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On Theses without Iterated Modalities of Modal Logics Between C1 and S5. Part 2

100%

Pietruszczak A.

Bulletin of the Section of Logic

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2017

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tom 46

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nr 3/4

EN

This is the second, out of two papers, in which we identify all logics between C1 and S5 having the same theses without iterated modalities. All these logics can be divided into certain groups. Each such group depends only on which of the following formulas are theses of all logics from this group: (N), (T), (D), ⌜(T)∨☐q⌝, and for any n > 0 a formula ⌜(T) ∨ (altn)⌝, where (T) has not the atom ‘q’, and (T) and (altn) have no common atom. We generalize Pollack’s result from [1], where he proved that all modal logics between S1 and S5 have the same theses which does not involve iterated modalities (i.e., the same first-degree theses).

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On Theses Without Iterated Modalities of Modal Logics Between C1 and S5. Part 1

100%

Pietruszczak A.

Bulletin of the Section of Logic

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2017

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tom 46

|

nr 1/2

EN

This is the first, out of two papers, in which we identify all logics between C1 and S5 having the same theses without iterated modalities. All these logics canbe divided into certain groups. Each such group depends only on which of thefollowing formulas are theses of all logics from this group: (N), (T), (D), ⌜(T)∨ ☐q⌝,and for any n > 0 a formula ⌜(T) ∨ (altn)⌝, where (T) has not the atom ‘q’, and(T) and (altn) have no common atom. We generalize Pollack’s result from [12],where he proved that all modal logics between S1 and S5 have the same theseswhich does not involve iterated modalities (i.e., the same first-degree theses).

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A comparison of two systems of point-free topology

63%

Gruszczyński R., Pietruszczak A.

Bulletin of the Section of Logic

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2018

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tom 47

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nr 3

EN

This is a spin-off paper to [3, 4] in which we carried out an extensive analysis of Andrzej Grzegorczyk’s point-free topology from [5]. In [1] Loredana Biacino and Giangiacomo Gerla presented an axiomatization which was inspired by the Grzegorczyk’s system, and which is its variation. Our aim is to compare the two approaches and show that they are slightly different. Except for pointing to dissimilarities, we also demonstrate that the theories coincide (in the sense that their axioms are satisfied in the same class of structures) in presence of axiom stipulating non-existence of atoms.

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On the Definability of Leśniewski’s Copula ‘is’ in Some Ontology-Like Theories

63%

Łyczak M., Pietruszczak A.

Bulletin of the Section of Logic

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2018

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tom 47

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nr 4

EN

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 (which corresponds to the universal name ‘object’) and we prove its adequacy with respect to the set-theoretic interpretation (again using an epimorphism theorem). 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.

Ograniczanie wyników

4 Bulletin of the Section of Logic

4 Pietruszczak A.

1 Gruszczyński R.

1 Łyczak M.

2 2018

2 2017

On Theses without Iterated Modalities of Modal Logics Between C1 and S5. Part 2

On Theses Without Iterated Modalities of Modal Logics Between C1 and S5. Part 1

A comparison of two systems of point-free topology

On the Definability of Leśniewski’s Copula ‘is’ in Some Ontology-Like Theories