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
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, HelmutHelmut, 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, HelmutHelmut, 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, USAStructural Engineer, Synechist Philosopher, Lutheran ChristianOn 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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