Re: Conjunctions, conjunctive adverbs, subordinators

On 3/16/06, Jim Henry <jimhenry1973@...> wrote:
>
> I think I recall reading that all 16 truth-table logic functions
> can be derived with just AND and NOT.  To get the semantics
> of "but", "although" and so forth, we could add an +UNEXPECTEDLY
> morpheme (adverb, affix, inflection or what have you) to either the first
> or second clause.
>
> I can't recall offhand how all of them are derived, but XOR ("unless") is
> (I think):
>
> NOT ( NOT A AND NOT B ) AND NOT ( A AND B )
> i.e., it is not true that both are false and it is not true that
> both are true, i.e. one is true and the other is false.
>
> > Conditional markers:
> >    if, unless, only if, whether or not, even if, in case (that)
>
> if A then B: TTFT (IMP)
> A unless B: FTTF (XOR)
> A only if B: TFFT (EQV)
> A whether or not B: TTFF
> A even if B: TTFF plus surprise marker on B
> A in case that B: TFFT (EQV) plus subjunctive or causative B?
>
Your values for these are a bit strange; are you using a different
order for truth-value assignments than <T,T>,<T,F>, <F,T>,<F,F>?

Here are more standard values for these:

If A then B: TFTT (TTFT is A if B)
A unless B: TTTF (same as OR; XOR is indeed FTTF)
A only if B: also TFTT (whereas equivalence -- TFFT -- is "if and only if")

>
> A really minimal conlang would have AND and NOT, probably,
> and maybe a surprise/unexpectedness/mirative particle (that
> also works as a verb "to be surprised" and a noun "surprise" and so forth).
>
> I'm not sure how we could do with only "if" though.
>
You can make do with just IF and NOT; Lukasiewicz's axiomatization of
sentential logic has these as the primitives, iirc.

A OR B = IF NOT A THEN B
A AND B = NOT ( IF A THEN NOT B )

You can also have just IF and OR, and derive the rest from these.

You can actually get the whole shebang with just one operator; either
of the Sheffer Stroke ( | = NAND = FTTT) or the Dagger (NOR = FFFT)
suffices to define the rest.

NOT A = A | A
A AND B = ( A | B ) | ( A | B )
A OR B = ( A | A ) | ( B | B )
IF A THEN B = A | ( B | B )

You can do it with the Dagger, too.  This would be really unwieldy for
speech, or course.  To really use it in a language you'd have to have
a whole mess of pronouns, pro-verbs, pro-clauses, etc. etc. to keep
from repeating everything multiple times.  Probably easier to stick
with all of NOT, AND, OR, and IF.

Were I doing a minimal system for communication purposes (as opposed
to, say, axiomatizing), I'd probably have NOT (can't really get away
without negation), express AND by juxtaposition, and then choose OR as
my last one and use NOT A OR B for IF A THEN B.

-- Pat

Reply