From: pratt@Sunburn.Stanford.EDU (Vaughan R. Pratt)
Subject: Re: Multiple Truth Values
Organization: Computer Science Department,  Stanford University.
Date: Tue, 12 Jan 1993 20:15:45 GMT

In article <TORKEL.93Jan12110752@lludd.sics.se> torkel@sics.se (Torkel Franzen) writes:
>In article <1993Jan12.083955.19685@CSD-NewsHost.Stanford.EDU> pratt@Sunburn.
>Stanford.EDU (Vaughan R. Pratt) writes:
>
>   >So if you
>   >truthfully 'fess right up with "I broke the vase" then it cannot be
>   >that you did not break the vase.  But if you say "I didn't not break
>   >the vase" then while this tips us off to some degree of culpability on
>   >your part it does not permit us to infer intuitionistically that you
>   >actually broke the vase.
>
>  In your terminology then, a confession on your part in the form
>"surely I broke the vase" must not be taken to mean that you did break
>the vase, although an indignant denial in the form "surely I did not
>break the vase" or "I did not surely break the vase" implies that you
>did not break the vase.

That intuitionistic logic is in agreement with Boolean logic as to the
interpretation of triple negation as negation is an excellent point.
However one can hardly infer a preference for Boolean logic over
intuitionistic on the basis of their similarities.  Rather, if there
are any serious problems with the identification ~~~x = ~x then both
Boolean and intuitionistic logic are in trouble.  If I were working at
the boundary of linguistics and philosophical logic I'd be strongly
tempted to pursue this question further by looking for naturally
occurring counterexamples to ~~~x = ~x.

>If you say "surely, if I broke the vase I must
>have smashed the window" and later admit that you broke the vase, we
>must be careful not to claim that you stand convicted on your own
>evidence of having smashed the window, but only of having surely
>smashed the window. I think this whole thing is a very shrewd move on
>your part in that it practically guarantees that whatever misdemeanors
>you may commit in the future, you will be able to confuse your
>interrogators to the point where they prefer to drop all charges.
>Presuming, of course, that your lawyer can convince the court that
>this person, being an Intuitionist, must be interrogated in his own
>language.

Australian and American courts interrogate witnesses in a language that
presumes intuitionistic logic, in that they draw a distinction between
straight answers and hedged and insist on the former.  (There is
nothing special about "surely", you can substitute "perhaps" or
"possibly" or "conceivably" if you feel "surely" doesn't adequately
support my basic point that the modality weakens the logical strength
by *some* unquantified amount.)

Do Swedish courts assume that natural language is based on Boolean or
intuitionistic logic?  If the former, all answers to A-or-B questions
would mean either A or B, and "Conceivably" as a response to "Did you
break the vase?" would be treated as "yes."

The assumption of Boolean logic does seem to have the advantage of
efficiency, and its adoption in the courtroom might be a step towards
reducing the backlog of cases.  There remains only the problem of
educating each newcomer to court to use it in place of the
intuitionistic logic the rest of us use every day outside the
courtroom.  Someone accustomed to prefacing every negative answer with
"Maybe", on the reasonable expectation of being pressed for a definite
yes or no, would be taken aback at first to have his every "maybe" read
as "yes" in the courtroom.  Eventually he would learn, just as circuit
designers know to expect that "maybe" as input to a TTL gate is likely
to be read as either "yes" or "no."

The sign of the influence of the double negation rule on language is a
delicate point, and it is understandable that as competent a
mathematician as Roger Penrose would get it backwards.
-- 
Vaughan Pratt			There's safety in large condition numbers.
Newsgroups: sci.logic,soc.culture.indian