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: ab(a ∧ b) F F F F T F T F F T T T
ab(a → b) F F T F T T T F F T T TWe 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 ab(a ∧ b) F F F F T F T F F T T T ab(a ∨ b) F F F F T T T F T T T TIn 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. ab((a ∧ b) → (a → b)) F F T F T T T F T T T T ab((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 On Sat, 10 Apr 2021 at 09:46, Mauro Bertani <[email protected]> wrote: > Dear Helmut, > This is truth table > ab((a → b) ∧ (a ∧ b)) > F F F > F T F > T F F > T T Tregards > Mauro > > On Sat, 10 Apr 2021 at 09:12, Mauro Bertani <[email protected]> > wrote: > >> 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 >> <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. >> >> 2 MAIL: >> Dear Helmut, >> I'm not sure to have understood what you have said. >> Let's: >> A={n: n=4*i con i (1..infinity)} >> B={n: n=2*i 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 >> <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> >> ] >> 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 >> >> > > -- > "[..] 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 > -- "[..] 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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