<html><head><meta http-equiv="content-type" content="text/html; charset=utf-8"></head><body dir="auto">Take into account that this kind of “sequential” multilayered 2-sorted logic do not take into account the superposition of cross layer causality, and in particular top-down causality which interacts with the individual layers. This story cannot be solved logically in the usual way. Wave phenomena. <div><br><div dir="ltr"><br><blockquote type="cite">On Feb 19, 2024, at 10:05 PM, Plamen <plamen.l.simeonov@gmail.com> wrote:<br><br></blockquote></div><blockquote type="cite"><div dir="ltr"><meta http-equiv="content-type" content="text/html; charset=utf-8">I am sorry. This comes with some delay. <br>Ted Goranson from this group has tried this in his 2-sorted logic (2SL) changing stepwise periodically LH with RH equation sides and vice versa to model small step multilayer advancements. I have evaluated this approach and despite being locally applicable in theory, I could not validate it would work on a global all-layers scale. This can become indefinitely complex and diverge at the end if you do not introduce other control points in that switchable causality. It looks good, but practically may be strange. Multiple sorts between the layers might be necessary. Proved by 2-3 examples via induction, no theorems. I don’t know if Ted has come further in this quest.<div><br></div><div>Best,</div><div><br></div><div>Plamen<br><div dir="ltr"><br><blockquote type="cite">On Feb 17, 2024, at 2:02 AM, Carlos Gershenson <cgershen@gmail.com> wrote:<br><br></blockquote></div><blockquote type="cite"><div dir="ltr"><meta http-equiv="content-type" content="text/html; charset=utf-8"><div dir="auto" style="overflow-wrap: break-word; -webkit-nbsp-mode: space; line-break: after-white-space;">Dear Lou,<div><br></div><div>Thank you for your examples. </div><div><br></div><div>The Kleene argument precisely shows the relevance of the difference between theory and practice. In theory, there will always be undecidability in “powerful enough” formal systems. In practice, we can arbitrarily try out all algorithms that we want for a finite time, and come up with a table of whether they halted or not. </div><div><br></div><div>Now, the question would be whether with such a pragmatic approach we can overcome the limits of formal systems that prevent us from representing properly e.g. evolutionary innovations, and changes in function in general. </div><div><br></div><div>But maybe we just need two levels: one where rules don’t change, and another where we can allow rules and meaning to change. The thing is that the way in which rules and meaning change cannot change… but then if we add another layer… To avoid turtles all the way down, could we have layer A changing layer B and vice versa? A coevolution of formal systems?</div><div><br></div><div>Best wishes,<br><div>
<div style="color: rgb(0, 0, 0); letter-spacing: normal; orphans: auto; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: auto; word-spacing: 0px; -webkit-text-stroke-width: 0px; word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space;"><div style="color: rgb(0, 0, 0); letter-spacing: normal; orphans: auto; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: auto; word-spacing: 0px; -webkit-text-stroke-width: 0px; word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space;"><span class="Apple-style-span" style="border-collapse: separate; color: rgb(0, 0, 0); font-family: Helvetica; font-style: normal; font-variant-ligatures: normal; font-variant-position: normal; font-variant-caps: normal; font-variant-numeric: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: -webkit-auto; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; border-spacing: 0px; -webkit-text-decorations-in-effect: none; -webkit-text-stroke-width: 0px;"><div style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space;">Carlos</div></span></div></div>
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