# theory of intervals

• To: lojban-list@snark.thyrsus.com
• Subject: theory of intervals
• From: cbmvax!uunet!glia.biostr.washington.edu.biostr.washington.edu!jsp (Jeff Prothero)
• Posted-Date: Wed, 9 Jan 91 14:20:40 PST

```I've long wanted Loglan to apply the described "theory of intervals"
to tense construction etc.

One pet peeve I like to bring up annually: Time, in contemporary
physics, and increasingly in practical experience as well, is a
partial order, not a complete order.  That is, given two spacetime
events A and B, the possible orderings are:

A before B
A coincident with B
A after B
A incomparable with B  <--- frequently ignored case.

(If the fourth case doesn't make sense to you, you need to consult
the standard light-cone diagram in any standard "ABCs of Relativity"
popular-science book/article...)

Any linguistic facility naming interval relationships by enumerating
possible endpoint orderings should (IMHO) be based on the above
partial order rather than the traditional Newtonian full order.

Thus, the standard (?) interval enumeration for two intervals A=[a,a']
B=[b,b']

01)   a <  b     a' <  b      "A before B"
02)   a <  b     a' =  b      "A until B starts"
03)   a <  b     a' in b      "A until B"
04)   a <  b     a' =  b'     "A until B ends"
05)   a <  b     a' >  b'     "B during A"
06)   a =  b     a' in B      "A during B"
07)   a =  b     a' =  b'     "A while  B"
08)   a =  b     a' >  b'     "B during A"
09)   a in B     a' =  b'     "A during B"
10)   a in B     a' >  b'     "A starting in B"
11)   a =  b'    a' >  b'     "B until A starts"
12)   a >  b'    a' >  b'     "A after B"

should really be augmented with the following intervals ("?" ::=
"imcomparable to"), each of the above cases with a "=" giving rise to
a case with a "?":

13)   a <  b     a' ?  b
14)   a <  b     a' ?  b'
15)   a ?  b     a' in B
16)   a ?  b     a' =  b'
17)   a =  b     a' ?  b'
18)   a ?  b     a' ?  b'
19)   a ?  b     a' >  b'
20)   a in B     a' ?  b'
21)   a ?  b'    a' >  b'

Jeff Prothero
jsp@milton.u.washington.edu
Better a bicycle without wheels
than a loglan without formal semantics!

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