The aim of the present study is (1) to show, on the basis of a number of unpublished documents, how Heinrich Scholz supported his Warsaw colleague Jan ?ukasiewicz, the Polish logician, during World War II, and (2) to discuss the efforts he made in order to enable Jan ?ukasiewicz and his wife Regina to move from Warsaw to Münster under life-threatening circumstances. In the first section, we explain how Scholz provided financial help to ?ukasiewicz, and we also adduce evidence of (...) the risks incurred by German scholars who offered assistance to their Polish colleagues. In the second section, we discuss the dramatic circumstances surrounding the ?ukasiewiczes' move to Münster in the summer of 1944. (shrink)
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 axiomatic 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 £. (shrink)
The brief article of 1910 which is translated here is, as the prefatory note explains, significant for understanding both the way in which ?ukasiewicz came to many-valued logic and the influences under which he stood at the time.
Compulsion and Surprise Two phenomena conspire to convince people that the physical world exists independently of them. One is its recalcitrance, or insusceptibility to control. It resists and constrains our actions. Much as we might wish to do so, we cannot lift heavy boulders, walk through walls, jump rivers, breathe under water, or fly (unaided) over mountains. The other feature, which is connected to the first, is the world’s propensity to surprise us. The sights and sound, pressures and pains of (...) the world force themselves upon us in perception whether we want them to or not, and are often unexpected and surprising. An unusual bird appears in the garden, a stranger calls at the door and reveals he is a long-lost cousin, the post brings an invitation out of the blue, the car won’t start (surprises may be unpleasant as well as pleasant). These two phenomena, recalcitrance and surprise, form a large part of the platonist’s case for the existence of an independent mathematical reality. The recalcitrance of mathematical reality indeed appears to be stronger than that of the physical: the necessity with which mathematical results follow from assumptions is stricter than the physical necessity by which a wall resists attempts to walk through it. This has rarely been put more eloquently than by Jan Łukasiewicz. Speaking in particular of mathematical logic, he wrote.. (shrink)
The paper contains an attempt at formulating the project of logic comprised in Jan Łukasiewicz's article „On determinism” and a construction of a logic which would realise this project. Such a logic consists of three consequence-operations build upon a four-element algebra. The values of the algebra have been defined by means of the following set of sentences: true and true today, true but not true today, false but not false today and false and false today. It turns out that only (...) one of the consequences is different from the classical consequence and all of them are logically two-valued. It is proved moreover, that the assumption that tautologies consists of sentences-forms which are „always” true today, results in the non-existence of such tautologies. (shrink)
Traditional theism (in Christianity, Judaism and Islam) understands God as possessing certain attributes including omnipotence. God is omnipotent in the sense that God possesses unlimited (maximal) power. For some classical philosophers and theologians (PetrusDamiani, René Descartes) God’s omnipotence requires his being able to do absolutely anything, including the logically impossible. But in Thomas Aquinas’ opinion, to do what is logically impossible is not an act of power but is self-contradictory action. For Aquinas, a logically impossible action is not an action. (...) The contemporary British philosopher of religion, Richard Swinburne, considers omnipotence from an analytic perspective and, partially,within Aquinas’ tradition. For Swinburne, omnipotence includes the power to perform only logically possible and consistent tasks. In this paper, I discuss systematically (§§ 3-6) the philosophical and logical problems of omnipotence and the relation between God and logic from the perspective of Jan Łukasiewicz’s logical investigations. Keywords: Omnipotence, God, logic, rationality, Richard Swinburne, Jan Łukasiewicz. (shrink)
Jan Łukasiewicz distinguished three various formulations of the law of contradiction in Aristotle's considerations concerning axiomatic foundations of philosophia prima in the book Γ of Methaphysics. Łukasiewicz referred to these formulations as „ontological”, „logical”, and „psychological”, respectively. The author focuses his attention on the last of them, namely to the so called psychological approach. He finds this approach to be an inadequate interpretation of Aristotle's views and tries to show that the most appropriate interpretation is pragmatic-logical.
This paper presents two systems of natural deduction for the rejection of non-tautologies of classical propositional logic. The first system is sound and complete with respect to the body of all non-tautologies, the second system is sound and complete with respect to the body of all contradictions. The second system is a subsystem of the first. Starting with Jan Łukasiewicz's work, we describe the historical development of theories of rejection for classical propositional logic. Subsequently, we present the two systems of (...) natural deduction and prove them to be sound and complete. We conclude with a ‘Theorem of Inversion’. (shrink)
mathematical formulae are our “mother tongue”, thanks to which we are able to develop a creative dialogue with our physical environment. The application of the language of mathematics gives us access to valuable information about events which occurred billions years ago and so allows us to reconstruct the history of the universe. This amazing property of nature inspires a non-trivial philosophical question: Why are there the mathematically described universal laws of physics at all, when nature could have been only an (...) uncoordinated disorder? The existence of the universal laws of nature seems to constitute the essence of the ontological structure of the world. Various authors call this basic field of formal structures – the matrix of the universe, the field of rationality, the formal field, the Logos, the Absolute, etc. Jan Łukasiewicz, the well-known representative of the Polish School of Logic, argued that the reality of ideal mathematical structures independent of human experience could be regarded as an expression of God’s presence in nature. Regardless of our terminological preferences, this structure can be regarded as a basic level of physical reality where the necessitarian inter- pretation of the laws of nature is confirmed and the astonishing effectiveness of mathematics could be explained. (shrink)
The life and work of Moj?esz Presburger (1904?1943?) are summarised in this article. Although his production in logic was small, it had considerable impact, both his own researches and his editions of lecture notes of Adjukiewicz and ?ukasiewicz. In addition, the surviving records of his student time at Warsaw University provide information on a little-studied topic.
We construct a faithful interpretation of ukasiewicz's logic in product logic (both propositional and predicate). Using known facts it follows that the product predicate logic is not recursively axiomatizable.We prove a completeness theorem for product logic extended by a unary connective of Baaz . We show that Gödel's logic is a sublogic of this extended product logic.
This paper examines how the work of Frege was known and received in Poland in the period 1910?1935 (with one exception concerning the later work of Suszko). The main thesis is that Frege's reception in Poland was perhaps faster and deeper than in other countries, except England, due to works of Russell and Jourdain. The works of ?ukasiewicz, Le?niewski and Cze?owski are described.