# [r-t] Invariants of truth

Don Morrison dfm at ringing.org
Mon Aug 11 03:16:07 UTC 2008

```There are two properties that my intuition tells me should hold for
any definition of truth:

1) A is a composition. B is a rotation of A. A is true if and only if
B is true.

2) A composition A is, or I suppose more accurately generates, an
ordered list of rows. Let composition B be, or generate, a list
exactly the same rows, just in a different order. A is true if and
only if B is true.

For single stages I think both of these invariants hold for the
current Decisions and for all the various proposals that have been
floating around on this list.

For multiple stages it gets trickier:

i) Neither holds for the current Decisions, at least for mixed,
lower stages such as doubles and minor.

ii) Mark's proposal, with the amorphous "structure" restriction
probably maintains invariant (1), though I'm not positive. It doesn't
maintain invariant (2), and I think he thinks that's A Good Thing.

iii) The "sets" definition I'm pretty sure preserves both invariants.
If, as has been asserted, Iain's recursive definition, and at least
one version of Richard's formulation, are equivalent to the sets one,
presumably they also preserve both invariants.

Am I right about (i), (ii) and (iii)? More importantly, should these
invariants hold? Or is my intuition just broken, which is certainly a
possibility?

There's probably some extensions to invariant (1) that I've not
completely thought through:

1a) You can probably say something similar regarding reversals. I'm
just a little worried about how to accurately define reversals in all
cases, particularly if jump changes are allowed. I'm pretty sure it
can be done, and suspect Richard may supply such a thing within
minutes of reading this message.

1b) I think rotations of sub-chunks, such as contained extents may
also be interesting, but I've not quite figured it out. I maybe
completely mistaken about this being in any significant way different
or interesting, though.

--
Don Morrison <dfm at ringing.org>
"quid non mortalia pectora cogis,/ auri sacra fames?" -- Virgil, _Aeneid_
(To what crime does accursed gold-lust not drive the hearts of men?)

```