Seeking advice on sequence re: logic predicates.
J.S. Seneschal
Sequence A396832 was recently rejected on the basis that “Our readers will not understand the terminology here.” – which suggests to me a potentially easy fix and re-submission considering most, if not all, of the terminology is already used elsewhere by other OEIS sequences in one way or another.
At the risk of having anyone write the definition for me, as warned against in Mr. Sloane’s recent post, I would just like to ask if there is any particular sticking point(s) in the wording that should be changed; if incorrect terminology is being used, or used incorrectly; if I’m over-complicating and/or under-clarifying – assuming my sequence makes any sense at all, or is even relevant to OEIS, which also seems to be a potential issue.
The sequence regards the minimal application of implied predicates in higher-order logic, with the basic premise being: Any conceivable thing implies it having at least one property. Any conceivable multitude of things implies at least one relationship between each thing. In high-order logic, predicates (properties and relations) are considered ‘things’ themselves to which further predicates can be applied (quantification over predicates) – what then is the resulting sequence if one continues this process of applying the least number of implied predicates to each new ‘thing’?
Different sequences would, of course, result depending on whether the specific logic system allowed reflexivity (self-relations) and/or combinatorics (counting the various groups of things as ‘things’ in themselves). I began by submitting a sequence for a simple non-reflexive, non-combinatorial system (strictly predicative). Below is the sequence submission in its last form and a link to the discussion leading up to this form for further context:
NAME
Number
of elements at step n in a minimal free extension resulting in a
saturated model of non-reflexive higher-order predicates.
DATA
1,
2, 3, 5, 14, 25, 315, 616, 189736, 379157, 71880015483, 143759841230,
10333445975237404051319, 20666891950402928087155,
213560211444814672490933743423959229578004755,
427120422889629344981857153401943221751958191
OFFSET
1,2
COMMENTS
Beginning
with one element, since no non-reflexive relations can yet be
applied, apply one property to the initial element making 2 total
elements; now apply 1 relation between them making 3 total elements;
continue applying novel, non-redundant properties and relations in
alternate steps so that every element eventually has exactly 1
property and 1 relation with every other element.
FORMULA
a(1)
= 1, a(2) = 2, a(3) = 3; for n > 3: a(n) = 2*a(n-1) - a(n-3) if n
is even; a(n) = a(n-1) + a(n-1)*(a(n-1)-1)/2 - a(n-3)*(a(n-3)-1)/2 if
n is odd.
CROSSREFS
Cf.
A000217, A108225.
https://oeis.org/history/view?seq=A396832&v=18
To get the ball rolling I have devised an updated name and comment:
NAME
Number of terms at step n in a strictly predicative free construction of higher-order logic.
COMMENTS
Alternating generation of higher-order predicates as new terms, starting from one entity. Non-reflexivity prevents a relation at Step 1, so a property is applied at Step 2 (making 2 terms), followed by a relation between them at Step 3 (making 3 terms). Continue applying higher-order predicates thusly (even steps adding unary properties to existing terms; odd steps adding binary relations between all unique pairs of existing terms) so that every term eventually has exactly 1 property and 1 relation with every other term.
Better? Worse?
Thanks in advance for any help on this matter.
Neil Sloane
is, we don't have any logicians among our editors (as far as I know).
Members of this list: please correct me if I am wrong! But I do know
a couple of logicians and I'll try to get one of them to help. In
spite of the screams of agony from the editors who looked at your
submission, I'm hopeful that we can get some versions of it into the
OEIS.
Best regards
Neil
Neil J. A. Sloane, Chairman, OEIS Foundation.
Also Visiting Scientist, Math. Dept., Rutgers University,
Email: njas...@gmail.com
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Arthur O'Dwyer
Sequence A396832 was recently rejected on the basis that “Our readers will not understand the terminology here.” [...]
The sequence regards the minimal application of implied predicates in higher-order logic, with the basic premise being: Any conceivable thing implies it having at least one property. [...]
NAME
Number of elements at step n in a minimal free extension resulting in a saturated model of non-reflexive higher-order predicates.
[...]
NAME
Number of terms at step n in a strictly predicative free construction of higher-order logic.
COMMENTS
Alternating generation of higher-order predicates as new terms, starting from one entity. Non-reflexivity prevents a relation at Step 1 [...]
David desJardins
Arthur O'Dwyer
(Douglas Hofstadter somewhere (probably somewhere in Gödel, Escher, Bach) has drawn a diagram resembling your "relations" universe: he draws e.g. a blob for "Tall" and a blob for "Short," then connects them with a blob for "Antonym," and so on; except that in his case the relation between "Antonym" and "Synonym" is itself "Antonym," and so on, so his graph is very much not uniform.)
J.S. Seneschal
Addressing Arthur’s comments:
> Or do you allow properties to have properties?
Yes. As per the process outlined in the original comment: “continue applying novel, non-redundant properties and relations in alternate steps so that every element eventually has exactly 1 property and 1 relation with every other element.” So, essentially, every even step adds properties (“blue marks”) to all the new marks (blue and red) added in the previous 2 steps. Every odd step adds relations (“red marks”) between all new marks added in the previous 2 steps and every mark in the entire model.
> ...it's really two sequences interleaved.
Yes, sort of... But I’m not quite sure how to extract one from the other, especially without making the explanation(s) even more convoluted – with each sequence on its own being even more obscure and un-useful than the current combined sequence. I encountered this issue a few years back submitting a sequence for “oblong cuboid numbers” which interleaved 2 other sequences (A045991 & A011379). Fortunately these sequences already existed on their own merit, otherwise I can’t envision either of them being accepted as just “half of the oblong cuboid numbers”.
> Hofstadter diagram...
Below is my own crude diagram I previously made illustrating the first few terms of the sequence, if it makes things any more or less clear:
Addressing David’s comments:
> ...is this actually going to be useful to other users?
No idea. I can’t foresee anyone ever needing to construct an infinite model of increasingly abstract predicate relationships, myself. The sequence was submitted merely as an observation: Predicates (properties and relations) are a fundamental aspect of logic. Higher-order logic implies an infinite self-generating growth of predicates (quantification over predicates). What, then, is the nature of that growth?
I suppose I should ask here if anyone can see any purely mathematical utility to this sequence, divorced from any formal logic implications (I know, it’s a stretch)?
> I don’t know your motivations here...
As stated in the original post, the reason for rejection and the discussion leading up to it suggested (at least to me) that a simple issue of terminology/semantics was holding back an otherwise potentially acceptable sequence. Just trying to learn and improve my OEIS submission skills for any future contributions, while hopefully clarifying the wording for my own edification on this particular sequence.
I rarely contribute anything here, and about half of what I do contribute gets rejected so I expect that. I’m not a mathematician nor a logician so there’s no ego boost or professional accolades to be gained. I’m happily surprised that anything I ever submit gets approved. I’m not precious about this sequence nor anything else I submit, so feel free to bash it and bin it accordingly!
J.S. Seneschal
I have been reflecting more on Mr. O'Dwyer's comments – specifically how my sequence is actually 2 sequences, and how the “odd step” isn’t really a “step” at all – and it occurs to me that I could then simply condense the entire sequence into the “odd step” as an iteration of graphs since, in my model, new relations are basically just edges, while properties and previous relations are basically just vertices.
This allows me to express the definition in mathematical terminology more suited to OEIS readers, while moving all talk of logic predicates to a brief comment and a supplemental link/illustration (yet to be conceived). The result produces an interesting (at least to me) analogy to the formula for producing the number of edges in a complete graph (Kn): instead of n(n-1)/2, we get (|Kn| – |E(Kn-1|)/2. Below is an example of a possible new submission:
NAME
TBD (use just the formula, perhaps?)
DATA
0, 1, 3, 14, 315, 189736, 71880015483, 10333445975237404051319, 213560211444814672490933743423959229578004755
OFFSET
0,2
COMMENTS
a(n) gives the total number of elements |Gn| in an iteration of graphs whereby |Gn| equals the total number of elements in the corresponding complete graph |Kn| minus the number of edges in the previous complete graph |E(Kn-1)|, divided by 2: (|Kn| – |E(Kn-1|) / 2.
The number of vertices (V) in each graph (Gn) is determined by the total number of elements in the previous graph |Gn-1| plus the total number of new elements added to the ante-penultimate graph |Gn-2|, so that: V = |Gn-1| + (|Gn-1| – |Gn-2|). The number of edges (E) equals those of the corresponding complete graph (Kn) minus the total number of elements in the previous graph |Gn-1|, less 1: E = E(Kn) – (|Gn-1| – 1).
This sequence can be used to represent the number of terms at step n in a strictly predicative free construction of higher-order logic (see illustration).
LINKS
TBD
FORMULA
a(1) = 1, a(2) = 3 ; for n > 2: a(n) = (U*(U + 1) - V*(V - 1)) / 2, where U = 2*a(n-1) - a(n-2), and V = 2*a(n-2) – a(n-3).
CROSSREFS
Cf. A000217, A108225
Let me know if this new approach is preferred, and if anything needs correcting (I can already anticipate some grumbling about my use of “ante-penultimate”!).
J.S. Seneschal
The number of edges (E) equals those of the corresponding complete graph (Kn) minus the total number of new elements in the previous graph |Gn-1| – |Gn-2|, less 1: E = E(Kn) – ((|Gn-1| – |Gn-2|) – 1).