Whenever I refer to an axiom set, unless stated otherwise, I presume the rule of detachemnt {|-Cab, |-a} -> b, where "a" and "b" constitute metavariables, and uniform substitution for variables.  The notation Da.b stands for a condensed detachment with "a" as the major premise and "b" as the minor premise.

Logical Formula of the Day: Cpp

Since this consists of a new thing I plan to do I guess I'll start small with one of the shortest tautologies in existence.  Cpp sometimes appears as an axiom in some propositional calculi.  For instance, A. N. Prior's formal logic lists {Cpp, CCCpqrCCrpCsCtp} as a basis for implicational propositional calculus.  It also appeared as an axiom in Peirce's axiom set for C-0 (0 meaning "falsum") classical logic {Cpp, CCpCqrCqCpr, CCpqCCqrCpr, C0p, CCCpqpp}.  In a natural deduction setting you can prove it as follows:

hypothesis 1 | p
Ci 1-1        2   Cpp

where "Ci" stands for conditional introduction.  For positive implicational propositional calculus with the axiom set {CpCqp, CCpCqrCCpqCpr} we can prove it in two condensed detachments.

axiom 1 CpCqp
axiom 2 CCpCqrCCpqCpr
D2.1   3 CCpqCpp
D3.1   4 Cpp

In classical logic with the basis {CCpqCCqrCCpr, CCNppp, CpCNpq} we can prove it in two steps also.

axiom 1 CCpqCCqrCpr
axiom 2 CCNppp
axiom 3 CpCNpq
D1.3   4 CCCNpqrCpr
D4.2   5 Cpp

Here's another proof from another axiom set:

axiom 1 CCCpqrCqr
axiom 2 CCCpqpp
D 1.2  3 Cpp

Actually, let's say we fix CCCpqrCqr as the first axiom 1.  Now replace the first p in CCCpqpp with CCpqp.  This yields CCCCCpqpqpp.  D1. CCCCCpqpqpp yields Cpp.  In any system sufficient for the conditional of classical logic, the variable "p" can get replaced by "CCpqp" since CCCpqpp is a theorem and CpCCpqp is also a theorem.  Consequently, we can iterate the process of replacing a "p" in any formula obtained in this way with CCCpqpp as the starting point and still obtain a theorem in classical logic, provided that we do NOT replace the last two "p's".  Each theorem obtained in this way serving as the minor premise along with CCCpqrCqr as the major premise will yield Cpp after a condensed detachment.

One can also prove it in Wajsberg-Lukasiewicz three-valued from the following basis as follows:

axiom 1 CpCqp
axiom 2 CCpqCCqrCpr
axiom 3 CCCpNppp
axiom 4 CCNpNqCqp
D2.1   5 CCCqprCpr
D5.3   6 Cpp

Or Lukasiewicz infinite valued-logic:

axiom     1 CpCqp
axiom     2 CCpqCCqrCpr
axiom     3 CCNpNqCqp
axiom     4 CCCpqqCCqpp
D2.1       5 CCCpqrCqr
D2.2       6 CCCCqrCprsCCpqs
D6.6       7 CCpCqrCCsqCpCsr
D5.4       8 CpCCpqq
C8.7       9 CCpCqrCqCpr
D9.1     10 CpCqq
D10.10 11 Cpp

Or from double suffixing CCpqCCqrCpr and the double negation laws CpNNp, CNNpp.

axiom   1 CCpqCCqrCpr
axiom   2 CpNNp
axiom   3 CNNpp
D1.2     4 CCNNpqCpq
D4.3     5 Cpp

Cpp as a major premise will only give you back the minor premise that you used when doing a condensed detachment.  On the other hand, Cpp as a minor premise has some interesting uses.  For instance D[CCpCqrCqCpr.Cpp] yields "modus ponens" CpCCpqq.  D[CCpCqrCCpqCpr.Cpp] yields CCCpqpCCpqq.  Perhaps more interestingly D[CCNpNqCCNpqq.Cpp] yields the law of Clavius CCNppp.  Additionally, if we consider D[CCpqCCqrCpr.a], and D[CCpqCCqrCpr.a] exists (which is equivalent to saying that "a" is a conditional), then D[D[CCpqCCqrCpr.a]].[Cpp] yields "a".

Basic Rule of inference "From any wff of the form "Cab", and any other wff of the form "a", we may infer "b"."

Axioms:

Group 1:

1 CpCqp  Conditional-Simplification abbreviatied C-s
2 CCpCqrCCpqCpr Conditional-Self-Distribution  abbreviated C-d.

Group 2:

3 CCpKqNqNp  Negation-in  abbreviated Ni
4 CCNpKqNqp  Negation-out abbreviated No

Group 3:

5 CKpqp Conjunction-out left abbreviated Kol
6 CKpqq Conjunction-out right abbreviated Kor
7 CpCqKpq  Conjunction-in abbreviated Ki

Group 4:

8   CpApq Alternation-in left abbreviated Ail
9   CpAqp Alternation-in right abbreviated Air
10 CCpqCCrqCAprq Alternation-out abbreviated Ao

Group 5:

11 CEpqCpq Equivlaence-out right abbreviated Eor
12 CEpqCqp Equivalence-out left abbreviated Eol
13 CCpqCCqpEpq Equivalence-in abbreviated Ei

Lemma 0: Epp. It can end up helpful to use this as a minor premise.

1 ! p hypothesis
2 Cpp 1-1 C-in
3 Epp 2, 2 E-in

Lemma 1: {Epq, p} => q. Nickname: E-detach right.

1 Epq assumption
2 p assumption
3 Cpq 1 E-out left
4 q 3, 2 C-out

Lemma 2: {Epq, q} => p. Nickname: E-detach left.

1 Epq assumption
2 q assumption
3 Cqp 1 E-out right
4 p 2, 3 C-out

Lemma 3: |- EEpqEqp. Nickname: E-commutation.

1 !-1 Epq hypothesis
2 !-1 Cqp 1 E-out right
3 !-1 Cpq 1 E-out left
4 !-1 Eqp 2, 3 E-in
5 CEpqEqp 1-4 C-in
6 !-2 Eqp hypothesis
7 !-2 Cpq 6 E-out right
8 !-2 Cqp 6 E-out left
9 !-2 Epq 7, 8 E-in
10 CEqpEpq 6-9 C-in
11 EEpqEqp 6, 10 E-in

Lemma 4: |- ApNp. Nickname: law of the excluded middle.

1 ! NApNp hypothesis
2 @ Np hypothesis
3 @ ApNp 2 A-in
4 @ KApNpNApNp 3, 1 K-in
5 ! p 2-4 N-out
6 ! ApNp 5 A-in
7 ! KApNpNApNp 6, 1 K-in
8 ApNp 1-7 N-out

Lemma 5: {Cpq, Nq} => Np. Nickname: Modus Tollens.

1 Cpq assumption
2 Nq assumption
3 ! p hypothesis
4 ! q 3, 1 C-out
5 ! KqNq 2, 4 K-in
6 Np 3-5 N-out

Lemma 6: {Np, NEpq} => q. Nickname: Lemma 6.

1 Np assumption
2 NEpq assumption
3 !-1 Nq hypothesis
4 @-1 p hypothesis
5 #-1 Nq hypothesis
6 #-1 KpNp 4, 1 K-in
7 @-1 q 5-6 N-out
8 !-1 Cpq 4-7 C-in
9 @-2 q hypothesis
10 #-2 Np hypothesis
11 #-2 KqNq 9, 3 K-in
12 @-2 p 10-11 N-out
13 !-1 Cqp 9-12 C-in
14 !-1 Epq 8, 13 E-in
15 !-1 KEpqNEpq 14, 2 K-in
16 q 3-15 N-out

Lemma 7: {p, q} => Epq. Nickname: Lemma 7.

1 p assumption
2 q assumption
3 !-1 p hypothesis
4 !-1 q 2 Repitition
5 Cpq 3-4 C-in
6 !-2 q hypothesis
7 !-2 p 1 Repitition
8 Cqp 6-7 C-in
9 Epq 5, 8 E-in


Lemma 8: |- EEpEqrEEpqr. Nickname: E-association.

1 !-1 EpEqr hypothesis
2 @-1 Epq hypothesis
3 #-1 p hypothesis
4 #-1 Eqr 3, 1, E-detach right
5 #-1 q 3, 2, E-detach right
6 #-1 r 5, 4, E-detach right
7 @-1 Cpr 3-6 C-in
8 #-2 Np hypothesis
9 #-2 Cqp 2 E-out left
10 #-2 Nq 8, 9, Modus Tollens
11 #-2 CEqrp 1 E-out right
12 #-2 NEqr 8, 11, Modus Tollens
13 #-2 r 10, 12 Lemma 6
14 @-1 CNpr 8-13 C-in
15 @-1 ApNp law of the excluded middle
16 @-1 r 7, 14, 15 A-out
17 !-1 CEpqr 2-16 C-in
18 @-2 r hypothesis
19 #-3 p hypothesis
20 #-3 Eqr 19, 1, E-detach right
21 #-3 q 18, 20, E-detach left
22 @-2 Cpq 19-21 C-in
23 #-4 q hypothesis
24 #-4 CEqrp 1 E-out right
25 #-4 Eqr 23, 18, lemma 7
26 #-4 p 25, 24 C-out
27 @-2 Cqp 23-26 C-in
28 @-2 Epq 22, 27 E-in
29 !-1 CrEpq 18-28 C-in
30 !-1 EEpqr 17, 29 E-in
31 CEpEqrEEpqr 1-30 C-in
32 !-2 EEpqr hypothesis
33 @-3 p hypothesis
34 #-5 q hypothesis
35 #-5 Epq 33, 34 lemma 7
36 #-5 r 35, 32, E-detach right
37 @-3 Cqr 34-36 C-in
38 #-6 r hypothesis
39 #-6 Epq 38, 32, E-detach left
40 #-6 q 39, 33, E-detach right
41 @-3 Crq 38-40 C-in
42 @-3 Eqr 37, 41 E-in
43 !-2 CpEqr 33-42 C-in
44 @-4 Eqr hypothesis
45 #-7 r hypothesis
46 #-7 q 44, 45, E-detach left
47 #-7 Epq 45, 32, E-detach left
48 #-7 p 46, 47, E-detach right
49 @-4 Crp 45-48 C-in
50 #-8 Nr hypothesis
51 #-8 Cqr 44, E-out left
52 #-8 Nq 50, 51, Modus Tollens
53 #-8 CEpqr 32 E-out left
54 #-8 CEqpr 53, E-commutation
55 #-8 NEqp 50, 54, Modus Tollens
56 #-8 p 52, 55, lemma 6
57 @-4 CNrp 50-56 C-in
58 @-4 ArNr law of the excluded middle
59 @-4 p 49, 57, 58 A-out
60 !-2 CEqrp 44-59 C-in
61 !-2 EpEqr 43, 60 E-in
62 CEEpqrEpEqr 32-61 C-in
63 EEpEqrEEpqr 31, 62 E-in

Some Questions on a Class of Derived Axiom Sets

Define Σ as an abbreviation for CCpqCCqrCpr.

Suppose we have a logic L where all theses have the conditional C as their principal connective, the rules of modus ponens (more generally we could consider any rule of detachment... but modus ponens should suffice here) and uniform substitution, axiom set {α, ..., ω}, Σ is a thesis of L, and Cpp is a thesis of L. Since condensed detachment "D" is a derivable rule of a inference, it follows that

Theorem: {DΣ.α, ..., DΣ.ω, Cpp} is also an axiom set for L
(proof hint: DΣ.x exists, where x is any thesis, and D.DΣ.x.(Cpp) gives you...).

There exist plenty of corollary theorems also where the alternative axiom set gets obtained by the commuted variant of Σ; CCpqCCrpCrq (or CCqpCCpqCpr), or one of the simplest generalizations of Σ CCpqCCqrCCrsCps (Sorites), further generalizations of Σ than Sorites (e. g. CCpqCCqrCCrsCCstCpt) as well as commuted variants of Sorites and its generalizations, where a commuted variant V of a thesis T, means that we can obtain thesis V from T with the help of CCpCqrCqCpr and conversely.

Question 1: The theorem stated here clearly bears a relation to BCI logic. Does there exist a name for the theorem I've stated above?

I'll try and explain this as follows:

An example of how things work here to clarify the questions:
{1-CCpqCCqrCpr or Σ, 2-CCNppp, 3-CpCNpq} is a known axiom set for two-valued propositional calculus which satisfies independence. To make things clear, Cpp and the other axioms of {DΣ.α, ..., DΣ.ω} in this case can get derived as follows:

1 CCpqCCqrCpr axiom
2 CCNppp axiom
3 CpCNpq axiom

   1 q/CNpq * C3-4

4 CCCNpqrCpr (this is DΣ.3)

   4 q/p, r/p * C2-5

5 Cpp

   1 p/Cpq, q/CCqrCpr, r/s * C1-6

6 CCCCqrCprsCCpqs (this is DΣ.1)

   1 p/CNpp, q/p, r/q *C2-7

7 CCpqCCNppq (this is DΣ.2)

Now the theorem here tells us that {CCCCqrCprsCCpqs, CCpqCCNppq, CCNpqrCpr, Cpp} is also an axiom set for two-valued propositional calculus.

Proof: In this example case since each thesis got derived from the axioms of {1-CCpqCCqrCpr or Σ, 2-CCNppp, 3-CpCNpq}, we only need show that from {1'-CCCCqrCprsCCpqs, 2'-CCpqCCNppq, 3'-CCNpqrCpr, 4'-Cpp} we can derive each of the axioms of {1-CCpqCCqrCpr or Σ, 2-CCNppp, 3-CpCNpq} as follows.

1' CCCCqrCprsCCpqs axiom
2' CCpqCCNppq axiom
3' CCNpqrCpr axiom
4' Cpp axiom

    1' s/CCqrCpr * 4 p/CCqrCpr-1 

1 CCpqCCqrCpr

     2' q/p * 4-2

2 CCNppp

     3' r/CNpq * 4 p/CNpq-3

3 CpCNpq
Example of question 2: {1-CCpqCCqrCpr or Σ, 2-CCNppp, 3-CpCNpq} satisfies independence. So, does the set {CCCCqrCprsCCpqs, CCpqCCNppq, CCNpqrCpr} also satisfy independence? Added: Mace4 has given me 3 models which taken together indicate that {CCCCqrCprsCCpqs, CCpqCCNppq, CCNpqrCpr} satisfies independence.

Question 2 more generally: If the axioms {α, ..., ω} satisfy independence, will the axioms of {DΣ.α, ..., DΣ.ω} also satisfy independence? Will the theses belonging to the set obtained from {α, ..., ω} using generalizations of Σ, or commutated variants of those generalizations, or the commuted variant of Σ also satisfy independence?

To perhaps aid in comprehensibility, I'll present the following proof of the theorem above. I'll let letters in bold stand for metalinguistic variables:

Proof: First, all members of {DΣ.α, ..., DΣ.ω} will exist, since all members of the original postulate set {α, ..., ω} come as conditionals, and the antecedent of Σ comes as the conditional Cpq. Next, each instance of DΣ.k, where k belongs to {α, ..., ω} will come as the consequent of a substitution instance of Σ. So, we have a class of formulas of the form CC q rC p r where r does not appear as a subformula in q, nor does r appear as a subformula in p, since r did not appear in the antecedent of Σ. Thus, a unifier of Cpp and the antecedent of whatever formula we choose DΣ.k at a particular time will exist. The unifier will have form C q q. Consequently, D.DΣ.k.(Cpp) will have form C p q. This form matches the original form for k. Since k consisted of an arbitrary name for a variable belonging to {α, ..., ω}, it follows that {DΣ.α, ..., DΣ.ω, Cpp} will also qualify as an axiom set for the same logic.

It might come as some interest to note that this proof implies a uniform method to obtain an unending sequence of alternative postulate sets for a logic under certain conditions. The proofs of the corollaries will also give us another method. Using CpCqp in place of how Σ can get used also consists of another method of obtaining alternative axioms.

Does Cantorian Set Theory Have Limitations for Evaluating the Size of Sets?      Cantorian set theory would have us believe that all countable infinites are the same size (or at least, that's the conceptual metaphor commonly used.)  I will try and argue that this makes little sense, in at least one significant case.      Let's look at classical propositional logic from the semantic point-of-view.  There exist three classes of well-formed formulas (hereafter formulas): tautologies, contradictions, and contingencies.  The class of tautologies is well known to be countably infinite, given countably many infinite variables, so via negation the class of contradictions is also countably infinite.  Though I don't know how to demonstrate it, the class of contingencies also seems countably infinite.      But, it does not make much sense to believe contingencies as big as tautologies.  Nor does it make much sense to believe contingencies as big as contradictions.  I will say that I do believe that a skilled set theorist could sufficiently describe two bijections: one between contingencies and tautologies, and another between contingencies and contradictions (and negation gives us a bijection between tautologies and contradictions).      However, if we had a machine which gave us propositional formulas at random, I assert that the class of contingencies would have the greatest frequency of occurrence after a sufficiently large amount of time (if needed, we might postulate 10 billion propositional formulas as enough.  My idea here goes that we have enough propositional formulas so that it becomes all too unlikely that the frequency distribution will change much when getting more propositional formulas at random.)  This would seem, at the very least, to imply the class of contingencies as greater than the class of contradictions.  It also would imply the class of contingencies greater than the class of tautologies.  Of course, that does not say that the frequency of contingencies obtained at random would come as greater than the class of tautologies and the class of contradictions combined.  That said, truth tables might seem to suggest this also, since there exist 14 truth functions of two variables which are contingencies, but only 2 which are either tautologies or contradictions, only 2 truth functions of three variables which are tautologies or contradictions and the rest of the truth functions of three variables are contingencies, and in general only 2 truth functions of n-variables and the rest of the truth functions of n-variables are contingencies.

The Inconvenience of the Lipid Hypothesis for Many Vegan Advocates

     The lipid hypothesis of atherosclerosis says that high serum cholesterol levels come as a causative factor in the development of atherosclerosis.  In other words, there exists a significant, real, statistical relationship between high cholesterol levels in the blood and atherosclerosis.  It does not say that high blood cholesterol levels (hypercholesterolemia) causes atherosclerosis and consequent atherosclerotic events (heart attacks, strokes, angina, etc.) necessarily, but rather that blood cholesterol levels come as a causative factor.  Since usually foods high in saturated fats and dietary cholesterol can raise serum cholesterol levels to a high level, or keep them at a high level, and since plant foods are not generally high in saturated fat and have no cholesterol in them, it would seem to follow that the lipid hypothesis would work out well for those who wish to advocate for a vegan diet.

      However, the easiest and simplest way to advocate for a vegan diet comes as to advocate for a diet that qualifies as vegan while having as few other restrictions as possible.  But, a diet developed with the lipid hypothesis in mind does not allow for this.  First off, trans-fatty foods can significantly elevate blood cholesterol levels.  Thus, the lipid hypothesis makes french fries, many brands of potato chips, foods with partially hydrogenated vegetable oil, and many other conventional vegan snack food products all potentially, if not actually, poor choices for your heart, your brain, and the blood flow in your body.  Second, foods high in simple sugars can significantly elevate blood cholesterol levels.  So, via the lipid hypothesis many, if not most brands of soft drinks, energy drinks, foods with high fructose corn syrup, fruit juices, etc. can elevate serum cholesterol levels.  Third, refined sources of carbohydrates, such as white bread, semolina pasta, white rice, etc. can elevate serum cholesterol levels.  Fourth, many oils due in part to their concentrated level of saturated fat can elevate serum cholesterol levels.  There no doubt exist many more inconvenient "diet-heart" ideas in accordance with the lipid hypothesis, which do NOT work out as convenient for those that would advocate veganism, since they end up implying potential problems with some vegan diets.  It is by no means impossible that, were they to understand it and its implications, some vegans might intensely dislike the lipid hypothesis for this very reason. 

Caldwell Esselstyn's Study and Its Follow-Up

Harriet Hall here writes "

Caldwell Esselstyn

Esselstyn did an uncontrolled interventional study of patients with angiographically documented severe coronary artery disease who were not hypertensive, diabetics, or smokers. He wanted to test how effective one physician could be in helping patients achieve a total cholesterol level of 150 mg/dL or less, and what effect maintaining that level would have on coronary disease. Patients agreed to follow a plant-based diet with <10% of calories derived from fat. They were asked to eliminate oil, dairy products (except skim milk and no-fat yogurt), fish, fowl, and meat. They were encouraged to eat grains, legumes, lentils, vegetables, and fruit. Cholesterol-lowering medication was individualized.

There were originally 24 patients: 6 dropped out early on, 18 maintained the diet, one of these 18 died of an arrhythmia and 11 completed a mean of 5.5 years followup. Repeat angiography showed that of 25 coronary artery lesions, 11 regressed and 14 remained stable. At 10 years, 11 patients remained: 6 continued the diet and had no further coronary events; 5 resumed their pre-study diet and reported 10 coronary events.

In a 12 year followup report, the 6 who had maintained the diet at 10 years and the 5 who had gone off it and had coronary events had apparently somehow morphed into 17 patients who had remained adherent to the diet and who had had no coronary events. I couldn’t understand the discrepancy in numbers; perhaps readers can explain it to me if I missed something."

First off, it may seem that another number doesn't match in that the follow-up study says that the original study had 24 patients, while the first study above mentions 22.  But, the first study says "The study included 22 patients with angiographically documented, severe coronary artery disease that was not immediately life threatening."  So, there isn't any necessarily contradiction in numbers there.

Second, the first study says "Of the 22 participants, 5 dropped out within 2 years, and 17 maintained the diet, 11 of whom completed a mean of 5.5 years of follow-up."  This does NOT say that 11 of them, implying that 6 quit the diet, but that 11 of them engaged in follow-up.  As the follow-up study says: "At 5 years, 11 of these patients underwent angiographic analysis by the percent stenosis method, which demonstrated disease arrest in all 11 (100%) and regression in 8 (73%)."  So, the 11 who completed the follow-up appears to refer to the patients who completed angiograms.  6 of the 17 didn't complete follow-up by not engaging in angiograms.

Another discrepancy in the numbers may appear to exist in that the follow study says "The remaining 18 patients adhered to the study diet and medication for 5 years.", while the first indicates 17 maintaining the diet.  But, that 17 number appears in the passage about 5.5 years of follow-up and the second study says "One patient admitted to the study with <20% left ventricular output died from a ventricular arrhythmia just weeks after the 5-year follow-up angiogram had confirmed disease regression.  Autopsy revealed no myocardial infarction. "  So, there doesn't exist any contradiction there.
A further discrepancy may seem to exist in that the first study says "Among the 11 remaining patients after 10 years, six continued the diet and had no further coronary events, whereas the five dropouts who resumed their prestudy diet reported 10 coronary events."  This might seem like 5 more patients (5 out of the 22 who had angiographically documented coronary artery disease) dropped out of the study, which would contradict 17 patients in the follow study.  However, the five dropouts here probably refers to the original 5 of the original 22 angiographically documented patients who dropped out of the dietary program withing 2 years.  Esselstyn may have only mentioned 6 continuing the diet here, because he was only able to verify 6 still on the dietary plan when he published this.  The other 11 (not all angiographically documented patients) he may not have verified as still on the dietary plan until later... and that's perhaps part of the reason why he provided the update to the study in the first place.

                                                          Mimic Operations

The author writes informal proofs in the following.  The author uses Reverse Polish Notation also.

Suppose we have an infinite set S, and a operation "*". A mimic operation *' of * consists of an operation on S such that for all but a finite number of points a₁, ..., a_n *=a₁, ..., a_n *'. So, for a unary operation F on the natural numbers N, a mimic operation F' satisfies xF=xF' except for a finite set of natural numbers {a₁,...,a_n}. For a binary operation G on N, a mimic of B, B' satisfies xyB=xyB' except for a finite number of pairs of natural numbers {(a, b)₁,...,(a, b)_n} and so on.

Theorem 1: For any n-ary operation O on an infinite set, there exists an infinity of mimic operations, O'.

Proof: An n-ary operation O can get defined by a set of n 1 + tuples. E. G. the binary operation of addition can get defined by the triples (2, 4, 6), (1, 1, 2) such that for (x, y, z), xy+=z. Now consider the set of such tuples T for O. Vary the n 1 + part of the tuples in T for some finite number of tuples. This forms a set of tuples T' which describes an n-ary operation distinct from O in but a finite number of points, and thus forms a mimic operation O'. Note that since there exist an infinity of tuples belonging to T, there exists an infinity of ways to form T' given T. Thus, given an n-ary operation O, there exists an infinity of mimic operations O'.

Define a k-mimic operation M^k of an operation M as an operation which differs from M by but k points, where both operations operate over an infinite set. In other words, for an operation M_n of arity n with mimic M^k_n, a₁...a_nM_n=a₁...a_nM^k_n, except for k tuples {(b₁, ..., b_n), ..., (l₁, ..., l_n)} where this equality fails.

Theorem 2: For any n-ary operation O on an infinite set, there exists an infinity of k mimic operations.

Proof: Since given the set of tuples T for an operation O there exist an infinity of ways to form T' (see the proof of Theorem 1). From T' we have an infinity of k-mimic operations O' of O.

Theorem 3: If a binary operation O satisfies commutativity, then if xxO=xxO¹,  O¹ does not satisfy commutativity.

Proof: If O¹ is a 1-mimic of O, then xyO=xyO¹ for all but one pair (a, b). Suppose that for all x and y, if xyO=yxO, and xyO¹=yxO¹.  Suppose that (a, b) consists of the point where O and O¹ differ, by say letting abO=c, and letting abO¹=d, where d does not equal c.  It follows then that abO¹=d=baO¹ by supposition of commutation for O¹. But, this implies that the pair (b, a) consists of a second point where O and O¹ differ, which contradicts the supposition of O¹ as a 1-mimic of O.  Consequently, the theorem follows and if O¹ satisfies commutation it at least consists of a 2-mimic.

Conjecture: If Oⁿ and O^k are mimics of operation O, then Oⁿ and O^k are mimics of each other.

The author thinks theorem 3 generalizes to o-mimics, where "o" indicates any positive odd number. The author also thinks that converse of the theorem holds in that if a 1-mimic does satisfy commutation, the operation which it mimics will not satisfy commutation also.  Do there exist any relationships (or lack thereof) between an associative operation and its mimics?  Between an idempotent operation and its mimics?