Mauro, List: That particular paper by Łukasiewicz is about classical propositional logic. However, Zdzisław Dywan presumably had its title in mind when he wrote "The Shortest Axioms of Łukasiewicz's Modal Logic" ( https://ojs.tnkul.pl/index.php/rf/article/view/12253/12316, 2016), referring to the four-valued Ł-modal system that I am currently exploring in the other thread. Unfortunately for me, Dywan's text is in Polish, but at least I can understand the logical formulas.
Dywan combines the two axioms that both he ( http://www.filozof.uni.lodz.pl/bulletin/pdf/41_34_4.pdf, 2012) and Marcin Tkaczyk ( https://apcz.umk.pl/czasopisma/index.php/LLP/article/view/LLP.2011.012/803, 2011) previously show to be sufficient, ◻p → p and ◻p → (q → ◻q), to obtain ◻p → (q ↔ ◻q). Tkaczyk also offers his own single-axiom alternative, (◻A ∧ B) → (A ∧ ◻B), calling it "jumping necessity." In both cases, the practical effect is precluding any proposition from being asserted as necessarily true, since it would entail that *all *true propositions are necessarily true. Regards, Jon Alan Schmidt - Olathe, Kansas, USA Structural Engineer, Synechist Philosopher, Lutheran Christian www.LinkedIn.com/in/JonAlanSchmidt - twitter.com/JonAlanSchmidt On Fri, Apr 23, 2021 at 6:58 AM Mauro Bertani <[email protected]> wrote: > Dear Schmidt, > I'm not sure but paraphrasing: I'm not talking about sets defined by > inequalities. The even are {x: x% 2 = 0} the odd {x: x% 2 = 1}. The odd is > not the !even (not even). A denial makes the model false. One can speak > only with an affirmative structure in this way the truth of the model is > given simply by the conjunction of all the variables. > Regards > Mauro > > Il ven 23 apr 2021, 08:59 Mauro Bertani <[email protected]> ha > scritto: > >> I correct me. With odd numbers of negation. >> Regards >> Mauro >> >> On Fri, 23 Apr 2021 at 08:50, Mauro Bertani <[email protected]> >> wrote: >> >>> 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. >>> >>
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