all.

It was only in my Oxford period in the 1970s working with Mike
Fourman that I got the right understanding.  Even so, some
people in Category Theory had quite different ways of taking
partial elements into account in formal theories.  But I found
my way "more relaxed" -- as Mike and I tried to explain in
our joint paper.