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Rola aksjomatu w matematyce współczesnej oraz w perspektywie dociekań nad aksjomatem wyboru

100%
Lewandowska K.
Zagadnienia Filozoficzne w Nauce
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2011
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issue 48
148-165
PL
We show how philosophy effected the shape of mathematics when the proof of Well-Ordering Principle was formulated by Ernst Zermelo. We also consider the significance of philosophy of mathematics today. We concentrate on Solomon Feferman and Penelope Maddy attitude in the recent debate on the need of new axioms in mathematics.
2

On Axiomatization of Łukasiewicz’s Four-Valued Modal Logic

100%
Tkaczyk M.
Logic and Logical Philosophy
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2011
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vol. 20
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issue 3
215-232
EN
Formal aspects of various ways of description of Jan Łukasiewicz’s four-valued modal logic are discussed. The original Łukasiewicz’s description by means of the accepted and rejected theorems, together with the four-valued matrix, is presented. Then the improved E. J. Lemmon’s description based upon three specific axioms, together with the relational semantics, is presented as well. It is proved that Lemmon’s axiomatics is not independent: one axiom is derivable on the base of the remanent two. Several axiomatizations, based on three, two or one single axiom are provided and discussed, including S. Kripke’s axiomatics. It is claimed that (a) all substitutions of classical theorems, (b) the rule of modus ponens, (c) the definition of “⋄” and (d) the single specific axiom schema: ⌐□A ∧ B→A ∧□ B⌐, , called the jumping necessity axiom, constitute an elegant axiomatics of the system.
3

Two Weak Lambek-Style Calculi: DNL and DNL

100%
Zielonka W.
Logic and Logical Philosophy
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2012
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vol. 21
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issue 1
53-64
EN
The calculus DNL results from the non-associative Lambek calculus NL by splitting the product functor into the right (⊳) and left (⊲) product interacting respectively with the right (/) and left (\) residuation. Unlike NL, sequent antecedents in the Gentzen-style axiomatics of DNL are not phrase structures (i.e., bracketed strings) but functor-argument structures. DNL− is a weaker variant of DNL restricted to fa-structures of order ≤￢ 1. When axiomatized by means of introduction/elimination rules for / and \, it shows a perfect analogy to NL which DNL lacks.
4
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Najkrótsze aksjomaty modalnej logiki Łukasiewicza

100%
Dywan Z.
Roczniki Filozoficzne
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2016
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vol. 64
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issue 2
5-9
PL
Pokazujemy, że formuła CLpEqLq aksjomatyzuje logikę modalną Łukasiewicza i nie istnieje krótszy aksjomat dla tej logiki.
EN
We show that the formula CLpEqLq axiomatizes Łukasiewicz’s modal logic and that there exists no shorter axiom for it.
5
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Axioms, axiomatization and law

75%
Madej M., Horák F.
The Lawyer Quarterly
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2018
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vol. 8
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issue 3
254-270
EN
This paper examines the possibility and the desirability of axiomatization in law. In the first part, the paper examines the notion of axiom and the ways how it was or could be introduced into law. It is here where the authors openly invite the reader to lose the conventional approach and think about alternative ways to build basic legal concepts. In the second part, the paper continues by presenting several theories which endeavored (or appeared to endeavor) to show that law can (and should be) axiomatized and which even attempted to axiomatize it. After establishing whether these theories were successful at all, the authors add some of their own ideas on the topic of axiomatization.
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