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 bertanimauro@gmail.com 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 bertanimauro@gmail.com 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 jawbrey@att.net 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 bertanimauro@gmail.com 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 https://github.com/bertanimauro/Ranganathan_APUPA/issues/5#issuecomment-1047587996 . 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

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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

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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]

https://pilot.list.iupui.edu/sympa/arc/peirce-l/2021-04/msg00095.html

Hi Helmut and Schmidt, I would unify this thread of discussion with this https://pilot.list.iupui.edu/sympa/arc/peirce-l/2021-04/msg00092.html about modal logic. We name it the modality of implication. First we can see two truth table: a b (a ∧ b) F F F F T F T F F T T T

a	b	(a → b)
F	F	T
F	T	T
T	F	F
T	T	T

We can see that these two truth tables have two lines in common. The third and the fourth and we can see that the first and the second row of the first table evaluate to false while in the second table they evaluate to true. Now we can compare another two table

a b (a ∧ b) F F F F T F T F F T T T

a b (a ∨ b) F F F F T T T F T T T T

In these two tables two lines are in common, the first and the fourth. The others evaluate to false in the first table and evaluate to truth in the second. If we postulate that implication is similar to inclusion we can say that (a&&b) -> (a->b) and (a&&b)->(a||b). In fact a&&b evaluates to false, so is included in the other.

a	b	((a ∧ b) → (a → b))
F	F	T
F	T	T
T	F	T
T	T	T

a	b	((a ∧ b) → (a ∨ b))
F	F	T
F	T	T
T	F	T
T	T	T

Now we can imagine possibility and necessity as a function of knowledge of information about the case where the truth table diverges. If I know only that a=T and b=T I have the possibility of saying that a->b but I don't have the necessity to say that a->b. For saying that, I have to know also that if a=F and b=F or a=F and b=V the proposition a->b is true. Implication is an inclusion of a truth table (possibility) where more information can bring to the necessity of the conclusion. It's similar to intention and extension where more information can change the proposition. The proposition that I finally evaluate differs in function of information on a part of the universe that I don't Know and for this reason I imagine it false. regards Mauro

[bertanimauro]

(a&b => (( a || b) =>((a=>b)&a)) && (( ((a ||b) =>((a=> b)&a))=> (a&b)))

(a&b => (( a || b) =>(a&b)) && (((a ||b) =>((a&b))=> (a&b))

(a&b => (( a || b)&a&b) && ((a ||b) &a&b)=> (a&b))

Per essere vero nei due versi l'or implica la verità di entrambi mentre l'implicazione richiedeva solo la verità di a

[bertanimauro]

Mettere a vera vale in tutte e due le direzioni

(a&b => (( a => b)&a)) && (((a =>b) &a)=> (a&b))

È una tautologia

[bertanimauro]

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 22 Apr. 2021.

CCpqCCCprqq (C(Cpq)(C(C(Cpr)q)q)) p q r ((p → q) → (((p → r) → q) → q)) 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

Axiom1: CCCpqrCCrpCsp ((p->q)->r)->((r->p)->(s->p)) p q r s (((p → q) → r) → ((r → p) → (s → p))) F F F F T F F F T T F F T F T F F T T T F T F F T F T F T T F T T F T F T T T T T F F F T T F F T T T F T F T T F T T T T T F F T T T F T T T T T F T T T T T T

Three thesis: 1) CpCqp p->(q->p) p q (p → (q → p)) F F T F T T T F T T T T

2) CCCpqpp ((p->q)->p)->p p q (((p → q) → p) → p) F F T F T T T F T T T T

3) CCpqCCqrCpr (p->q)->((q->r)->(p->r)) p q r ((p → q) → ((q → r) → (p → 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 T

https://link.springer.com/chapter/10.1007%2F978-3-319-29300-4_4

B',K,Peirce [circa 1926; ibid.]

Two-bases: p→(q→(r→p)) , ((p→q)→r)→((s→r)→((p→r)→r)) [Wajsberg (1932) but circa 1925–1926; cf. Prior (1962, p. 302)]

[bertanimauro]

Dear list, 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.

SoI think that implication, being an universal affirmative proposition, came before negation.

In another book [2] Peano describe the continetur and the deducibur ( inclusion and implication) as the same operator

maybe regards Mauro

Ref: [1] Peano, Giuseppe. Calcolo geometrico secondo l'Ausdehnungslehre di H. Grassmann: preceduto dalla operazioni della logica deduttiva. Vol. 3. Fratelli Bocca, 1888. [2] Peano, Giuseppe. Arithmetices principia, nova methodo exposita. Roma, Fratres Bocca, 1889.

Mauro, Jon, List,

again, reflection goes in the opposite direction than real action and evolution do, but "primitive" is a concept about evolution, not about reflection, isn´t it?

In logical reflection, inference comes before double negation, but anyway, first there were no As without being Bs, then somebody experienced, that if A, then B, then this somebody reflected "why is that so? Ah, because there are no As without being Bs!" So the situation of the nonexistence of As being not Bs was first, is more primitive, than the if-then-experience.

Best, Helmut

---

A={n: n=i4 con i..n} B={n: n=i2 con i..n}

Mi accorgo che tutte le a sono b, ma ad un certo punto mi accorgo che, per esempio 6, appartiene a B ma non ad A. Così implico che A è incluso in B e quindi A implica B se a appartiene ad A, apparterrà anche a B. Se a non appartiene ad A non apparterrà neanche a B.

[bertanimauro]

Cf: Logical Graphs, Truth Tables, Venn Diagrams • 4 https://inquiryintoinquiry.com/2021/05/30/logical-graphs-truth-tables-venn-diagrams-4/

... All we are saying is give Peirce a chance ...

Re: Laws of Form https://groups.io/g/lawsofform/topic/logical_graphs_truth_tables/82270207 • John Mingers ( https://groups.io/g/lawsofform/message/278 ) • Lyle Anderson ( https://groups.io/g/lawsofform/message/279 )

Dear John, Lyle,

I’ve seen too many ways of interpreting and implementing If‑Then‑Else clauses to know what any one person or processor means by them until they give me the truth table they have in mind, so if you write out the truth table you like for them I’ll be able to work with that and say something more definite about it.

More importantly, once we get the full power of Peirce’s logical graphs, Spencer Brown’s calculus of indications, and the extensions to cactus graphs and differential logic in gear we’ll find there are better, clearer, more efficient ways of handling Boolean Expansions and Case Analysis and more generally applying propositional logic to real problems.

Here’s the NKS Forum link again:

[NKS Forum] Cactus Rules https://web.archive.org/web/20041025093703/http://forum.wolframscience.com/archive/topic/256-1.html

The anchor post of that series used to have a file attached with the full set of cactus graphs for propositions on three variables … but it looks like the file was not preserved. There’s a couple of links to other copies below.

[Inquiry List] Cactus Rules https://web.archive.org/web/20141210144230/http://stderr.org/pipermail/inquiry/2004-April/001322.html

[Ontology List] Cactus Rules https://web.archive.org/web/20081012033302/http://suo.ieee.org/ontology/msg05518.html

Regards,

Jon

[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]

https://github.com/bertanimauro/Ranganathan_APUPA/issues/8#issuecomment-1047589799 [&] _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.