Helmut, List: In logic, "primitive" simply means "not derived"; or as Wikipedia puts it ( https://en.wikipedia.org/wiki/Primitive_notion), "not defined in terms of previously-defined concepts." In classical logic, negation can be treated *as if* it were a primitive because "A implies B" happens to be equivalent to "not (A and not B)" in that particular formal system. In intuitionistic/synechistic logic, implication is a primitive and negation is derived from the implication of falsity, which is exactly what Peirce advocates repeatedly in his writings about both algebraic and graphical systems--even though he is always talking about classical logic. Again, his *philosophical *point is that we can draw inferences without a notion of falsity--in fact, we can only *learn *anything, including that there is such a thing as falsity, by drawing inferences--and implication corresponds to inference, while negation corresponds to falsity.
Regards, Jon S. On Fri, Apr 9, 2021 at 1:40 AM 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 >>>> >>>
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