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Due to many applications, the Choquet integral as a powerful tool for modeling non-deterministic problems needs to be further extended. Therefore the paper is devoted to a generalization of the Choquet integral. As a basis, the pseudo-integral for bounded integrand is extended to the case for nonnegative integrands at first, and then the generalized Choquet integral is defined. As special cases, pseudo-Choquet Stieltjes integrals, pseudo-fuzzy Stieltjes integrals, g-Choquet integrals, pseudo-(N)fuzzy integrals and pseudo-(S)fuzzy integrals are obtained, and various kinds of properties and convergence theorems are shown, meanwhile Markov, Jensen, Minkowski and Holder inequalities are proved. In the end, the generalized discrete Choquet integral is discussed. The obtained results for the generalized Choquet integral cover some previous results on different types of nonadditive integrals.

In this paper, we consider the problem of aggregating a collection of transitive fuzzy binary relations in such a way that the aggregation process preserves transitivity. Specifically, we focus our efforts on the characterization of those functions that aggregate a collection of fuzzy binary relations which are transitive with respect to a collection of t-norms preserving the transitivity. We characterize them in terms of triangular triplets. Further, the relationship between triangular triplets, the monotonicity of the aggregation function and an appropriate dominance notion is explicitly stated. Special attention is paid to a few classes of transitive fuzzy binary relations that are relevant in the literature. Concretely, we describe, in terms of triangular triplets, those functions that aggregate a collection of fuzzy pre-orders, fuzzy partial orders, relaxed indistinguishability relations, indistinguishability relations and equalities. A surprising relationship between functions that aggregate transitive fuzzy relations into a T M -transitive fuzzy relation and those that aggregate relaxed indistinguishability relations is shown. A few consequences of the new results are also provided for those cases in which all the t-norms of the given collection are the same. Some celebrated results are retrieved as a particular case from the exposed theory.

The paper studies the aggregation of pairs of T-indistinguishability operators. More concretely we address the question whether A ( E , E ′ ) is a T-indistinguishability operator if E , E ′ are T-indistinguishability operators. The answer depends on the aggregation function, the t-norm T, and the chosen T-indistinguishability operators. It is well-known that an aggregation function preserves T-transitive relations if and only if it dominates the t-norm T. We show the important role of the minimum t-norm T M in this preservation problem. In particular we develop weaker forms of domination that are used to provide characterizations of T M -indistinguishability preservation under aggregation. We also prove that the existence of a single strictly monotone aggregation that satisfies the indistinguishability operator preservation property guarantees all aggregations to have the same preservation property.

In this paper we introduce and study the algebra of almost strongly fuzzy bounded operators on fuzzy normed linear spaces. Some connections with the algebra of strongly fuzzy bounded operators are presented. The completeness of these algebras is also established. A characterization of bounded elements of the algebra of almost strongly fuzzy bounded operators is given. We also introduce two fuzzy spectral radii for almost strongly fuzzy bounded operators. For bounded elements of the considered algebra the equality between these fuzzy spectral radii is proved.

In this paper, the notions of regularity and normality of ( L , M ) -fuzzy topological spaces are introduced by using residual implication, where L and M are completely distributive De Morgan algebras. It is shown that ( L , M ) -fuzzy interior operator and ( L , M ) -fuzzy closure operator can be used to characterize regularity and normality. The relationships among separation axioms of an ( L , M ) -fuzzy topological space are discussed. Moreover, it is proved that the four separation axioms are equivalent to one another in an ( L , M ) -fuzzy metric space.

It is well known that metric spaces are an instance of categorical enrichment in a particular quantale. We show that in a categorically natural way a notion of Lipschitz norm arises in the context of an arbitrary diagram of quantales, instead of just one particular quantale. The generalised Lipschitz norm we present depends functorially on the diagram and is itself a functor to the indexing category of the diagram. The entire process is, in a way we make precise, an instance of a concrete Grothendieck construction.

Inspired by the notion of quasi-homogenous copulas, we introduce a new class of functions with a given curved section. The convexity of curved sections plays a key role in characterizing the corresponding copulas. This class of copulas generalizes the class of quasi-homogenous copulas.

There are two cases of the (conditional) distributivity of semi-t-operators over S-uninorms that have not been discussed by Fang and Hu in 2019. This paper further characterizes the necessary condition for the left (resp. right) distributivity of semi-t-operators over S-uninorms in these two cases. The sufficient conditions for the left (resp. right) distributivity of semi-t-operators over S-uninorms are also discussed. The outcome of one case is different from the conclusion obtained by Fang and Hu, whereby it is impossible to obtain the necessary and sufficient conditions for semi-t-operators distributed over S-uninorms as usual. In particular, this paper points out that the distributivity of semi-t-operators over S-uninorms is equivalent to their conditional distributivity over S-uninorms under the two cases mentioned above. Some examples are also presented to support our conclusions.

In this work, we introduce the notion of dG-Choquet integral, which generalizes the discrete Choquet integral replacing, in the first place, the difference between inputs represented by closed subintervals of the unit interval [0,1] by a dissimilarity function; and we also replace the sum by more general appropriate functions. We show that particular cases of dG-Choquet integral are both the discrete Choquet integral and the d-Choquet integral. We define interval-valued fuzzy measures and we show how they can be used with dG-Choquet integrals to define an interval-valued discrete Choquet integral which is monotone with respect to admissible orders. We finally study the validity of this interval-valued Choquet integral by means of an illustrative example in a classification problem. © 2021 This work was supported in part by the Spanish Ministry of Science and Technology, under project PID2019-108392GB-I00 (AEI/10.13039/501100011033), by the project PJUPNA-1926 of the Public University of Navarre and by the project VEGA 1/0267/21 .

An EQ-algebra has three basic binary operations (meet, multiplication and a fuzzy equality) and a top element. An lEQ-algebra is a lattice-ordered EQ-algebra satisfying the substitution property of the join operation. In this article, we study bounded residuated lEQ-algebras (BR-lEQ-algebras for short). We introduce a subvariety RL-EQ-algebras of BR-lEQ-algebras, and prove that the categories of RL-EQ-algebras and residuated lattices are categorical isomorphic. We also prove that RL-EQ-algebras are precisely the BR-lEQ-algebras that can be reconstructed from residuated lattices. We further show the existence of a closure operator on the poset of all BR-lEQ-algebras with the same lattice and multiplication reduct, the existence of the maximum element in the poset. Then we introduce filters in BR-lEQ-algebras and give a lattice isomorphism between the filter lattice and the congruence lattice. Finally, we prove that the category of residuated lattices is isomorphic to a reflective subcategory of BR-lEQ-algebras.

The purpose of this article is to establish a new result on the existence of a solution for a new class of fuzzy fractional differential inclusions with Clarke's subdifferential via the resolvent operator in Banach spaces. First, we introduce a new class of fuzzy fractional differential inclusions with Clarke's subdifferential via the resolvent operator in Banach spaces. Then, we establish existence conditions for this inclusion system using Bohnenblust-Karlin's fixed point theorem under some suitable conditions. Finally, we give an example to illustrate the main results. Our results are new even when applied to the corresponding fractional differential inclusions in the literature.

This article is a continuation of our research on a one-to-one correspondence between n-dimensional spectral resolutions and n-dimensional observables on lexicographic types of quantum structures which started in Dvurecenskij and Lachman ( https://doi.org/10.1016/j.fss.2021.05.005 ). There we presented the main properties of n-dimensional spectral resolutions and observables, and we studied in depth characteristic points which are crucial for our study. Here we present the main body of our research. We investigate a one-to-one correspondence between n-dimensional observables and n-dimensional spectral resolutions with values in a lexicographic form of quantum structures such as perfect MV-algebras or perfect effect algebras. The multidimensional version of this problem is more complicated than a one-dimensional one because if our algebraic structure is k-perfect for k > 1 , then even for the two-dimensional case of spectral resolutions we have more characteristic points. The results obtained are applied to the existence of an n-dimensional meet joint observable of n one-dimensional observables on a perfect MV-algebra and a sum of n-dimensional observables.

This paper further develops the study of construction of uninorms on bounded lattices. First, by using the fact that triangular norms (t-norms) and triangular conorms (t-conorms) on an arbitrary bounded lattice always exist, we present two new constructions of uninorms on an arbitrary bounded lattice L with an additional constraint on the neutral element. In addition, some illustrative examples for the new constructions of uninorms on bounded lattices are provided. Then, we illustrate how our new construction methods are different from some existing methods for the construction of uninorms on bounded lattices. Finally, we provide an answer to the open problem presented by Cayli.

This paper provides a complete classification of all t-norms on a family of lattices in terms of t-norms on discrete chains. Moreover, the cardinal of some classes on discrete chains is computed. Therefore, the number of t-norms on the family of lattices is obtained. Also, new results involving Archimedean and divisible t-norms are presented. Finally, we bring out dual results for t-conorms.

Based on both a given semi-t-norm and semi-t-conorm, we propose some methods to obtain semi-t-operators on bounded lattices, which are illustrated by some examples. We also discuss the relationship among semi-t-operators constructed.

We propose a new method for constructing nullnorms on bounded lattices based on the simultaneous use of a closure operator and an interior operator. This method requires as input a given underlying t-conorm on a join sub-semilattice (depending on the closure operator), a given underlying t-norm on a meet sub-semilattice (depending on the interior operator) and an auxiliary mapping. It is shown that a nullnorm constructed by Karacal et al. in 2015 and a nullnorm constructed by Ertugrul in 2018 can be retrieved as special cases. We also explain how to obtain the dual results.

We present a complete representation theorem for nullnorms on bounded lattices. Explicitly, any nullnorm on a bounded lattice can be represented in terms of two order-preserving maps, a triangular conorm, a triangular norm and a conditionally associative function. As a particular case, we retrieve the known representation of nullnorms only taking values that are comparable with the annihilator (in terms of two order-preserving maps, a triangular conorm and a triangular norm). As an illustration of the representation theorem, we generate construction methods for nullnorms, especially those taking at least one value that is incomparable with the annihilator.

The main aim of this paper is to characterize two wide classes of nullnorms on bounded lattices. First, we construct two methods of nullnorms on bounded lattices and find sufficient and necessary conditions for nullnorms that possess an annihilator. Moreover, we prove that the conditions are equivalent and illustrate them with examples. Finally, we present another two nullnorms on bounded lattices and illustrate that they differ from nullnorms constructed by Cayli in 2020.

First, we present construction methods for interior operators on a meet semilattice. Second, under the assumption that the underlying meet semilattices constitute the range of an interior operator, we prove an ordinal sum theorem for countably many (finite or countably infinite) triangular norms on bounded meet semilattices, which unifies and generalizes two recent results: one by Dvořak and Holcapek and the other by some of the present authors. We also characterize triangular norms that are representable as the ordinal sum of countably many triangular norms on given bounded meet semilattices.

The purpose of this paper is to establish some fixed point results for a pair of fuzzy dominated mappings satisfying contractive conditions on closed ball in multiplicative metric space. Some new fixed point results with graphic contraction on closed ball for a pair of fuzzy graph dominated mappings on multiplicative metric space have been established. Furthermore, we find a unique common solution for a system of non linear Voltera type integral equations and lastly we give an application to ensure the existence of common bounded solution of a functional equation in dynamic programming.

The aim of this paper is to investigate some laws of large numbers for multidimensional arrays of level-wise negatively associated and level-wise pairwise negatively dependent random upper semicontinuous functions under various settings. We also provide some Rosenthal's type and Hajek-Renyi's type maximal inequalities for multi-dimensional structure. Our results are extensions of corresponding ones in the literature.

This paper establishes the relationship between the linearly correlated difference and the generalized Hukuhara difference of fuzzy numbers. Specifically, these two types of differences are slightly different when the basic fuzzy number is non-symmetric. But they are completely coincident when the basic fuzzy number is symmetric. The main difference of two types of differences is that the linearly correlated difference always exists in the space of linearly correlated fuzzy numbers, while the generalized Hukuhara difference does not necessarily exist. Furthermore, the relationship between the linearly correlated derivative and the generalized Hukuhara derivative is also examined with the help of the relationship between two types of differences. It is interesting that, under certain appropriate conditions, the linearly correlated differentiability and the generalized Hukuhara differentiability are equivalent for a linearly correlated fuzzy number-valued function regardless of whether the basic fuzzy number is symmetric or not. Compared with the generalized Hukuhara difference and the generalized Hukuhara derivative, the calculation of the linearly correlated difference and the linearly correlated derivative is easier by using the corresponding representation functions.

In this paper, we propose new concepts for differentiability of fuzzy number-valued functions. These derivatives are based on the Hausdorff distance between fuzzy numbers. We study some properties of the proposed notions of differentiability and compare these derivatives with other well-known derivatives. We also present the characterization theorems of the new derivatives in terms of the differentiability of their endpoint functions.

Recently, Zhang et al. studied the distributivity between uni-nullnorms and overlap (grouping) functions [68] . They obtained the sufficient and necessary conditions for the distributivity equations between them. However, the distributivity between null-uninorms and overlap (grouping) functions is missing. To fill this gap, in this paper, we explore some new results on the distributivity equations between overlap (grouping) functions and null-uninorms, which are the generalization of uninorms and nullnorms. Meanwhile, we give the full characterization of any idempotent null-uninorm.

We introduce the concept of maximal directions of increasingness (resp. decreasingness) of an aggregation function. In the bivariate case, we derive these maximal directions with respect to points on the main diagonal of the unit square for a symmetric aggregation function that has either piecewise convex or piecewise concave level curves and is differentiable up to second order. With any bivariate aggregation function of the latter type we associate another bivariate aggregation function that has the same maximal directions of increasingness (resp. decreasingness) while having straight lines as level curves. We explore under which conditions the latter aggregation function is a semi-copula, a quasi-copula or a copula. As a by-product we establish a new construction method for aggregation functions with given diagonal section.

The (conditional) distributivity plays an important role in the construction of integrals. In this work, we aim to characterize the conditional distributivity equation only involving uninorms, where the second uninorm belongs to the well-known classes.

This paper is a sequel to \cite{Tz10}, where a local version of ZFC, LZFC, was introduced and examined and transitive models of ZFC with properties that resemble large cardinal properties, namely Mahlo and $\Pi_1^1$-indescribable models, were considered. By analogy we refer to such models as "large models", and the properties in question as "large model properties". Continuing here in the same spirit we consider further large model properties, that resemble stronger large cardinals, namely, "elementarily embeddable", "extendible" and "strongly extendible", "critical" and "strongly critical", "self-critical'' and "strongly self-critical", the definitions of which involve elementary embeddings. Each large model property $\phi$ gives rise to a localization axiom $Loc^{\phi}({\rm ZFC})$ saying that every set belongs to a transitive model of ZFC satisfying $\phi$. The theories ${\rm LZFC}^\phi={\rm LZFC}$+$Loc^{\phi}({\rm ZFC})$ are local analogues of the theories ZFC+"there is a proper class of large cardinals $\psi$", where $\psi$ is a large cardinal property. If $sext(x)$ is the property of strong extendibility, it is shown that ${\rm LZFC}^{sext}$ proves Powerset and $\Sigma_1$-Collection. In order to refute $V=L$ over LZFC, we combine the existence of strongly critical models with an axiom of different flavor, the Tall Model Axiom ($TMA$). $V=L$ can also be refuted by $TMA$ plus the axiom $GC$ saying that "there is a greatest cardinal", although it is not known if $TMA+GC$ is consistent over LZFC. Finally Vop\v{e}nka's Principle ($VP$) and its impact on LZFC are examined. It is shown that ${\rm LZFC}^{sext}+VP$ proves Powerset and Replacement, i.e., ZFC is fully recovered. The same is true for some weaker variants of ${\rm LZFC}^{sext}$. Moreover the theories LZFC$^{sext}$+$VP$ and ZFC+$VP$ are shown to be identical., Comment: 32 pages

The present paper extends some previous works studying automorphisms in type-2 fuzzy sets. The framework for the analysis is the set of convex and normal functions from [ 0 , 1 ] to [ 0 , 1 ] (fuzzy truth values). The paper concentrates on those automorphisms that, in this framework, leave the constant function 1 fixed. This function is quite important since it defines the boundary between the functions that represent “TRUE” (increasing functions) and those that represent “FALSE” (decreasing functions), being at the same time the only normal function that is simultaneously increasing and decreasing. While C.L. Walker, E.A. Walker and J. Harding introduced in 2008 a family of functions leaving the constant function 1 fixed, the main goal of this paper is to prove that the functions of that family are in fact automorphisms, and moreover, that they are the only automorphisms (in the mentioned set of convex and normal functions from [ 0 , 1 ] to [ 0 , 1 ] ) that preserve the function 1.

Decomposition integrals provide a framework for non-linear integrals that include Choquet, Shilkret, the PAN, and the concave integrals. All of these integrals found their applications in mathematics, notably in decision-making and economy. An important class of decomposition integrals is the class of integrals extending Lebesgue integral in the sense that the decomposition integral with respect to classical measures coincides with Lebesgue integral. In this paper, we consider finite spaces X only and discuss some necessary and sufficient conditions for this property. Also, some construction methods are given and exemplified.

In this study, we generalize the concept of the k-order additivity of a set function. First, we discuss the Mobius transform for a non-discrete set function. Next, we generalize the definition of the k-order additivity of a set function using the Mobius transform and provide the equivalent conditions for the k-order additivity. Furthermore, we consider the k-order additivity of the distorted monotone measure. We prove that under certain conditions, a distorted measure is k-order additive if and only if the distortion function is a polynomial of k-th order.

We investigate several consequences of inclusion relations between quantified provability logics. Moreover, we give a necessary and sufficient condition for the inclusion relation between quantified provability logics with respect to $\Sigma_1$ arithmetical interpretations., Comment: 22 pages

We examine sufficient and necessary conditions for the Choquet integral on the support of its integrand to be equal to the integral on a set which includes this support (we consider the problem in the case of a locally compact Hausdorff space).

In this paper we provide a full characterization of functions G : [ 0 , y ¯ ] 2 → [ 0 , y ¯ ] satisfying the distributivity equation G ( a ∘ b , a ∘ c ) = a ∘ G ( b , c ) for any a , b , c with a fixed binary operation ∘ : [ 0 , y ¯ ] 2 → [ 0 , y ¯ ] generalizing minimum and maximum, where y ¯ ∈ ( 0 , ∞ ] . The above functional equation with an admissible function G and ∘ being minimum [resp. maximum] has appeared when studying properties of minitive [resp. maxitive] homogeneity for the recently introduced upper n-Sugeno integral.

It is remarked that the generalization of Jensen's inequality for pseudo-integrals (Pap and Strboja [14] ) is not a complete generalization of the classical Jensen's inequality, and a generalized Jensen's inequality for pseudo-integral with respect to ( ⊕ , ⊗ ) -pseudo-convex function is given in [28] . The present article is a continuation of the previous work. A new notion of ( ⊕ 1 , ⊗ 1 ) , ( ⊕ 2 , ⊗ 2 ) -pseudo-convex function is introduced, which generalize the notion of ( ⊕ , ⊗ ) -pseudo-convex function and many other previously generalizations. Motivated by the work of Kaluszka et al. [6] , related to Jensen's inequality with respect to different generalized fuzzy integrals, a new generalized Jensen's inequality between a pseudo-integral and general fuzzy integral, as well as between two different pseudo-integrals, with respect to ( ( ⊕ 1 , ⊗ 1 ) , ( ⊕ 2 , ⊗ 2 ) ) -pseudo-convex functions are proved. These results cover all previously obtained Jensen's inequalities for pseudo-integrals (Zhang and Pap [28] ) as well as the classical Jensen's inequality.

In this paper, we consider a new family of multivariate copulas described by a sequence of functions, named as AMO copula. The set of AMO copulas corresponds to a class of multivariate shock models with the Archimedean type of dependence. Sufficient conditions on the involved sequence of functions to obtain a multivariate copula are given and the probabilistic structure of the AMO copula is provided. We show that the family of AMO copulas is a generalization of Archimedean and Marshall-Olkin copulas family, and it includes some well-known copulas as specific cases. An alternative method for generating random vectors from AMO copulas via distortion functions is provided. In addition, a singular component along the main diagonal of the AMO copula is also verified. Finally, the tail behaviors of AMO copulas are discussed and some numerical illustrations are provided.

In this paper, construction methods of overlap and grouping functions on complete lattices via complete homomorphisms and complete 0 L , 1 L -endomorphisms are investigated. At first, we propose the notion of O-generator triple of overlap functions. Then, properties such as ( α , B , C ) -migrativity, ( B , C ) -homogeneity, idempotency and cancellation law for the overlap functions obtained by such generator triples are discussed. Finally, we give an analogous discussion on grouping functions on complete lattices.

In 2018, Qiao and Hu [48] studied the α-migrativity of uninorms over overlap and grouping functions when the uninorm U belongs to one certain class (e.g., U min , U max , the family of idempotent uninorms, representable uninorms or uninorms continuous on ] 0 , 1 [ 2 ). In addition, they obtained some equivalent characterizations of them. This paper will continue to consider the characterizations of this kind of migrativity equations by means of the ordinal sum of overlap and grouping functions. We give the necessary and sufficient conditions for the solutions of the ( α , O ) -migrativity equation when the uninorm U becomes a t-norm or a conjunctive uninorm locally internal on the boundary and the ( α , G ) -migrativity equation when the uninorm U becomes a t-conorm or a disjunctive uninorm locally internal on the boundary, respectively.

The omnipotent instrument for modeling multivariate dependence of random variables in standard probability theory has become copulas discovered by A. Sklar in 1959. Only recently Omladic and Stopar prove that in the bivariate case an analogous role is played by exactly the same copulas for random variables coming from finitely additive probability spaces. One of the main results of this paper is that this is true also in the general multivariate case. The extension to n dimensions requires a better understanding of quasi-copulas, the lattice closure of copulas with respect to the pointwise order. We need to develop a new equivalent definition of this notion that should be useful in other applications as well. Another tool we introduce and seems to be new even in the standard probability approach, is multivariate quasi-distributions. We also expand the coherence theory for quasi-copulas and quasi-distributions to the multivariate situation. Finally, our main result is a multivariate Sklar type theorem in the imprecise setting.

Nullnorms with a zero element being at any point of a bounded lattice are an important generalization of triangular norms and triangular conorms. This paper obtains an equivalent characterization for the existence of idempotent nullnorms with the zero element a on any bounded lattice containing only two distinct elements incomparable with a. Furthermore, some basic properties for the bounded lattice containing only two distinct elements incomparable with a are presented.

This paper continues to study the construction of uni-nullnorms on bounded lattices. At first, we introduce a new method for constructing the smallest uni-nullnorm on an arbitrary bounded lattice L having the elements e , a ∈ L , based on the existence of a uninorm on [ 0 , a ] 2 with the neutral element e and a triangular norm on [ a , 1 ] 2 . And then, we propose another new approach to obtain the largest uni-nullnorm on L via a uninorm on [ 0 , a ) 2 with the neutral element e and a triangular norm on [ a , 1 ] 2 . Furthermore, we provide some corresponding examples to illustrate that our construction methods differ from the existing ones.

Uninorms on bounded lattices have recently attracted widespread attention. In this study, we first propose the necessary structure for uninorms with Archimedean underlying t-norms and t-conorms on bounded lattices. We also discuss the characterization of more general classes of uninorms. We then study the necessity for a particular structure of uninorms that is required by many existing methods for constructing uninorms on bounded lattices. Finally, we propose a construction approach as an example.

We study the extreme points (in the Krein-Milman sense) of the class of semilinear copulas and provide their characterization. Related results into the more general setting of conjunctive aggregation functions (i.e., semi–copulas and quasi–copulas) are also presented.

The construction of uninorms on bounded lattices has been a attractive research area. In this paper, we introduce some new classes of uninorms on bounded lattices via t-subnorms and t-subconorms with a neutral element e under some constraints. We also give some illustrative examples to help understand the structure of these new uninorms. Moreover, we provide an answer to the open problem raised by Cayli in 2019. Finally, the differences between our methods and existing approaches are assessed.

We introduce three broad classes of nullnorms on a bounded lattice and lay bare the structure of their members. For that purpose, we introduce particular subsets of a bounded lattice, called upper (resp. lower) beams, and appropriate associative operations on them, called beam (resp. dual beam) operations, which conveniently generalize triangular norms (resp. triangular conorms). It is shown that nullnorms in the first (resp. second) class are characterized by a triangular conorm (resp. triangular norm) and a beam operation (resp. dual beam operation), while nullnorms in the third class are characterized by a triangular conorm and a triangular norm. We also discuss the relationships among these three classes.

We show that the weak monadic second order theory of the structure $({\mathbb Q}

In the literature, predicativism is connected not only with the Vicious Circle Principle but also with the idea that certain totalities are inherently potential. To explain the connection between these two aspects of predicativism, we explore some approaches to predicativity within the modal framework for potentiality developed in Linnebo (2013) and Linnebo and Shapiro (2019). This puts predicativism into a more general framework and helps to sharpen some of its key theses.

We prove two theorems concerning indestructibility properties of the first two strongly compact cardinals when these cardinals are in addition the first two measurable cardinals. Starting from two supercompact cardinals $\kappa _1 < \kappa _2$ , we force and construct a model in which $\kappa _1$ and $\kappa _2$ are both the first two strongly compact and first two measurable cardinals, $\kappa _1$ ’s strong compactness is fully indestructible (i.e., $\kappa _1$ ’s strong compactness is indestructible under arbitrary $\kappa _1$ -directed closed forcing), and $\kappa _2$ ’s strong compactness is indestructible under $\mathrm {Add}(\kappa _2, \delta )$ for any ordinal $\delta $ . This provides an answer to a strengthened version of a question of Sargsyan found in [17, Question 5]. We also investigate indestructibility properties that may occur when the first two strongly compact cardinals are not only the first two measurable cardinals, but also exhibit nontrivial degrees of supercompactness.