Are any terms of BCI combinator calculus eligible for copyright?
<p>In logic, a <a href="https://en.wikipedia.org/wiki/Combinatory_logic#Combinatory_calculi" rel="nofollow noreferrer">combinatory calculus</a> is a system for expressing proofs whose conclusions are combinations of their premises. Such proofs are called <em>combinators</em>. Premises can be proofs, and a combinator can apply one of its premises to another.</p>
<p>Today, I'm curious about the BCI combinator calculus. This calculus has three axioms, called <strong>B</strong>, <strong>C</strong>, and <strong>I</strong>:</p>
<ul>
<li><strong>B</strong>: (y → z) → (x → y) → (x → z)</li>
<li><strong>C</strong>: (x → (y → z)) → (y → (x → z))</li>
<li><strong>I</strong>: x → x</li>
</ul>
<p>The reader may recognize these; <strong>B</strong> is <a href="https://en.wikipedia.org/wiki/Modus_ponens" rel="nofollow noreferrer">modus ponens</a>, <strong>C</strong> swaps two premises, and <strong>I</strong> is a <a href="https://en.wikipedia.org/wiki/Tautology_(logic)" rel="nofollow noreferrer">tautology</a>. Now, it is well-known folklore in computer science that BCI is a complete basis for simple linear logic (e.g. see <a href="https://ncatlab.org/nlab/show/combinatory+logic#LinearAndAffine" rel="nofollow noreferrer">nLab</a>). In more familiar language, this means that for any proof which only uses each premise exactly once (&quot;linear logic&quot;), there is a term which only uses <strong>B</strong>, <strong>C</strong>, and <strong>I</strong> as its ingredients and faithfully represents that proof. For a demonstration, consider Algorithm B at <a href="https://www.cantab.net/users/antoni.diller/brackets/intro.html" rel="nofollow noreferrer">this interactive Web page</a>; if the input only uses each letter once, then the output will only use <strong>B</strong>, <strong>C</strong>, and <strong>I</strong>.</p>
<p>Which terms, if any, are eligible for copyright? In particular, is there a smallest copyrightable term of BCI? I know that this is a jurisdiction-dependent question.</p>
<p>If this question seems silly, note that for the BI calculus (just containing <strong>B</strong> and <strong>I</strong>) the answer is hopefully &quot;none&quot; and for the SK calculus (which is equivalent to typical programming languages in expressive power) the answer is hopefully &quot;some&quot;; I figure that asking about a non-trivial intermediate calculus is interesting.</p>

<h1>Short Answer</h1>
<p>No meaningful intellectual property protection is available for BCI calculus, BI calculus, or SK calculus.</p>
<h1>Long Answer</h1>
<h3>Copyright Protection</h3>
<blockquote>
<p>There are several categories of things that can’t be copyrighted. A
common thread to most of them is that they lack the necessary
creativity to be eligible for copyright.</p>
<ol>
<li>Ideas    This includes procedures, processes, systems, methods of operation, concepts, principles, or discoveries. Plans, methods, and
devices, are also not protectable. However the particular manner in
which they are expressed or described in a writing can be copyrighted.</li>
</ol>
</blockquote>
<p>(<a href="https://www.newmediarights.org/business_models/artist/what_can%E2%80%99t_be_copyrighted" rel="nofollow noreferrer">Source</a>)</p>
<p>Copyright can protect a particular expression of an idea (e.g. the layout and exact language of a page in a textbook about BCI calculus that is not the only possible way to present it), but not a concept or system like BCI calculus.</p>
<p>The same reasoning means that it is probably not possible to copyright BI calculus or SK calculus, even though a particular textbook expressing them in a particular way might be possible to protect with copyright.</p>
<p>Also, since BCI calculus is derived from publications in mathematics journals made in 1924 and 1930 by its two lead creators, anything derived from the original publications would probably be in the public domain by now.</p>
<p>In the United States:</p>
<blockquote>
<p>every book and tale published before 1927 is in the public domain;
American copyrights last for 95 years for books originally published
between 1927 and 1978 if the copyright was properly registered and
maintained.</p>
</blockquote>
<p>It is likely that the copyright for the 1930 journal article in question was not registered and maintained, as it was not routine practice for mathematical journals to do so at the time. Even if it was, the copyright for the 1930 paper would be almost spent and would expire in the year 2025.</p>
<h3>The Alternative Of Patent Protection</h3>
<p>Patents can protect ideas of ways to implement something, but also has limitations that probably exclude BCI calculus as something that could be patented:</p>
<blockquote>
<p>Certain things can never be patented, regardless of how well they meet
these four standards. They include the elements, theoretical plans,
laws of nature, physical phenomena, and abstract ideas. So patenting
fire or the wheel is out, though some people have tried.</p>
</blockquote>
<p>(<a href="https://www.rocketlawyer.com/business-and-contracts/intellectual-property/patents/legal-guide/why-cant-you-patent-an-idea" rel="nofollow noreferrer">Source</a>)</p>
<p>Any patent received for BCI calculus prior to the year 2002 would also have expired bringing the subject matter of the patent into the public domain.</p>
<p>Also, since BCI calculus is something that has already been disclosed to the public, it is barred from patentability as a form of existing prior art. The existence of prior art would also prevent BI calculus and SK calculus from being patented.</p>
<h3>Trade Secret Protection Considered</h3>
<p>It also probably isn't possible to protect BCI calculus (or BI calculus or SK calculus) as a trade secret, because it has been disclosed to the public and hence isn't secret.</p>
<p>(<a href="https://www.wipo.int/tradesecrets/en/tradesecrets_faqs.html" rel="nofollow noreferrer">Source</a>)</p>
<h3>Trade Mark Protection Considered</h3>
<p>It might be possible to protect the name &quot;BCI calculus&quot; as a trademark under which services teaching people to use it are marketed, but that wouldn't protect the underlying system or concepts.</p>
<p>But there is a strong argument that this would not even qualify as a trademark because it is merely descriptive and does not have a &quot;secondary meaning&quot; particular to a particular user of the label for the system.</p>

copyright