Marcin Łyczak

The paper investigates the formal distinction between logical necessity and metaphysical necessity. After K. Fines terminology, we differentiate between ‘logical necessity’ (or ‘absolute necessity’) and ’metaphysical necessity’ in the debate between modal monists, who believe these modalities are reducible to one another, and pluralists, who argue for their irreducibility. This is one of the key and long-discussed issues in modal metaphysics, examined through the works of notable philosophers and logicians such as D. Chalmers, P. Van Inwagen, S. Kripke, D. Lewis, G. Rosen, R. Stalnaker, and, of course, Fine. We adopt a formal approach to model these modalities using Kripke structures. We argue that metaphysical necessity, constrained by the formal strength of at most $$\textsf {S5}$$ S 5, cannot be considered as absolute as the formal strength of absolute necessity in a proper extension of modal logic $$\textsf {S5}$$ S 5. This is because in all models, absolute necessary truths are logical truths, and there are no additional accidental or possible truths in this sense. This does not hold even for necessity in modal logic $$\textsf {S5}$$ S 5. The paper is organized as follows: we begin by outlining pre-theoretical intuitions. Next we present the semantics of non-accidental Kripke models for metaphysical and absolute necessity. The first one will be considered as a modality dependent on the accessibility relation between possible worlds, while the second will be independent of it and forces truth in all possible worlds across all models. We prove that our absolute necessity, contrary to metaphysical necessity, always captures only logically valid formulas. Finally we develop a formal system, prove its completeness and discuss the implications and limitations of the approach. We aim to provide new insights into the ongoing debate between modal monists and pluralists, which turns out to be connected with R. Carnap’s old idea of logical necessity, expressed in terms of possible worlds semantics.

The use of the primitive notion of mereological fusion (also known as composition and sum) has been considered by various philosophers and logicians, including Aristotle, G. Leibniz, S. Leśniewski, K. Fine, J. Ketland, T. Schindler, and S. Kleishmid. The problem of finding an axiomatization of Classical Mereology with primitive fusion, instead of the primitive notion of being a part, is quite old and was formally considered by C. Lejewski. Lejewski somehow axiomatized classical mereology using primitive fusion (1962, and also later in 1984). However, his mereology is formulated in a very rich nonclassical language of Leśniewski’s ontology with the essential use of quantification over function symbols, including function symbols “part of” and “sum of.” Lejewski’s approach is not expressible in modern mereologies, and we do not use it in any way. We prove that classical mereology is axiomatizable using primitive mereological fusion within the framework of two-sorted logic. The theory presented here is the first contemporary axiomatization of classical mereology with primitive mereological fusion.

We take into account two areas of the logical research of the Lvov-Warsaw School. First, we consider a new approach to research in the history of logic introduced and practiced by Łukasiewicz and some of his followers. In this style of doing history of logic, the knowledge of original philosophical and logical texts was combined with competence in modern logic. This method resulted in many important discoveries both in history and in logic and philosophy. At the same time, we pay attention to contemporary historical research devoted to the logic created by the School itself. In this context, we present an overview of the six papers included in the special issue ‘Logic and its History in the Lvov-Warsaw School’. These papers were presented at an international symposium, ‘History in/of the Lvov-Warsaw School’, held in October 2022 in Warsaw, and they cover various aspects of historical-logical and logical achievements of the School. They concern topics from history of ancient logic, history of modern proof theory, and the issues of applying logic to philosophy and theology in the same way as the one implemented at the Lvov-Warsaw School.

We present a temporal logic of branching time with four primitive operators: |$\exists {\mathcal {C}}$| – it may change whether; |$\forall {\mathcal {C}} $| – it must change whether; |$\exists \Box $| – it may be endlessly unchangeable that; and |$\forall \Box $| – it must be endlessly unchangeable that. Semantically, operator |$\forall {\mathcal {C}}$| expresses a change in the logical value of the given formula in every state that may be an immediate successor of the one considered, while |$\exists {\mathcal {C}}$| expresses a change in the logical value of the given formula in some state that is a possible immediate successor. |$\forall {\mathcal {C}} $| and |$\exists {\mathcal {C}}$| are not normal operators, they are not mutually definable and have unusual properties: |$\forall {\mathcal {C}} A\leftrightarrow \forall {\mathcal {C}} \neg A$| and |$\exists {\mathcal {C}} A \leftrightarrow \exists {\mathcal {C}} \neg A$| are axioms. Operators |$\exists \Box $| and |$\forall \Box $| express endless unchangeability in some and all paths, respectively. Our axiomatization contains two infinitary rules that allow one to obtain |$\exists \Box A$| from infinitely many combinations of formulas with |$A, \neg, \forall {\mathcal {C}}, \land $|⁠, and to get |$\forall \Box A$| from infinitely many combinations of formulas with |$A, \neg, \exists {\mathcal {C}}, \land $|⁠. We show that our formalism is complete by presenting a bridge between our logic and logic |$\textsf {UB}$|⁠, which is a fragment of |$\textsf {CTL}$| in which until operators does not occur.

We formulate a certain subtheory of Ishimoto’s [1] quantifier-free fragment of Leśniewski’s ontology, and show that Ishimoto’s theory can be reconstructed in it. Using an epimorphism theorem we prove that our theory is complete with respect to a suitable set-theoretic interpretation. Furthermore, we introduce the name constant 1 and we prove its adequacy with respect to the set-theoretic interpretation. Ishimoto’s theory enriched by the constant 1 is also reconstructed in our formalism with into which 1 has been introduced. Finally we examine for both our theories their quantifier extensions and their connections with Leśniewski’s classical quantified ontology.

The so-called Principle of Plenitude was ascribed to Leibniz by A. O. Lovejoy in The Great Chain of Being: A Study of the History of an Idea (1936). Its temporal version states that what holds always, holds necessarily (or that no genuine possibility can remain unfulfilled). This temporal formulation is the subject of the current paper. Lovejoy’s idea was criticised by Hintikka. The latter supported his criticisms by referring to specific Leibnizian notions of absolute and hypothetical necessities interpreted in a possible-worlds semantics. In the paper, Hintikka’s interpretative suggestions are developed and enriched with a temporal component that is present in the characteristics of the real world given by Leibniz. We use in our approach the Leibnizian idea that change is primary to time and the idea that there are possible laws that characterize worlds other than the real one. We formulate a modal propositional logic with three primitive operators for change, temporal constancy, and possible lawlikeness. We give its axiomatics and show that our logic is complete with respect to the given semantics of possible worlds. Finally, we show that the counterparts of the considered versions of the Principle of Plenitude are falsified in this semantics and the same applies to the counterpart of Leibnizian necessarianism.

We present the propositional logic LEC for the two epistemic modalities of current and stable knowledge used by an agent who system-atically enriches his language. A change in the linguistic resources of an agent as a result of certain cognitive processes is something that commonly happens. Our system is based on the logic LC intended to formalize the idea that the occurrence of changes induces the passage of time. Here, the primitive operator C read as: it changes that, defines the temporal succession of states of the world. The notion of current knowledge concerns variable components of the world and it may change over time. We represent it by the primitive operator k read as: the agent currently knows that, and assume that it has S5 properties. The second type of knowledge, symbolized by the primitive operator K read as: the agent stably knows that, relates to constant components of the world and it does not change. As a result of the axiomatic entanglement of C, K and k we show that stable knowledge satisfies axioms of S4.3. K and k modalities are not mutually definable, stable knowledge implies the current one and if the latter never changes, then it comes to be stable. The combination of K and k with the idea of an expanding language allows questioning of the so-called perfect recall principle. It cannot be maintained for both types of knowledge just because of changes in the vocabulary of the agent and possibly the growing spectrum of possible states of the world. We interpret LEC in the semantics of histories of epistemic changes and show that it is complete. Finally, we compare our logic with selected epistemic logics based on the concept of linear discrete time.

We present a study of unpublished fragments of Jan F. Drewnowski’s manuscript from the years 1922–1928, which contains his own axiomatics for mereology. The sources are transcribed and two versions of mereology are reconstructed from them. The first one is given by Drewnowski. The second comes from Leśniewski and was known to Drewnowski from Leśniewski’s lectures. Drewnowski’s version is expressed in the language of ontology enriched with the primitive concept of a (proper) part, and its key axiom expresses the so-called weak super-supplementation principle, which was named by Drewnowski “the postulate of the existence of subtractions”. Leśniewski’s axiomatics with the primitive concept of an ingrediens contains the axiom expressing the strong super-supplementation principle. In both systems the collective class of objects from the range of a given non-empty concept is defined as the upper bound of that range. From a historical point of view it is interesting to notice that the presented version of Leśniewski’s axiomatics has not been published yet. The same applies to Drewnowski’s approach. We reconstruct the proof of the equivalence of these two systems. Finally, we discuss questions stemming from their equivalence in frame of elementary mereology formulated in a modern way.

We present the logic$${\mathsf {LCB}}$$LCBwhich is expressed in a propositional language constantly enriched by new atomic expressions. Our formal framework is the propositional doxastic logic$${\mathsf {KD45}}$$KD45with the belief operator$${\mathcal {B}}$$B, extended by the$${\mathcal {C}}$$Coperator, to be readit changes that.... We describe the changing beliefs of an agent who uses progressively expanding language. The approach presented here allows us to weaken pragmatic objections to the so-called principle ofnegative retrospectionaccepted in$${\mathsf {KD45}}$$KD45and the problem oflogical omniscience. In what follows, we present the expanding propositional language used in our formalism, interpreted using an epistemic version of Kripke semantics. Next, we give a syntactic characterization of logic$${\mathsf {LCB}}$$LCBand prove the soundness and completeness of$${\mathsf {LCB}}$$LCBin respect to our semantics. Finally, we compare the idea of expanding language with the notion ofagent awarenessand we relate our formalism to two epistemic temporal logics.

The logic of change formulated by K. Świętorzecka, has its motivation coming from the Aristotelian theory of substantial change which is undrstood as a transformation consisting in the disappearing and becoming of individual substances. The transition: becoming/disapearing (and conversely) is expressed in by the primitive operator C, to be read: it changes that …, and it is mapped by the progressively expanding language. We are interested in attributive changes of individual substances. We consider a formalism with two non-normal and not mutually definable operators of possible and necessary change, inspired by Aristotelian distinction of accidental and essential attributes. From we adopt the idea that temporal concepts are defined via change operators, and the idea of an expanding language. In what follows, We axiomatise our new logic and describe its semantics, giving the proof of its completeness. We compare our formalism with selected modal logics.