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: pqr((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 Fis 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. On Wed, 21 Apr 2021 at 03:32, Jon Alan Schmidt <[email protected]> wrote: > Mauro, Terry, List: > > Thanks for your posts. I just sent one of my own in the other thread, > which includes some discussion of truth tables for the four-valued modal > logic that Łukasiewicz developed, with more to come in the near future. I > will be sure to take a look at Nute's book since I still hope ultimately to > connect my ongoing study of Peirce's Existential Graphs with his > pragmatism, where *subjunctive *conditionals are especially important. > > Regards, > > Jon Alan Schmidt - Olathe, Kansas, USA > Structural Engineer, Synechist Philosopher, Lutheran Christian > www.LinkedIn.com/in/JonAlanSchmidt - twitter.com/JonAlanSchmidt > > On Tue, Apr 20, 2021 at 9:42 AM Terry L Rankin <[email protected]> > wrote: > >> Formal conditional logics and corresponding algebras appear to be the >> best approach to sorting these matters. Like modal logics, they rely on >> possible world semantics, while further invoking “conditional selection >> functions” across ranges of possible worlds, in addition to more familiar >> “accessibility relations” applied in modal logics. An excellent work on >> formal conditional logics and their algebras is Donald Nute’s *Topics in >> Conditional Logic <https://www.amazon.com/dp/902771049X/> *(Philosophical >> Studies Series, 20) from Kluwer (1980), >> https://www.amazon.com/dp/902771049X/. >> >> >> >> Terry R. >> >> >> >> *From:* [email protected] < >> [email protected]> *On Behalf Of *Mauro Bertani >> *Sent:* Tuesday, April 20, 2021 3:08 AM >> *To:* Helmut Raulien <[email protected]>; Peirce List < >> [email protected]>; [email protected] >> *Subject:* Re: [PEIRCE-L] Paradisaical logic (was Phaneroscopy... >> >> >> >> 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 >> <https://na01.safelinks.protection.outlook.com/?url=https%3A%2F%2Fpilot.list.iupui.edu%2Fsympa%2Farc%2Fpeirce-l%2F2021-04%2Fmsg00092.html&data=04%7C01%7C%7C31f93843d9134359caf608d903cb104f%7C84df9e7fe9f640afb435aaaaaaaaaaaa%7C1%7C0%7C637544992952321054%7CUnknown%7CTWFpbGZsb3d8eyJWIjoiMC4wLjAwMDAiLCJQIjoiV2luMzIiLCJBTiI6Ik1haWwiLCJXVCI6Mn0%3D%7C1000&sdata=963%2BklUG2hIqi37kxJ4kVGC5CLElF1sdL0JbfHhPTtM%3D&reserved=0> >> 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 >> > _ _ _ _ _ _ _ _ _ _ > ► 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. 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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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