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 <https://books.google.it/books?hl=en&lr=&id=5LJi3dxLzuwC&oi=fnd&pg=PA9&dq=peano+calcolo+geometrico&ots=4xikZ7toBC&sig=yvGzsGadG6UgBZ7pBHp9SA6wvLs&redir_esc=y#v=onepage&q=peano%20calcolo%20geometrico&f=false> ] 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 <https://books.google.it/books?hl=en&lr=&id=5LJi3dxLzuwC&oi=fnd&pg=PA14&dq=peano+calcolo+geometrico&ots=4xikZ7toBC&sig=yvGzsGadG6UgBZ7pBHp9SA6wvLs&redir_esc=y#v=onepage&q=peano%20calcolo%20geometrico&f=false> ] 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. On Fri, 9 Apr 2021 at 08:40, Helmut Raulien <[email protected]> wrote: > > > > Supplement: Now I am wondering, how "primitive" is defined. I said, that > the consequent in > *IF* "IF-THEN", *THEN* "NOT WITHOUT" > like in > *IF* "tennis ball" *THEN* "ball" > must be more primitive than the antedecent, because it has fewer traits > (distinction marks), is more general than it, is its superclass. On the > other hand, it provides more possibilities for variations: A ball may have > the size of a planet, may be of gold covered with diamonds...., while a > tennis ball has a certain size and material. > But possibilities are not implications, but the opposite of them, like a > distinction mark is the opposite of a possibility. In my understanding, > "primitive" means to have less distinction marks and so more possibilities. > > Jon, List, > > I was not talking about the implication/inference, from which can be > inferred a double negation, but about the inference, which infers a double > negation from an inference. The underlined one, not the other: > > *IF* "IF-THEN", *THEN* "NOT WITHOUT" > > Best, Helmut > > > 09. April 2021 um 02:22 Uhr > "Jon Alan Schmidt" <[email protected]> > *wrote:* > Helmut, List: > > No one is disputing that the antecedent of an implication is logically > prior to its consequent. Rather, at issue is whether the logical relation > of implication is more primitive than the logical relation of negation. > That is Peirce's consistent position, and it is central to his entire > philosophy of logic because implication mirrors inference. We can judge one > proposition (the consequent B) to be true *because *another proposition > (the antecedent A) is true, *without *necessarily taking the additional > step of recognizing it to be false that A is true and B is false. > > > CSP: For in reasoning, at least, when we first affirm, or affirmatively > judge, the conjugate of premisses, the judgment of the conclusion has not > yet been performed. There then follows a real movement of thought in the > mind, in which that judgment of the conclusion comes to pass. Now surely, > speaking of the same A and B as above, it were absurd to say that a real > change of A into a sequent B consists in a state of things that should > consist in there not being an A without a B. For in such a state of things > there would be no change at all. (R 300:49[48], 1908) > > > Negation is not required to infer that if something is a tennis ball, then > it is a ball, or that if something is a kangaroo, then it is a vertebrate. > In each case, the consequent necessarily *follows *from the antecedent > because the concept of "tennis ball" *includes *the concept of "ball" and > the concept of "kangaroo" *includes *the concept of "vertebrate." > Negation comes into play only with the further inference that there is no > tennis ball that is not also a ball, or that there is no kangaroo that is > not also a vertebrate. While these different formulations are equivalent in > classical logic, they are not so in intuitionistic/synechistic logic--the > latter can be derived from the former, but not vice-versa. > > Regards, > > Jon S. > > On Thu, Apr 8, 2021 at 1:04 PM Helmut Raulien <[email protected]> wrote: > >> >> Jon, List, >> >> ok, another argument, then I probably will have run out of them: In a >> no-identity- one-way- implication/inferation the implied/inferred-to is >> more primitive than the implier/inferred-from, as it must have fewer traits >> than the implier/inferred-from. The implier has more traits, some of which >> distinguish in the way, that the implied is not a kind of every kind of the >> implier. >> >> Example: "Tennis ball" implies "ball". "Ball" merely has the trait of >> being roughly spherical. This is more primitive than "tennis ball", which >> has additional traits, that lead to the fact, that not every ball is a >> tennis ball. >> >> Best, Helmut >> 08. April 2021 um 18:51 Uhr >> "Jon Alan Schmidt" <[email protected]> >> *wrote:* >> Helmut, List: >> >> >> HR: ... implication itself has been derived from double negation in a >> quasi-evolution, so double negation is more primitive. >> >> >> No, implication mirrors inference, which is more primitive than negation >> (single or double). >> >> Regards, >> >> Jon S. >> >> On Thu, Apr 8, 2021 at 9:27 AM Helmut Raulien <[email protected]> wrote: >> >>> Jon, List, >>> >>> I think, this is an example for reflection going the other way than >>> action, evolution: >>> >>> We can derive "vertebrate" from "kangaroo", but kangaroos are derived >>> (deriven?) from vertebrates, and vertebrates are (resp. were, when they >>> were not a class but merely a species) more primitive, than kangaroos are. >>> >>> In the same way, we, by reflection, can derive double negation from >>> implication, but implication itself has been derived from double negation >>> in a quasi-evolution, so double negation is more primitive. >>> >>> Best, Helmut >>> 08. April 2021 um 15:25 Uhr >>> "Jon Alan Schmidt" <[email protected]> >>> *wrote:* >>> Helmut, List: >>> >>> No, you had it right the first time. In intuitionistic/synechistic >>> logic, we can derive double negation (nested cuts/ovals) from implication >>> (scroll), but we cannot derive implication (scroll) from double negation >>> (nested cuts/ovals). Implication is primitive, single negation is derived >>> from it as the implication of falsity or absurdity, and double negation is >>> derived from it as a composite of two single negations. >>> >>> Regards, >>> >>> Jon Alan Schmidt - Olathe, Kansas, USA >>> Structural Engineer, Synechist Philosopher, Lutheran Christian >>> www.LinkedIn.com/in/JonAlanSchmidt - twitter.com/JonAlanSchmidt >>> >>> On Thu, Apr 8, 2021 at 12:15 AM Helmut Raulien <[email protected]> wrote: >>> >>>> >>>> Supplement: Oh-oh, I have mixed it completely up. Every implication is >>>> a double negation, but not every double negation is an implication, so >>>> double negation is the superclass, and thus more primitive than >>>> implication. >>>> Just like every kangaroo is a vertebrate, but not every vertebrate is a >>>> kangaroo, so vertebrate is the superclass, and more primitive. The first >>>> vertebrate is said to have been a fish-like creature with a single >>>> spine-bone. >>>> Jon, List, >>>> >>>> if we distinguish at all between implication and double negation, we >>>> are in the field of nonclassical logic. In intuitionalistic logic, every >>>> implication is a double negation, but not every double negation is an >>>> implication. So implication is the superclass, and double negation the >>>> subclass. >>>> the folllowing I derive from Stanley N. Salthe´s paper >>>> "Salthe12Axiomathes": In a subsumption (classification), the subclass has >>>> inherited the classifying traits from the superclass, and then developed >>>> additional traits, that distinguish it from other subclasses of the same >>>> superclass. So the subclass is more complex, less primitive than the >>>> superclass. So implication is more primitive than double negation, and >>>> double negation is derived from implication. But is single negation derived >>>> from double negation, or is single negation even more primitive than >>>> implication, and is just the combination of two negations derived from >>>> implication? >>>> >>>> Best, Helmut >>>> >>> _ _ _ _ _ _ _ _ _ _ ► PEIRCE-L subscribers: Click on "Reply List" or > "Reply All" to REPLY ON PEIRCE-L to this message. PEIRCE-L posts should go > to [email protected] . ► To UNSUBSCRIBE, send a message NOT to > PEIRCE-L but to [email protected] with UNSUBSCRIBE PEIRCE-L in the > SUBJECT LINE of the message and nothing in the body. More at > https://list.iupui.edu/sympa/help/user-signoff.html . ► PEIRCE-L is owned > by THE PEIRCE GROUP; moderated by Gary Richmond; and co-managed by him and > Ben Udell. > _ _ _ _ _ _ _ _ _ _ ► PEIRCE-L subscribers: Click on "Reply List" or > "Reply All" to REPLY ON PEIRCE-L to this message. PEIRCE-L posts should go > to [email protected] . ► To UNSUBSCRIBE, send a message NOT to > PEIRCE-L but to [email protected] with UNSUBSCRIBE PEIRCE-L in the > SUBJECT LINE of the message and nothing in the body. More at > https://list.iupui.edu/sympa/help/user-signoff.html . ► PEIRCE-L is owned > by THE PEIRCE GROUP; moderated by Gary Richmond; and co-managed by him and > Ben Udell. > _ _ _ _ _ _ _ _ _ _ > ► PEIRCE-L subscribers: Click on "Reply List" or "Reply All" to REPLY ON > PEIRCE-L to this message. PEIRCE-L posts should go to > [email protected] . > ► To UNSUBSCRIBE, send a message NOT to PEIRCE-L but to > [email protected] with UNSUBSCRIBE PEIRCE-L in the SUBJECT LINE of the > message and nothing in the body. More at > https://list.iupui.edu/sympa/help/user-signoff.html . > ► PEIRCE-L is owned by THE PEIRCE GROUP; moderated by Gary Richmond; and > co-managed by him and Ben Udell. > -- "[..] events are primarily linguistic or cognitive in nature. That is, the world does not really contain events. Rather, events are the way by which agents classify certain useful and relevant patterns of change." Allen and Fergusson "No, no. *History of Eternity*. At first I wanted to find every single one of the buyers to apologize because of the book and also to thank them for what they had done. There is an explanation for that. If you think of thirty-seven people—those people are real, I mean every one of them has a face of his own, a family, he lives on his own particular street. Why, if you sell, say two thousand copies, it is the same thing as if you had sold nothing at all because two thousand is too vast—I mean, for the imagination to grasp. While thirty-seven people—perhaps thirty-seven are too many, perhaps seventeen would have been better or even seven—but still thirty-seven are still within the scope of one's imagination." http://www.theparisreview.org/interviews/4331/the-art-of-fiction-no-39-jorge-luis-borges
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