Propositional calcolus, ontolog

[bertanimauro]

From APUPA created by bertanimauro: bertanimauro/APUPA#10

https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_Part_1#Casual_Introduction

[bertanimauro]

Cf: All Process, No Paradox • 8
http://inquiryintoinquiry.com/2021/03/16/all-process-no-paradox-8/

| These are the forms of time, which imitates eternity
| and revolves according to a law of number.
|
| Plato • Timaeus 38 A
| Benjamin Jowett (trans.)

Re: Laws of Form ( https://groups.io/g/lawsofform/topic/81284216 )

Dear Seth, James, Lyle, All ...

Nothing about calling time an abstraction makes it a nullity.
I'm too much a realist about mathematical objects to ever mean
that. As a rule, on the other hand, I try to avoid letting
abstractions leave us so absent-minded as to forget the concrete
realities from which they are abstracted. Keeping time linked to
process, especially the orders of standard process we call “clocks”,
is just part and parcel of that practice.

Synchronicity being what it is, this very issue came up just last night in
a very amusing Facebook discussion about “windshield wipers slappin' time …”
( https://www.youtube.com/watch?v=Mc7qmE5CiuY )

At any rate, this thread is already moving too fast for the pace
I keep these days but maybe I can resolve remaining confusions about
the game afoot by recycling a post I shared to the old Laws of Form list.
This was originally a comment on Lou Kauffman's blog back when he first
started it. Sadly, he wrote only a few more entries there in the time since.

Re: Lou Kauffman
https://homepages.math.uic.edu/~kauffman/
::: Iterants, Imaginaries, Matrices
http://kauffman2013.wordpress.com/2013/12/27/iterants-imaginaries-and-matrices/

As serendipity would have it, Lou Kauffman, who knows a lot about
the lines of inquiry Charles Sanders Peirce and George Spencer Brown
pursued into graphical syntaxes for logic, just last month opened a blog
and his very first post touched on perennial questions of logic and time —
Logos and Chronos — puzzling the wits of everyone who has thought about
them for as long as anyone can remember. Just locally and recently
these questions have arisen in the following contexts:

[Links omitted here. Please see the blog post linked above for the list.]

Kauffman's treatment of logic, paradox, time, and imaginary truth values
led me to make the following comments I think are very close to what I'd
been struggling to say before.

Let me get some notational matters out of the way before continuing.

I use B for a generic 2-point set, usually {0, 1} and typically but
not always interpreted for logic so that 0 = false and 1 = true.
I use “teletype” parentheses (...) for negation, so that (x) = ¬x
for x in B. Later on I’ll be using teletype format lists
(x_1, ..., x_k) for minimal negation operators.

[ See https://oeis.org/wiki/Minimal_negation_operator ]

As long as we’re reading x as a boolean variable x in B
the equation x = (x) is not paradoxical but simply false.
As an algebraic structure B can be extended in many ways
but it remains a separate question what sort of application,
if any, such extensions might have to the normative science
of logic.

On the other hand, the assignment statement x := (x) makes perfect sense
in computational contexts. The effect of the assignment operation on the
value of the variable x is commonly expressed in time series notation as
x' = (x) and the same change is expressed even more succinctly by defining
dx = x' − x and writing dx = 1.

Now suppose we are observing the time evolution of a system X
with a boolean state variable x : X → B and what we observe is
the following time series.

Table. Time Series 1 (also attached)
https://inquiryintoinquiry.files.wordpress.com/2021/03/all-process-no-paradox-e280a2-2-e280a2-time-series-1.png

Computing the first differences we get:

Table. Time Series 2 (also attached)
https://inquiryintoinquiry.files.wordpress.com/2021/03/all-process-no-paradox-e280a2-2-e280a2-time-series-2.png

Computing the second differences we get:

Table. Time Series 3 (also attached)
https://inquiryintoinquiry.files.wordpress.com/2021/03/all-process-no-paradox-e280a2-2-e280a2-time-series-3.png

This leads to thinking of the system X as having an extended state
(x, dx, d²x, ...), and this additional language gives us the facility
of describing state transitions in terms of the various orders of
differences. For example, the rule x' = (x) can now be expressed
by the rule dx = 1.

The following article has a few more examples along these lines.

Differential Analytic Turing Automata (DATA)
https://oeis.org/wiki/Differential_Analytic_Turing_Automata_%E2%80%A2_Overview

Resources

Differential Logic and Dynamic Systems
https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Overview

Regards,

Jon

[bertanimauro]

rewrite compound sobject in first order logic

with and , not, a,b ,c,e,f as postulate. G is a compound subject

Dear Jon,
To prove but the idea is good:
d= a b c
g= e f
h = d g
l= ( d ( h))
m =(g (h))
n = l m
g = h n
Regards
Mauro
Nascondi testo citato

Il gio 25 mar 2021, 21:23 Mauro Bertani [email protected] ha scritto:
Dear Jon,
d=(a b c)
f= (d e)
h (d (f))
Maybe
Regards
Mauro

Il gio 25 mar 2021, 21:14 Mauro Bertani [email protected] ha scritto:
Dear Jon,
Maybe
"a b c"
Where a,b,c is the part
"(a b c)"
The negation of the whole
Regards
Mauro

Il gio 25 mar 2021, 21:02 Jon Awbrey [email protected] ha scritto:
Dear Mauro,

Sorry, I'm not sure what you're saying. Here I'm using
a parenthesized rendition of the forms Spencer Brown and
Peirce used in their logical graphs. In contexts where
I have better formatting I use a different typeface for
the logical parentheses to avoid confusing them with the
ordinary sort.

For example, in the so-called existential interpretation
Peirce eventually settled on for propositional calculus:

" " is "true".

"( )" is "false".

"a b" is "a and b"

"(a)" is "not a"

"((a)(b)) is "a or b"

"(a(b))" is "a implies b"

etc.

See the following article:

https://inquiryintoinquiry.com/2008/09/19/logical-graphs-2/

Regards,

Jon

On 3/25/2021 3:38 PM, Mauro Bertani wrote:

Dear Jon,
Maybe (a) = 0 = false.
a = n = true?
Regards
Mauro

[bertanimauro]

In universal algebra, an algebra (or algebraic structure) is a set A together with a collection of operations on A. An n-ary operation on A is a function that takes n elements of A and returns a single element of A. Thus, a 0-ary operation (or nullary operation) can be represented simply as an element of A, or a constant, often denoted by a letter like a. A 1-ary operation (or unary operation) is simply a function from A to A, often denoted by a symbol placed in front of its argument, like ~x. A 2-ary operation (or binary operation) is often denoted by a symbol placed between its arguments, like x ∗ y. Operations of higher or unspecified arity are usually denoted by function symbols, with the arguments placed in parentheses and separated by commas, like f(x,y,z) or f(x1,...,xn). Some researchers allow infinitary operations, such as {\displaystyle \textstyle \bigwedge {\alpha \in J}x{\alpha }}\textstyle \bigwedge {\alpha \in J}x{\alpha } where J is an infinite index set, thus leading into the algebraic theory of complete lattices. One way of talking about an algebra, then, is by referring to it as an algebra of a certain type {\displaystyle \Omega }\Omega , where {\displaystyle \Omega }\Omega is an ordered sequence of natural numbers representing the arity of the operations of the algebra.

Ref: https://en.m.wikipedia.org/wiki/Universal_algebra

[bertanimauro]

In mathematics, especially in the fields of universal algebra and graph theory, a graph algebra is a way of giving a directed graph an algebraic structure. It was introduced in (McNulty & Shallon 1983), and has seen many uses in the field of universal algebra since then.

Let D = (V, E) be a directed graph, and 0 an element not in V. The graph algebra associated with D has underlying set {\displaystyle V\cup {0}}V\cup {0}, and is equipped with a multiplication defined by the rules

xy = x if {\displaystyle x,y\in V}x,y\in V and {\displaystyle (x,y)\in E}{\displaystyle (x,y)\in E},
xy = 0 if {\displaystyle x,y\in V\cup {0}}{\displaystyle x,y\in V\cup {0}} and {\displaystyle (x,y)\notin E}{\displaystyle (x,y)\notin E}

Ref:
https://en.m.wikipedia.org/wiki/Graph_algebra
Kelarev, A.V.; Sokratova, O.V. (2001), "Directed graphs and syntactic algebras of tree languages", J. Automata, Languages & Combinatorics, 6 (3): 305–311, ISSN 1430-189X, MR 1879773

[bertanimauro]

After the operations have been specified, the nature of the algebra is further defined by axioms, which in universal algebra often take the form of identities, or equational laws. An example is the associative axiom for a binary operation, which is given by the equation x ∗ (y ∗ z) = (x ∗ y) ∗ z. The axiom is intended to hold for all elements x, y, and z of the set A.

[bertanimauro]

Von KARGER, B. (1998). Temporal algebra. Mathematical Structures in Computer Science, 8(3), 277-320. doi:10.1017/S0960129598002540

Abstract
We develop temporal logic from the theory of complete lattices, Galois connections and fixed points. In particular, we prove that all seventeen axioms of Manna and Pnueli's sound and complete proof system for linear temporal logic can be derived from just two postulates, namely that ([oplus ], &[ominus ]tilde;) is a Galois connection and that ([ominus ], [oplus ]) is a perfect Galois connection. We also obtain a similar result for the branching time logic CTL.

A surprising insight is that most of the theory can be developed without the use of negation. In effect, we are studying intuitionistic temporal logic. Several examples of such structures occurring in computer science are given. Finally, we show temporal algebra at work in the derivation of a simple graph-theoretic algorithm.

This paper is tutorial in style and there are no difficult technical results. To the experts in temporal logics, we hope to convey the simplicity and beauty of algebraic reasoning as opposed to the machine-orientedness of logical deduction. To those familiar with the calculational approach to programming, we want to show that their methods extend easily and smoothly to temporal reasoning. For anybody else, this text may serve as a gentle introduction to both areas.

[bertanimauro]

Il gio 25 mar 2021, 22:51 Mauro Bertani [email protected] ha scritto:
Dear Jon,
To prove but the idea is good:
d= a b c
g= e f
h = d g
l= ( d ( h))
m =(g (h))
n = l m
g = h n

This imply

a b c d e = l m
= (d (h)) (g(h))
= (a b c( d g)) (e f ( d g))
= (a b c ( a b c e f)) ( e f ( a b c e f))

It's too late to think good. There are definitely too much error
Regards
Mauro

[bertanimauro]

https://inquiryintoinquiry.files.wordpress.com/2019/07/box-q-que-pqq.jpg

: Animated Logical Graphs • 19
https://inquiryintoinquiry.com/2019/07/11/animated-logical-graphs-19/

All,

We have encountered the question of how to extend our
formal calculus to take account of operator variables.

In the days when I scribbled these things on the backs of
computer punchcards, the first thing I tried was drawing big
loopy script characters, placing some inside the loops of others.
Lower case alphas, betas, gammas, deltas, and so on worked best.
Graphics like these conveyed the idea that a character-shaped
boundary drawn around another space can be viewed as absent or
present depending on whether the formal value of the character
is unmarked or marked. The same idea can be conveyed by
attaching characters directly to the edges of graphs.

Here is how we might suggest an algebraic expression of the form “(q)”
where the absence or presence of the operator “( )” depends on the value
of the algebraic expression “p”, the operator “( )” being absent whenever p
is unmarked and present whenever p is marked.

Figure 1. Cactus Graph (q)_p = {q,(q)}
https://inquiryintoinquiry.files.wordpress.com/2019/07/box-q-que-pqq.jpg

It was obvious from the outset this sort of tactic would need
a lot of work to become a usable calculus, especially when it
came time to feed those punchcards back into the computer.

Regards,

Jon

p = isAChimicalProfessor(x)

[bertanimauro]

Rivedere figura ranganathan

In logic 9 combinazioni:
a = isProfessorOfLaw(x) and isBrillantProfessor(x)
b = isProfessorOfLaw(x) and isMediocreProfessor(x)

...

[bertanimauro]

So the power of negative thinking is NAND if all the variable are true the solution is false or say in an other way just a false variable make true the solution. In NNOR if all variable are false the solution is trueor in an other way if just a variable is true the solution is false.
So:
a b NAND NNOR (NNOR)
V V F F V
V F V F V
F V V F V
F F V V F

[bertanimauro]

dear Helmut,
in an old post of Jon, he said:
" " is "true".

"( )" is "false".

"a b" is "a and b"

"(a)" is "not a"

"((a)(b)) is "a or b"

"(a(b))" is "a implies b"

etc.
By these we can append:
NAND = (a b)
NNOR= (a)(b)
and then we can construct the duality, with the sign /=, like this:
NAND/=(NNOR)
maybe I think.
regards
Mauro

[bertanimauro]

Paradisaical Logic

Negative operations (NOs), if not more important than
positive operations (POs), are at least more powerful
or generative, because the right NOs can generate all
POs, but the reverse is not so.

Which brings us to Peirce’s amphecks, NAND and NNOR,
either of which is a sole sufficient operator for
all boolean operations.

Amphecks ( https://oeis.org/wiki/Ampheck )
NAND ( https://oeis.org/wiki/Logical_NAND )
NNOR ( https://oeis.org/wiki/Logical_NNOR )

In one of his developments of a graphical syntax for logic,
that described in passing an application of the Neither-Nor
operator, Peirce referred to the stage of reasoning before
the encounter with falsehood as “paradisaical logic, because
it represents the state of Man’s cognition before the Fall.”

Here’s a bit of what he wrote there —

C.S. Peirce • Relatives of Second Intention
https://inquiryintoinquiry.com/2012/04/07/c-s-peirce-relatives-of-second-intention/

Resources

Logic Syllabus ( https://oeis.org/wiki/Logic_Syllabus )

Peirce’s 1870 Logic Of Relatives
https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Overview

Regards,

Jon

[bertanimauro]

lambda calcolus

Ref: https://en.m.wikipedia.org/wiki/Lambda_calculus

[bertanimauro]

Cf: Animated Logical Graphs • 67
http://inquiryintoinquiry.com/2021/03/27/animated-logical-graphs-67/

Re: Differential Propositional Calculus • Discussion 4
https://inquiryintoinquiry.com/2021/03/20/differential-propositional-calculus-discussion-4/

Re: Laws of Form • Lyle Anderson
https://groups.io/g/lawsofform/message/198

Re: Peirce List • Mauro Bertani
https://list.iupui.edu/sympa/arc/peirce-l/2021-03/msg00134.html

Dear Lyle,

Yes, the ability to work with functions as “first class citizens”,
as we used to say, is one of the things making lambda calculus at
the theoretical level and Lisp at the practical level so nice.
All of which takes us straight into Curry-Howard-ville ...

In other words, the Curry–Howard correspondence is the observation that two families of seemingly unrelated formalisms—namely, the proof systems on one hand, and the models of computation on the other—are in fact the same kind of mathematical objects.

If one abstracts on the peculiarities of either formalism, the following generalization arises: a proof is a program, and the formula it proves is the type for the program. More informally, this can be seen as an analogy that states that the return type of a function (i.e., the type of values returned by a function) is analogous to a logical theorem, subject to hypotheses corresponding to the types of the argument values passed to the function; and that the program to compute that function is analogous to a proof of that theorem. This sets a form of logic programming on a rigorous foundation: proofs can be represented as programs, and especially as lambda terms, or proofs can be run.

Ref: https://en.m.wikipedia.org/wiki/Curry%E2%80%93Howard_correspondence#CITEREFDe_Groote1995

[bertanimauro]

https://www.ebi.ac.uk/chebi/searchId.do?chebiId=CHEBI:52625

[bertanimauro]

Thanks Schmidt,
I have read Łukasiewicz. I would replace his axiom [1]:
CCCpqrCCrpCsp
with these three axioms:

1. (p&&q)->(p->q)
2. (p&&q)->(p||q)
3. p->!p->p
   but I have some problems with sentence like this:
   p q r ((r ∧ (p ∧ q)) → ¬(¬(p → ¬q) → r))
   F F F T
   F F T T
   F T F T
   F T T T
   T F F T
   T F T T
   T T F T
   T T T F
   is like as when there is a negation in the conseguent the antecedent p&&q&&r not could be positive.
   so the third axiom would be something similar to:
   3a) (p&&F)->!p

Regards
Mauro

[1] Łukasiewicz, Jan. “The Shortest Axiom of the Implicational Calculus of Propositions.” Proceedings of the Royal Irish Academy. Section A: Mathematical and Physical Sciences, vol. 52, 1948, pp. 25–33. JSTOR, www.jstor.org/stable/20488489. Accessed 23 Apr. 2021.

[bertanimauro]

In tutti i sistemi dipende dalla verità di a

[bertanimauro]

Dear Helmut,
I go back to my writings of last year and I reread the reasoning. I keepin a whole all the last two mail:

1 MAIL:
Last year I read part of the book of Peano [1]. In this book Peano explains the state of art of logic in 1888. He explains in this way the rudimental concept of implication:
[link to pag 9 of book]
a < b or b > a the class [proposition] defined by the condition a is part of by those defined by b, or in another way a has as a consequence b
a = b if a is true and also b, and viceversa
a ^ b the condition assuming that both a and b are true
a U b the condition assuming that or a or b are true
(a) the condition that we obtain negating a
F the absurd condition
T the identical condition

Than the book explains the calculus of proposition and terminates with this 4 type of proposition:
[link to pag 14 of book]
I) All a are b
II) No a is b
III) Some a is b
IV) Some a is not b
And he transforms the first proposition in
a ^ (b) = F
that is more similar at
(a(b)) the cactus formula for implication
Peano named these propositions in this way:
The I) and II) are Universal. The III) and IV) that are negations of universal preposition, he named them particular. The I) and the III) that contain an even number of negations, he named them proposition affirmative. The II) and IV) that contains an odd number of negations, he named them negative.

2 MAIL:
Dear Helmut,
I'm not sure to have understood what you have said.
Let's:
A={n: n=4i con i (1..infinity)}
B={n: n=2i con i (1..infinity)}

I see that all a are also b. But at one moment I will see that there are some b, like for example 6,that are not a. So the not existence of a that are not b and the existence of b that are not a, drive me to conclude that A is included in B and A implies B. So if..then come after negation.
It's right?

NOW:
so we can say that not only a->b is All A are B but also Some B is not A. We can write:
[book pag 14]
Some B is not A: ([B ^ A] =F) remember that the square brackets are separation and the brackets "()" are negation.
Now we can write:
(a^(b)) ^ (b^a)
This is a new concept of implication: we can prove say that is included in implication concept more abstract:
([(a^(b)) ^ (b^a)] ([(a(b))]))
I rewrite this in another notation. Put the sign "->" as implication:
((a^(b)) ^ (b^a)) -> (a->b)
((a->b)^(b^a))->(a->b)
This is a tautology

In few word: implication is: All A are B and some B are not A

regards
Mauro

[bertanimauro]

Forse potrebbe essere utile avere due concetti con stessa chiave ma due principi costruttivi diversi. Vedi #5 (comment) .
Così sono in grado di parlare di un "quadro da quattro soldi" o di un "pezzo di carta straccia" per rappresentare il concetto un "quadro di nessun valore".
È la seconda volta che ritorna la non necessità dell'unicità della chiave come base al pensiero pragmatico

[bertanimauro]

Hi List,
this is my opinion. I use an example:
Marco P 01
Lara S 02
Mauro P 03
Giovanna P 04

Male 00
Female 01

father 00
mather 01
brother 02
sister 03
son 04
wife 05
husband 06

Marco is a male 0100
Lara is a female 0201
Mauro is a male 0300
Giovannais a female 0401

Marco is husband of Lara 0100060201
Marco is father of Mauro 0100000300
Marcois father of Giovanna 0100000401

The concept of family of Marco is 0100060201 0100000300 0100000401

Lara is wife of Marco 0201050100
Lara is mother of Mauro 0201010300
Lara is mother of Giovanna 0201010401

The concept of family of Lara is 0201050100 0201010300 0201010401

Now, we have the same sign "my family", the same object "Marco, Lara, Mauro , Giovanna" but two different interpretation " 0100060201 0100000300 0100000401", "0201050100 0201010300 0201010401 " that represent different views of relation between the objects

regards
Mauro

ref: https://list.iupui.edu/sympa/arc/peirce-l/2021-04/msg00129.html

[bertanimauro]

Hi Helmut,
I think that with the xor we go out by the paradisiacal logic. We say that if a then not b or if not a then b. We go out by the heavens and we go into the hierarchies, into the trees where we don't know the value of the variable. If in paradisiacal logic, with the and, we reason on definition, with the xor we reason on indetermination.
For example:
__ philosophy ------------------------- a
| |____ epistemology ---------- c
| |____ analogy ---------------- d
|__ natural Science ------------------ b
|____ algebra------------------- e
|____ cosmogony ------------- f

If we don't know if a is true or if b is true but not both, but we know that if a is true then it will be or c or d but not both. If b is true than it will be or e or f but not both, we write this:

((a&&!b&&((c&&!d)||(!c&&d))&&!e&&!f)||(b&&!a&&((e&&!f)||(!e&&f))&&!c&&!d))

With indetermination we need the operator not, but here we are out from paradisiacal logic. It's similar to determining f=1/x for x=0.
But if I know that my book is about epistemology I can write c&&(c->a). I don't need either not or xor. In paradisiacal logic we can only talk about the determined and independently of our proposition there is always the limit case where the truth of our proposition is the conjunction of the variable. The hypothetical necessitates negation. Discovering necessitates negation.

Thanks in advance

[bertanimauro]

Hi Helmut,
I find this way of solution that is more useful also for other cases.
The First step is to find all the variables of the situation.
IF a THEN b ELSE c
So the variables are a,b,c
The second step is to find the case that evaluates to true ours sentences.
IF a=true THEN b= true, c=false ----------- 1 case
ELSE a=false,b=false,c=true --------------- 2 case
the Third step is to conjunct the variabile and disjunct the case :
(a&&b&&!c)||(!a&&!b&&c) -------------> (1 case) || (2 case)

https://www.wolframalpha.com/input/?i=%28%28a%26%26b%26%26%21c%29%7C%7C%28%21a%26%26%21b%26%26c%29%29
Rule 66 cellular automata

the solution with the xor is:
(a xor b) and (a xor c)
https://www.wolframalpha.com/input/?i=%28a+xor+c%29+and+%28b+xor+c%29

I hope I help you
Thanks in advance
Mauro

[bertanimauro]

Neanche nor e nand

[bertanimauro]

L'unicità dei numeri, la chiave, sta nella costruzione dei teoremi e nell'usare un' unica classificazione. No teoremi con stesso numeri di cifre. Da dimostrare...

[bertanimauro]

https://www.wolframalpha.com/input/?i=%28+not+a+%26+not+b+%26+c%29+%7C%7C+%28+a+%26+not+b+%26+not+c%29+%7C%7C+%28+not+a+%26b+%26+not+c%29

[bertanimauro]

Io a->b l'ho scritto come a&b ma non è bidirezionale. Ossia (a->b) non -> (a&b)

[bertanimauro]

Lo xor non si può rappresentare perché c'è una negazione ( a & !b) or (!a & b)

[bertanimauro]

cactus language

With an eye toward the aims of the NKS Forum, I've begun to work out
a translation of the "elementary cellular automaton rules" (ECAR's),
in effect, just the boolean functions of abstract type q : B^3 -> B,
into cactus language, and I'll post a selection of my working notes
here. By way of the briefest possible reminder, this cactus syntax,
in its existential interpretation and its traverse-string redaction,
uses just two series of k-adic connectives, first, the concatenation
of k expressions is read as their k-adic logical conjunction, second,
a bracket of the form (e_1, ..., e_k) is read to say that exactly one
of the k expressions e_1, ..., e_k is false. I may sometimes refer to
this bracket as a k-adic "boundary operator" or a k-place "cactus lobe".

1 --------------------------------------------------------
(p,q) = p xor q

p = 1100
q= 1001
(p,q) = 0101

(p,q,r)
(p xor q) xor r
p =1,q=0,r=1 da falso

Solo uno deve essere vero come le fette di una torta in un pie chart. Ogni variabile rappresenta una realtà disgiunta. Come un ramo di un albero

ref: http://web.archive.org/web/20070823095525/http://suo.ieee.org/ontology/msg05491.html

2 ------------------------------------------
p q = p and q
(p q) = not(p and q)

Tutte le variabili devono essere vere. E' il tutto. Almeno un elemento deve appartenere a tutti gli insiemi
http://web.archive.org/web/20070708170445/http://suo.ieee.org/ontology/msg05493.html
prev:
Jon Awbrey, "Differential Logic and Dynamic Systems"
| http://stderr.org/pipermail/inquiry/2003-May/thread.html#478
| http://stderr.org/pipermail/inquiry/2003-June/thread.html#553

3 -------------------------------------
(p (q)) = p->q
Conjunctive Implications and Their Complements
(p (q))(q (r)) = p->q and q->r
ref: http://web.archive.org/web/20070823094023/http://suo.ieee.org/ontology/msg05494.html
http://web.archive.org/web/20070225030529/http://suo.ieee.org/ontology/msg05496.html

---

4 -------------------------------------------------
From now on, the terms "thematic extension" and "thematization" will be used to
describe both the process and degree of explication that progresses through this
series of pictures, both the operation of increasingly explicit symbolization and
the dimension of variation that is swept out by it. To speak of this change in
general, that takes us in our current example from J to !j!, I introduce a class
of operators symbolized by the Greek letter theta, writing !j! = theta(J) in the
present instance. The operator theta, in the present situation bearing the type
theta : [u, v] -> [u, v, x], provides us with a convenient way of recapitulating
and summarizing the complete cycle of thematic developments.

Figure 21 shows how the thematic extension operator theta acts on two
further examples, the disjunction ((u)(v)) and the equality ((u, v)).
Referring to the disjunction as f<u, v> and the equality as g<u, v>,
I write the thematic extensions as !f! = theta(f) and !g! = theta(g)
theta(f) e theta(g) è ilrisultato di un ramodell'alberologico del cactus graph

the functions f<u, v> = ((u)(v)) and g<u, v> = ((u, v))

ref:
http://web.archive.org/web/20070304145333/http://suo.ieee.org/ontology/msg04827.html
http://web.archive.org/web/20070304145343/http://suo.ieee.org/ontology/msg04828.html
http://web.archive.org/web/20070304145353/http://suo.ieee.org/ontology/msg04829.html
http://web.archive.org/web/20070304145413/http://suo.ieee.org/ontology/msg04832.html

---

5 ----------------------------------
all the 256 possible s proposition with 3 variable and relative cactus graph
ref:
http://web.archive.org/web/20061230212330/http://suo.ieee.org/ontology/msg05512.html
http://web.archive.org/web/20061230212443/http://suo.ieee.org/ontology/msg05518.html

---

Reference Material:

http://atlas.wolfram.com/
http://atlas.wolfram.com/01/01/
http://atlas.wolfram.com/01/01/views/3/TableView.html
http://atlas.wolfram.com/01/01/views/172/TableView.html

l'xor va bene per le gerarchie o è in un ramo o è nell'altro dell'albero. L'and è per quei casi in cui l'oggetto si trova in più rami dell'albero

[bertanimauro]

Dear Jon,
with the construct of Lyle Anderson we can say:

1. (a=>(b=>(c))) & (not(a)=>0) & (not(b)=>0)= (a&b&c) - link
   but I prefer the opinion of Helmut (link):
2. (a=>b) &a = (a&b)
   and for 3 variable:
3. (a=>(b=>(c))) &a &b = (a&b&c) - link
   Where = is a metalogical symbol representing "can be replaced in a proof with".
   The 2. formula don't need the use of negation and it seems to imply the truth of a.
   Thanks in advance
   Mauro

reff:
https://list.iupui.edu/sympa/arc/peirce-l/2021-05/msg00142.html
https://groups.io/g/lawsofform/topic/logical_graphs_truth_tables/82270207

[bertanimauro]

2 3 8 256
2^3=8
2^8=256
Diversi operatori stesse possibilità
Nasce l'abstract algebra

[bertanimauro]

Il mondo si basa su una logica matematica basata sulle gerarchie. Questo voleva dire sowa. Bisogna cambiare la logica matematica se si vuole imporre la classificazione a faccette. Il segreto forse sta nell'and

[bertanimauro]

Dear Mauro, List

I think, that has nothing to do with "if then else", and my opinion was false, I had later in the thread corrected it due to the "ex falso quod libet" rule.

If I have understood "If A then B else C" correctly ("else" meaning either B or C, not both), it implies that A = B, and is equal with "(A and B) xor C".

Best
Helmut

[bertanimauro]

Epistemologicamente lo xor è la prima scelta binaria che si distacca dall'idea del tutto. O è a o è b

[bertanimauro]

https://inquiryintoinquiry.com/2019/07/28/animated-logical-graphs-26/

Dear Jon,
I have found this rule:
a b c (c → ((a → b) → c))
F F F T
F F T T
F T F T
F T T T
T F F T
T F T T
T T F T
T T T T
What do you think about?
Regards
Mauro B.

[bertanimauro]

a	b	c	((a ∧ (b ∧ ¬c)) ∨ (¬a ∧ (¬b ∧ c)))
F	F	F	F
F	F	T	T
F	T	F	F
F	T	T	F
T	F	F	F
T	F	T	F
T	T	F	T
T	T	T	F

[bertanimauro]

#8 (comment)
[&] _i una falsa rende falso. Catene di and
! [|] _i una vera rende falso. Negazione di catene di or
Rete neurali very simple

[bertanimauro]

La mia è una forma semplice di arithmetization of a syntax tracciaOntolog/arithmetization-syntax.pdf simile ai numeri di Godel. Ma ha una metrica tra i numeri e usa la funzione BcomplexObject per diminuire le distanze. Qualcuno nella mailing-list aveva detto che senza metrica non aveva significato. Guardare Carnap .
Penso che sia un gruppoide su X (Subject,Object), Y (Predicate) con un'unica funzione f che è BcomplexObject . Crea un algebra totalmente ordinata

[bertanimauro]

In logic, especially mathematical logic, a Hilbert system, sometimes called Hilbert calculus, Hilbert-style deductive system or Hilbert–Ackermann system, is a type of system of formal deduction attributed to Gottlob Frege[1] and David Hilbert. These deductive systems are most often studied for first-order logic, but are of interest for other logics as well.

Most variants of Hilbert systems take a characteristic tack in the way they balance a trade-off between logical axioms and rules of inference.[1] Hilbert systems can be characterised by the choice of a large number of schemes of logical axioms and a small set of rules of inference. Systems of natural deduction take the opposite tack, including many deduction rules but very few or no axiom schemes. The most commonly studied Hilbert systems have either just one rule of inference – modus ponens, for propositional logics – or two – with generalisation, to handle predicate logics, as well – and several infinite axiom schemes. Hilbert systems for propositional modal logics, sometimes called Hilbert-Lewis systems, are generally axiomatised with two additional rules, the necessitation rule and the uniform substitution rule.

In a Hilbert-style deduction system, a formal deduction is a finite sequence of formulas in which each formula is either an axiom or is obtained from previous formulas by a rule of inference. These formal deductions are meant to mirror natural-language proofs, although they are far more detailed.

Ref:
https://en.wikipedia.org/wiki/Hilbert_system
https://en.wikipedia.org/wiki/List_of_Hilbert_systems

[bertanimauro]

my work experience with various military and intelligence community bureaucracies, I noted that they seemed to think that actually solving a problem was a bad thing. When I tried to explain how there would always be something to do do after we had solved the problem, they often fired me. Five years ago I fooled them and retired. I recently got a call from a friend who is still in the business who asked were I had put the work we were doing before we retired because they were revisiting the same thing and he wanted to have the answer ready when it was time to give it to the customer, again.

[bertanimauro]

my first automatic reasoner

https://docs.google.com/spreadsheets/d/1bkmy2FxPvdR4kC-YzWYF0wX5z-OA80c5vt_7QtBHe8Q/edit?usp=sharing

[bertanimauro]

Hi Helmut,
more difficult:
IF a THEN
---- IF b THEN c
---- ELSE d
ELSE
---- IF e THEN f
---- ELSE g

(a && b && c&&!d&&!e&&!f&&!g)||(a&&!b&&!c&&d&&!e&&!f&&!g) ||(!a&&!b&&!c&&!d&&e&&f&&!g)||(!a&&!b&&!c&&!d&&!e&&!f&&g)

https://www.wolframalpha.com/input/?i=%28a+%26%26+b+%26%26+c%26%26%21d%26%26%21e%26%26%21f%26%26%21g%29%7C%7C%28a%26%26%21b%26%26%21c%26%26d%26%26%21e%26%26%21f%26%26%21g%29+%7C%7C%28%21a%26%26%21b%26%26%21c%26%26%21d%26%26e%26%26f%26%26%21g%29%7C%7C%28%21a%26%26%21b%26%26%21c%26%26%21d%26%26%21e%26%26%21f%26%26g%29

7 variabili, 2 valori-> 2^7:128 possibilità
2^128: 340282366920938463463374607431768211456 possibili combinazioni

I think that It would be the propositional calculus of causality. Jon, what do you think about this idea?
Mauro

Ref: https://list.iupui.edu/sympa/arc/peirce-l/2021-06/msg00020.html

[bertanimauro]

List,

I think my previous post on this slide may have overemphasized the difference between Peirce’s 1867 view of the categories and his later “phaneroscopic” view of them, and I’d like to correct that before we leave slide 6, which refers to Peirce’s early discovery that the “set of genuinely universal categories is small and gradually ordered.”

It may not be clear what “genuinely universal categories” are — i.e. how they differ from other sets of categories — nor is it clear what André means by “gradually ordered.” The next few slides (dealing with prescission) will probably clarify this; but before we get to them, I’d like to provide some relevant text from a letter Peirce wrote c. 1905 to a “Signor Calderoni”:

[[ … on May 14, 1867, after three years of almost insanely concentrated thought, hardly interrupted even by sleep, I produced my one contribution to philosophy in the “New List of Categories” in the Proceedings of the American Academy of Arts and Sciences, Vol. VII, pp. 287-298. Tell your friend Julian that this is, if possible, even less original than my maxim of pragmatism; and that I take pride in the entire absence of originality in all that I have ever sought to bring to the attention of logicians and metaphysicians. My three categories are nothing but Hegel's three grades of thinking. I know very well that there are other categories, those which Hegel calls by that name. But I never succeeded in satisfying myself with any list of them. We may classify objects according to their matter; as wooden things, iron things, silver things, ivory things, etc. But classification according to structure is generally more important. And it is the same with ideas. Much as I would like to see Hegel's list of categories reformed, I hold that a classification of the elements of thought and consciousness according to their formal structure is more important. I believe in inventing new philosophical words in order to avoid the ambiguities of the familiar words. I use the word phaneron to mean all that is present to the mind in any sense or in any way whatsoever, regardless of whether it be fact or figment. I examine the phaneron and I endeavor to sort out its elements according to the complexity of their structure. I thus reach my three categories. ]] (CP 8.213, c. 1905).

Peirce’s assertion that his “three categories are nothing but Hegel's three grades of thinking” might be misleading in some ways, but it confirms André’s statement that they are “gradually ordered.” It also shows that Peirce’s method of “reaching” his three categories did not undergo a complete change when he renamed it “phaneroscopy” in 1904.

Gary f.

[bertanimauro]

https://github.com/bertanimauro/cellularAutomata

[bertanimauro]

(a->b)&& a

(a&&b)->(a->b)

Although the Philonian views lead to such inconveniences as that it
| is true, as a consequence 'de inesse', that if the Devil were elected
| president of the United States, it would prove highly conducive to the
| spiritual welfare of the people (because he will not be elected), yet
| both Professor Schroeder and I prefer to build the algebra of relatives
| upon this conception of the conditional proposition. The inconvenience,
| after all, ceases to seem important, when we reflect that, no matter
| what the conditional proposition be understood to mean, it can always
| be expressed by a complexus of Philonian conditionals and denials of
| conditionals. It may, however, be suspected that the Diodoran view
| has suffered from incompetent advocacy, and that if it were modified
| somewhat, it might prove the preferable one.
|
| C.S. Peirce, 'Collected Papers', CP 3.443,
|"The Regenerated Logic", 'Monist', vol. 7,
| pp. 19-40, 1896.

[bertanimauro]

 helmut, jon, list,

I have not yet had a way of reading the thesis of Lyle Anderson, and is very much related to the inclusion of truth tables, but what I want to express is a simpler concept.

  

 01. Juni 2021 um 12:34 Uhr
"Mauro Bertani" <[email protected]>
wrote:

Hi Helmut,

I think that with the xor we go out by the paradisiacal logic. We say that  if a then not b or if not a then b. We go out by the heavens and we go into the  hierarchies, into the trees where we don't know the value of the variable. If in paradisiacal logic, with the and, we reason on definition, with the xor we reason on indetermination.

For example:

__ philosophy ------------------------- a

|        |____ epistemology ---------- c

|        |____ analogy  ---------------- d

|__ natural Science ------------------ b

         |____ algebra------------------- e

         |____ cosmogony ------------- f

 

If we don't know if a is true or if b is true but not both, but we know that if a is true then it will be or c or d but not both. If b is true than it will be or e or f but not both, we write this:

 

((a&&!b&&((c&&!d)||(!c&&d))&&!e&&!f)||(b&&!a&&((e&&!f)||(!e&&f))&&!c&&!d))

 

With indetermination we need the operator not, but here we are out from paradisiacal logic. It's similar to determining f=1/x for x=0.

But if I know that my book is about epistemology I can write c&&(c->a). I don't need either not or xor. In paradisiacal logic we can only talk about the determined and independently of our proposition there is always the limit case where the truth of our proposition is the conjunction of the variable. The hypothetical necessitates negation. Discovering necessitates negation.

 

Thanks in advance

Mauro

 

In this email I talk about epistemology

  I think that with the xor we go out by the paradisiacal logic. We say that  if a then not b or if not a then b  

 so c is true. I have talk about analogy 

  With indetermination we need the operator not, but here we are out from paradisiacal logic. It's similar to determining f=1/x for x=0.

so c is true.

I talk about algebra (variable) 

If we don't know if a is true or if b is true but not both, but we know that if a is true then it will be or c or d but not both. If b is true than it will be or e or f but not both, we write this:   

so e is true. And I talk of  cosmogony (theory of hole)

 It's similar to determining f=1/x for x=0.

so f is true. So we can say:

c&&d&&e&&f&&(c->a)&&(d->a)&&(e->b)&&(f->b)

that is to say:

a&&b&&c&&d&&e&&f

In paradisiacal logic we can only talk about the determined and independently of our proposition there is always the limit case where the truth of our proposition is the conjunction of the variable. 

We can say that paradisiacal logic is a theory of all. No category excludes another category. It's simply a conjunction of category and his relation of inclusion.Is a more simple logic but more time efficient.

For Helmut:

I think that this run:

((a&&b&&!c)||(!a&&!b&&c))

a | b | c | ((a ∧ (b ∧ ¬c)) ∨ (¬a ∧ (¬b ∧ c))) -- | -- | -- | -- F | F | F | F F | F | T | T F | T | F | F F | T | T | F T | F | F | F T | F | T | F T | T | F | T T | T | T | F

Thanks in advance

Mauro

[bertanimauro]

Sillogismi

Traslare una frase con quantificatori in una senza
https://docs.google.com/document/d/1dccRNoaIVoH_S-tFFdVSBfd4QcNnG5ROScas9oKgV2Q/edit?usp=sharing
https://docs.google.com/document/d/1Vwr6IeU16Z4HBtpLNQKjBPyQJQVH_Ic4drHteHzpoQk/edit?usp=sharing
https://groups.io/g/lawsofform/message/1065

[bertanimauro]

Modus ponens (or "the fundamental rule of inference"[5]) is often written as follows: The two terms on the left, P → Q and P, are called premises (by convention linked by a comma), the symbol ⊢ means "yields" (in the sense of logical deduction), and the term on the right is called the conclusion:

P → Q, P ⊢ Q

This is a sophisticated concept. Russell and Whitehead (2nd edition 1927) in their Principia Mathematica describe it this way: "The trust in inference is the belief that if the two former assertions [the premises P, P→Q ] are not in error, the final assertion is not in error . . . An inference is the dropping of a true premiss [sic]; it is the dissolution of an implication" (p. 9). Further discussion of this appears in "Primitive Ideas and Propositions" as the first of their "primitive propositions" (axioms): *1.1 Anything implied by a true elementary proposition is true" (p. 94). In a footnote the authors refer the reader back to Russell's 1903 Principles of Mathematics §38.

[bertanimauro]

Hi Helmut,

the secret of propositional calculus with 3 variables is behind this four number:

2 3 8 256

3 variable : {a,b,c}

2 value per variable : {0,1} or {T,F}

the combination of 2 value for 3 variable are 2^3 = 8 rows:

a	b	c
F	F	F
F	F	T
F	T	F
F	T	T
T	F	F
T	F	T
T	T	F
T	T	T

Now came in action theVenn Diagram: for 3 variable the space is divided in 8 parts: each part is  tuple of 3 variables when the tupla began true for the evaluation of a proposition.

For example here the tupla (a&&b&&c):

https://www.wolframalpha.com/input/?i=%28a%26%26b%26%26c%29

For example here the two tuple   (a&&b&&c) e (a&&!b&&c):

https://www.wolframalpha.com/input/?i=%28a%26%26b%26%26c%29+%7C%7C+%28a%26%26%21b%26%26c%29

In fact the venn diagram has two parts colored and the truth table has two rows that evaluate a true.

8 variables that can be coloured or less is the same as 8 variables with two values. so 2^8=256 combinations.

Now we can express 256 concepts with three variables, but there are many more prepositions that express this concept. This is due to the use of different operators and the number of operators. Born abstract algebra.

The last step is to order the 256 concept. Also in this case there are many ways to do this. This creates a lattice

I'm not sure but we come into group theory.

I hope I help you

Mauro

[bertanimauro]

Nand e nor

a nand ( b nand c)
!a or !(!b or !c)
!(a && !( b&&c))
!( a and b)
!a or !b

https://www.wolframalpha.com/input/?i=+%28%21a++or+%21%28%21b+or+%21c%29%29

a nor (b nor c)
(!a and !(!b and !c))
!(a or !( b or c))
!( a or b)
!a and ! b

https://www.wolframalpha.com/input/?i=+%28%21a++and+%21%28%21b+and+%21c%29%29

De Morgan law

[bertanimauro]

Laws of form translation

QUOTE Armahedi Mazhar : https://groups.io/g/lawsofform/message/1066
So, I prefer the Kauffman notation
Aab = [a]b, Eab=[a][b], Iab=[[a][b]] and Oab=[[a]b], since it preserves the relation A=[O], O=[A], E=[I] and I=[E]
QUOTE

a b = a or b
[a] = !a

A:[O]: !a || b = [a]b
E:[I]: !(a&&b) = [a][b]
I:[E]: (a&&b) = [[a][b]]
O:[A]: a&& !b = [[a] b]

[bertanimauro]

Aggiungiamo la relazione R come:
R(x,r,y,z) => yEx&&(yEx => z!Ex)&&(yEr && zEr)&&(yEx=>yEr)&&(z!Ex=>yEr) we can say:
R(q,x,y,z) && R(s,x,z,y)
(yEq&&(yEq => z!Eq)&&(yEx && zEx)&&(yEq=>yEx)&&(z!Eq=>yEx)) && (zEs&&(zEs => y!Es)&&(yEx && zEx)&&(yEs=>yEx)&&(z!Es=>yEx))
(d&&(d=>!e)&&(a&&b)&&(d=>a)&&(!e=>b)) && (f&&(f =>!g)&&(a&&b)&&(f=>a)&&(!g=>b))
(a &&b&&d&&!e&&f&&!g)

R(q,x,y,z) && R(s,x,z,y) => T(x,y,z)
(d&&(d=>!e)&&(a&&b)&&(d=>a)&&(!e=>b)) && (f&&(f =>!g)&&(a&&b)&&(f=>a)&&(!g=>b))=>(((a&&b)||(!a&&!b)))
La differenza implica l'uguaglianza

REF:
https://docs.google.com/document/d/1gMXicxgMJHjNjS1qyXx5WAEr6QlzzJiFxZ3j4xkRQn8/edit?usp=sharing

[bertanimauro]

Transformation between logical operator
https://docs.google.com/document/d/1eZOqm_rIea8GZ-i81TzVcVqhvvEd1cHtzZNW9D6ILOU/edit?usp=sharing

[bertanimauro]

Sistemato primo assioma che era sbagliato
4 Assiomi

1. ((a)||(!a)) &&
2. ((((a&&b)||(!a&&!b)) => ((b&&a)||(!b&&!a)))&&(((b&&a)||(!b&&!a))=>((a&&b)||(!a&&!b)))) &&
3. ((((c&&a)||(!c&&!a))&&((a&&b)||(!a&&!b)))=>((c&&b)||(!c&&!b))) &&
4. (((a=>((b&&a)||(!b&&!a)))=> (a=>b)) && ((b=>(((b&&a)||(!b&&!a))))=>(b=>a))) TAUTOLOGIA

1. T(x, y, y)
2. T(x, y, z) ⇐⇒ T(x, z, y)
3. T(x, w, y) ∧ T(x, y, z) =⇒ T(x, w, z)
4. ((yEx ⇒T(x,y,z)) ⇒ (yEx ⇒ zEx)) && ((zEx ⇒ T(x,y,z)⇒(zEx⇒yEx))
   (T(x,y,z) ⇒ (yEx ⇒ zEx)) && (T(x,y,z)⇒(zEx⇒yEx))
   Dire T(x,y,z) è come dire (yEx ⇒ zEx) && (zEx⇒yEx)
   Quindi
   T(x,y,z) = ((a&&b)||(!a&&!b))
   T(x,y,z) ⇒ ((yEx ⇒ zEx) && (zEx⇒yEx))
   ((a&&b)||(!a&&!b))⇒((a⇒b)&&(b⇒a)) TAUTOLOGIA

[bertanimauro]

Sistemata conclusione finale
4 Assiomi
((a)||(!a)) &&
((((a&&b)||(!a&&!b)) => ((b&&a)||(!b&&!a)))&&(((b&&a)||(!b&&!a))=>((a&&b)||(!a&&!b)))) &&
((((c&&a)||(!c&&!a))&&((a&&b)||(!a&&!b)))=>((c&&b)||(!c&&!b))) &&
(((a=>((b&&a)||(!b&&!a)))=> (a=>b)) && ((b=>(((b&&a)||(!b&&!a))))=>(b=>a))) TAUTOLOGIA

1. T(x, y, y)
2. T(x, y, z) ⇐⇒ T(x, z, y)
3. T(x, w, y) ∧ T(x, y, z) =⇒ T(x, w, z)
4. ((yEx ⇒T(x,y,z)) ⇒ (yEx ⇒ zEx)) && ((zEx ⇒ T(x,y,z)⇒(zEx⇒yEx))
   (T(x,y,z) ⇒ (yEx ⇒ zEx)) && (T(x,y,z)⇒(zEx⇒yEx))
   Dire T(x,y,z) è come dire (yEx ⇒ zEx) && (zEx⇒yEx)
   Quindi
   T(x,y,z) = ((a&&b)||(!a&&!b))
   T(x,y,z) ⇒ ((yEx ⇒ zEx) && (zEx⇒yEx))
   ((a&&b)||(!a&&!b))⇒((a⇒b)&&(b⇒a)) TAUTOLOGIA
   ((a⇒b)&&(b⇒a))⇒((a&&b)||(!a&&!b)) TAUTOLOGIA
   T(x,y,z) ⇐⇒ ((yEx ⇒ zEx) && (zEx⇒yEx))
   T(x,y,z) ⇐⇒ (yEx ⇐⇒ zEx)

[bertanimauro]

Minimal negation operator.
((a),(b),(c))
radio botton
f_22
https://docs.google.com/document/d/1gFzbMDW_zbqq452-VjWIlJ02dWtyxrNhWgpvMYBcM2I/edit?usp=sharing

[bertanimauro]

John, Jon, List

I think I have shown that one can already avoid, at first, the most repulsive and abstract aspect of the exposition of the axioms of CategoryTheory . Because it is the natural transformations of functors that are the essential obstacle opening towards categories of categories. These "relations of relations" which disturb no longer appear if we place ourselves in the field of Posets (Partially Ordered Set) which is easy to apprehend especially since the Poset that we use to begin with is just the order on the three numbers 1,2,3 that children count on their fingers! Then it's just a matter of understanding the notion of "structure-preserving application" which is easy to illustrate with diagrams. I must say that the mental operations of most philosophers who manipulate sequences of reasoning in the field of classical logic mobilize capacities far beyond those required to work with such a simple Poset. The resulting lattice structure is immediately put into a diagram and there is no need to know its exact definition.Moreover, to convince oneself of its natural relevance to semiotics, one need only read Peirce: CP 2.254 to 2.264.
Best regards

Robert Marty

Where a group may be thought of as a group of symmetry transformations that isomorphically relates one object to itself (the symmetries of one object, such as the isometries of a polyhedron) a groupoid is a collection of symmetry transformations acting between possibly more than one object.

Hence a groupoid consists of a set of objects x,y,z,⋯ and for each pair of objects (x,y) there is a set of transformations, usually denoted by arrows

x f⟶y
which may be composed if they are composable (i.e. if the first ends where the second starts)

...........Y
f↗.............↘g
X....⟶g∘f.....Z

such that this composition is associative

These last two properties are the decisive ones of a functor; they are called the functoriality conditions. They are a direct generalization of the notion of homomorphism (of monoids, groups, algebras, etc.) to the case that there are more objects. As a slogan we have

The notion of functor is a horizontal categorification of that of homomorphism.

Ref: https://ncatlab.org/nlab/show/category+theory+vs+order+theory

[bertanimauro]

Hi James,
I'm not good in proof but I prove to follow your speech. I find some problem. It seems that my conclusions are different. I propose a little modification to step 4.
Here my steps:

(D3) y ∈' x ⇐⇒ ¬((y ∈ x ∧ x ∈ x) ∨ (¬(y ∈ x) ∧ ¬(x ∈ x)))
y E' x <==> !((a&b)||(!a&&!b))
y E' x <==> a XOR b
y E' x <==> y XOR x
y E' x <==> y 0+ x where 0+ symbol of albebric normal form

If
T(x,y,z) is the predicate (y ∈ x ∧ z ∈ x) ∨ (y ∉ x ∧ z ∉ x)
then (with different definitions for x, y and z we can express the following formulas):

1. T(x, y, y)
2. T(x, y, z) ⇐⇒ T(x, z, y)
3. T(x, w, y) ∧ T(x, y, z) =⇒ T(x, w, z)
4. ¬(x ∈ x) =⇒ ¬T(x, x, y) ⇐⇒ x ∈ y

This last formula proves that ¬(x ∈ x) can be expressed with T instead of ∈.

1. ((a&b)||(!a&&!b))
2. ((a&b)||(!a&&!b)) <==> ((b&a)||(!b&&!a))
3. (((c&&a)||(!c&&!a))&&((a&b)||(!a&&!b)))=>((c&&b)||(!c&&!b))
   https://www.wolframalpha.com/input/?i=%28%28%28c%26%26a%29%7C%7C%28%21c%26%26%21a%29%29%26%26%28%28a%26b%29%7C%7C%28%21a%26%26%21b%29%29%29%3D%3E%28%28c%26%26b%29%7C%7C%28%21c%26%26%21b%29%29
4. !d=>!((d&&a)||(!d&&!a)) non è una tautologia
5. !((d&&a)||(!d&&!a))=>a non è una tautologia
   https://www.wolframalpha.com/input/?i=%21%28%28d%26%26a%29%7C%7C%28%21d%26%26%21a%29%29%3D%3Ea
   ??? --- Is the paradox of type?, x can't be member of himself? --- ???

1 e 2 . ((a&b)||(!a&&!b)) && ((((a&b)||(!a&&!b)) => ((b&a)||(!b&&!a)))&&(((b&a)||(!b&&!a))=>((a&b)||(!a&&!b))))
(a&b)||(!a&&!b)

1 e 2 e 3 . ((a&b)||(!a&&!b) && (((c&&a)||(!c&&!a))&&((a&b)||(!a&&!b)))=>((c&&b)||(!c&&!b)))
(!a || !b ||c)

1e 2 e 3 e 4 . nel caso ¬(x ∈ x) =⇒( ¬T(x, x, y) ⇐⇒ x ∈ y)
(((!a || !b ||c)) &&( !d=>!((d&&a)||(!d&&!a)))&& ((!((d&&a)||(!d&&!a))=>a)&&(a=>(!((d&&a)||(!d&&!a))))) )
(a && !b && !d)||(a && c &&!d)
(yEx e z!Ex e x!Ex) o (yEx e wEx e x!Ex)

1 e 2 e 3 e 4 nel caso (¬(x ∈ x) =⇒ ¬T(x, x, y) )⇐⇒ x ∈ y
(((!a || !b ||c))&&((!d=>!((d&&a)||(!d&&!a)))=>a) && (a=>((!d=>!((d&&a)||(!d&&!a))))))
(a && !b) || (a && c) || (!a && !d)
(yEx e z!Ex) o (yEx e wEx) o (y!Ex e x!Ex)

PROPOSTA:
The 4 step can re-write with:
(a=>((b&&a)||(!b&&!a)))=> (a=>b)
((a=>((b&&a)||(!b&&!a)))=> (a=>b)) && ((b=>(((b&&a)||(!b&&!a))))=>(b=>a))
y E x => (y E x => z E x)
y E x <=> z E x

So, x would be the equality that links the two differences y and z, similar to Etter's concept of "Membership and Identity". obviously y will not be a member of q (y!Eq) while z will be a member of q (zEq), thus forming the difference between y and z.
it should be proved that:
yEx <=> zEx
y!Eq <=>zEq
I found a very similar concept in "An Essay on Man" by Ernst Cassirer.
Thanks in advance
Mauro

REF:
https://groups.io/g/lawsofform/message/1122

[bertanimauro]

DUALE
a} DUALE:
---1] !(a || b || c) è vera quando sono tutte false
---2] (a && b && c) è vera quando sono tutte vere

b} DUALE
---1] !(!a && !b && !c) è falsa quando sono tutte false
---2] !a || !b || !c è falsa quando sono tutte vere

ref:
https://groups.io/g/lawsofform/message/337
https://groups.io/g/lawsofform/message/335
https://groups.io/g/lawsofform/topic/83557540
Passaggio:
((a xor !a)&&(b xor !b)) xor (a&&b)

((a xor b)=>(!a xor !b)) && ((!a xor !b)=>(a xor b)) tautologia
a xor b = !a xor !b
!(!a && !b) = a or b
!(!a && !b && !c) = a || b || c è falsa quando sono tutte false

(!a &&b) xor (a &&!b) xor (!a && !b) = !a or !b
!a || !b || !c = !(a && b && c) è falsa quando sono tutte vere

[bertanimauro]

Hi jon,Helmut,list,

I have discover that with the Disjunctive normal form (DNF) I can represent all true table:

This is an example:

I want a truth table be true if: a,b,c is true or if a,c is true and b false

(a&&b&&c) || (a&&!b&&c) 

a	b	c	((a ∧ (b ∧ c)) ∨ (a ∧ (¬b ∧ c)))
F	F	F	F
F	F	T	F
F	T	F	F
F	T	T	F
T	F	F	F
T	F	T	T
T	T	F	F
T	T	T	T

Now also  truth for: a,b,c false

  (a&&b&&c) || (a&&!b&&c)||(!a&&!b&&!c)   

a	b	c	((a ∧ (b ∧ c)) ∨ ((a ∧ (¬b ∧ c)) ∨ (¬a ∧ (¬b ∧ ¬c))))
F	F	F	T
F	F	T	F
F	T	F	F
F	T	T	F
T	F	F	F
T	F	T	T
T	T	F	F
T	T	T	T

so my hierarchy was:

For example:

__ philosophy ------------------------- a

|        |____ epistemology ---------- c

|        |____ analogy  ---------------- d

|__ natural Science ------------------ b

         |____ algebra------------------- e

         |____ cosmogony ------------- f

(a&&!b&&c&&!d&&!e&&!f)||(a&&!b&&!c&&d&&!e&&!f)||(!a&&b&&!c&&!d&&e&&!f)||(!a&&b&&!c&&!d&&!e&&f)

It's more power this DNF

Thanks in advance