Le vendredi 17 juillet 2015 00:45:03 UTC, guskant a écrit :
Le jeudi 16 juillet 2015 21:35:46 UTC, xorxes a écrit :On Thu, Jul 16, 2015 at 1:23 AM, guskant
<gusni...@gmail.com> wrote:
For the same reason, I prefer the interpretation of "all true or all false" for {go} and "one and only one of them is true" for {gonai} to the interpretation of grouping of binary connectives.
One problem with that interpretation for "gonai" is that it doesn't go well with "go ... ginai ... gi ..." and such. The natural Lojbanic interpretation is for "nai" to negate the following connectand. So "gonai A gi B gi C" would be true when A differs in truth value from B and from C.
"One and only one of them" is more useful, but hard to reconcile with the general pattern.
mu'o mi'e xorxes
In the case that we give those special meaning to n-ary forethought connectives, {go} and {gonai} should be defined as broader meaning than XNOR or XOR, so that they become XNOR and XOR respectively in the case of n=2.
If {go} and {gonai} are defined as "all true or all false" and "one and only one of them is true", there must be {gocu'i} for the other cases: {gonai ... gi ... gi ...} is not equal to {nago ... gi ... gi ...} in the case of n>2.
Le jeudi 16 juillet 2015 21:58:31 UTC, xorxes a écrit :On Thu, Jul 16, 2015 at 7:30 AM, Alex Burka
<dur...@gmail.com> wrote:
So, to the question of that {gu} and {go} mean with more than 2 arguments, we kind of find ourselves with tradeoff between logic and ease of use.
First I get nervous extending them to >2-ary at all, since it becomes harder to reason about the negations ({gonai X gi Y} is XOR, but {gonai X gi Y gi Z} is not, it's something strange instead -- and nobody can do truth tables in their head, even fluent speakers).
The interpretations you list are undoubtedly useful (particularly the "which one out of these X alternatives"-connective is often asked for,
"Which one(s) of the following" would be "ge'i ... gi ... gi ... gi ... ", right?
though there is a workable solution using {moi}). But they do not correspond to the mathematical interpretation. For example, n-ary XOR is supposed to be true when an odd number of the arguments are true, which is not obvious (or often useful in speech). Changing this can cause difficulties with composition, for example logical transformations like De Morgans' laws will not work.
The negation of "ga ... gi ... gi ..." is still "ge nai ... gi nai ... gi nai ..." and viceversa. And "go ... gi ... gi ..." should be equivalent to "go nai ... gi nai ... gi nai ... ", but that means "go nai" can't mean "one and only one of the following".
Yes, if {gonai} is defined as above, equivalence between {go ... gi ... gi ...} and {go nai ... gi nai ... gi nai ...} will be broken. We need to keep {go ... gi ...} and {go nai ... gi nai ...} be equivalent in the case of n=2. Would this change of meaning of {GA nai} hurt usefulness of binary connective system? If we carefully define {GA (nai)} of n-ary connective system, it might be possible to keep the binary connective system safe.
I am examining more reasonable semantics by adding a new syntactic feature "GA BO" to the recent unstable version of zantufa:
http://guskant.github.io/gerna_cipra/zantufa-1.9999.html
{go broda gi brode gi brodi}: all true or all false of {broda}, {brode} and {brodi}.
{gonai broda gi brode gi brodi}: all true or all false of {na broda}, {brode} and {brodi}.
{gonaibo broda gi brode gi brodi}: only one of {broda}, {brode} and {brodi} is true; the others are false.
Then in the case of binary, both {gonai} and {gonaibo} have usual XOR meaning:
{gonai broda gi brode}={gonaibo broda gi brode}: only one of {broda} and {brode} is true, the other is false.
Do you find any problem? If not, the feature will be added to the next stable release of zantufa.
mi'e la guskant