In many educational systems, ethnic minority students score lower in their academic achievement, and consequently, teachers develop low expectations regarding this student group. Relatedly, teachers’ implicit attitudes, explicit expectations, and causal attributions also differ between ethnic minority and ethnic majority students—all in a disadvantageous way for ethnic minority students. However, what is not known so far, is how attitudes and causal attributions contribute together to teachers’ judgments. In the current study, we explored how implicit attitudes and causal attributions contribute to (...) preservice teachers’ judgments of the low educational success of an ethnic minority student. Results showed that both implicit attitudes and causal attributions predicted language proficiency and intelligence judgments. Negative implicit attitudes, assessed with the IRAP, and internal stable causal attributions led to lower judgments of language proficiency, whereas lower judgments of intelligence were predicted by positive implicit attitudes and higher judgments of intelligence by external stable attributions. Substantial differences in the prediction of judgments could be found between the IRAP and BIAT as measures of implicit attitudes. (shrink)

Kleene’s strong three-valued logic extends naturally to a four-valued logic proposed by Belnap. We introduce a guard connective into Belnap’s logic and consider a few of its properties. Then we show that by using it four-valued analogs of Kleene’s weak three-valued logic, and the asymmetric logic of Lisp are also available. We propose an extension of these ideas to the family of distributive bilattices. Finally we show that for bilinear bilattices the extensions do not produce any new equivalences.

This paper presents a weak Kleene approach to conditionals that preserves some salient formal features of conditionals, particularly their interdefinability with Boolean logical connectives. I argue that such an approach fares better than other proposed weak logics of conditionals in this regard. In particular, it fares better than the logics proposed by Cooper, Cantwell, Farrell, De Finetti, Égré, Rossi, and Sprenger.

A structural theorem for Kleene algebras is proved, showing that an element of a Kleene algebra can be looked upon as an ordered pair of sets, and that negation with the Kleene property is describable by the set-theoretic complement. The propositional logic \ of Kleene algebras is shown to be sound and complete with respect to a 3-valued and a rough set semantics. It is also established that Kleene negation can be considered as a modal (...) operator, due to a perp semantics of \. Moreover, another representation of Kleene algebras is obtained in the class of complex algebras of compatibility frames. One concludes with the observation that \ can be imparted semantics from different perspectives. (shrink)

Kleene’s well-known strong three-valued logic is shown to be one of a family of logics with similar mathematical properties. These logics are produced by an intuitively natural construction. The resulting logics have direct relationships with bilattices. In addition they possess mathematical features that lend themselves well to semantical constructions based on fixpoint procedures, as in logic programming.

A many-one degree is functional if it contains the index set of some class of partial recursive functions but does not contain an index set of a class of r.e. sets. We give a natural embedding of the r.e. m-degrees into the functional degrees of 0'. There are many functional degrees in 0' in the sense that every partial-order can be embedded. By generalizing, we show there are many functional degrees in every complete Turning degree. There is a natural tie (...) between the studies of index sets and partial-many-one reducibility. Every partial-many-one degree contains one or two index sets. (shrink)

The Swedish artist and writer Tyra Kleen was a professional artist and a constant traveller who had a great interest in different religious questions. This article describes her Symbolist artistry, her interest in Theosophy and her journeys to India and Asia.

This little gem is stated unbilled and proved in the last two lines of §2 of the short note Kleene [1938]. In modern notation, with all the hypotheses stated explicitly and in a strong form, it reads as follows:Second Recursion Theorem. Fix a set V ⊆ ℕ, and suppose that for each natural number n ϵ ℕ = {0, 1, 2, …}, φn: ℕ1+n ⇀ V is a recursive partial function of arguments with values in V so that the (...) standard assumptions and hold with. Every n-ary recursive partial function with values in V is for some e. For all m, n, there is a recursive function : Nm+1 → ℕ such that.Then, for every recursive, partial function f of arguments with values in V, there is a total recursive function of m arguments such thatProof. Fix e ϵ ℕ such that and let.We will abuse notation and write ž; rather than ž() when m = 0, so that takes the simpler formin this case ). (shrink)

Paraconsistent Weak Kleene logic is the 3-valued logic with two designated values defined through the weak Kleene tables. This paper is a first attempt to investigate PWK within the perspective and methods of abstract algebraic logic. We give a Hilbert-style system for PWK and prove a normal form theorem. We examine some algebraic structures for PWK, called involutive bisemilattices, showing that they are distributive as bisemilattices and that they form a variety, \, generated by the 3-element algebra WK; (...) we also prove that every involutive bisemilattice is representable as the Płonka sum over a direct system of Boolean algebras. We then study PWK from the viewpoint of AAL. We show that \ is not the equivalent algebraic semantics of any algebraisable logic and that PWK is neither protoalgebraic nor selfextensional, not assertional, but it is truth-equational. We fully characterise the deductive filters of PWK on members of \ and the reduced matrix models of PWK. Finally, we investigate PWK with the methods of second-order AAL—we describe the class \ of PWK-algebras, algebra reducts of basic full generalised matrix models of PWK, showing that they coincide with the quasivariety generated by WK—which differs from \—and explicitly providing a quasiequational basis for it. (shrink)

Kleene algebras and action logic were proposed to be solutions to the finite axiomatization problem of the algebra of regular sets (of strings). They are treated here as nonclassical logics—with Hilbert-style axiomatizations and semantics. We also provide intuitive accounts in terms of information states of the semantics which provide further insights into the formalisms. The three types of "Kripke-style'' semantics which we define develop insights from gaggle theory, and from our four-valued and generalized Kripke semantics for the minimal substructural (...) logic. Soundness and completeness are proven each time. (shrink)

By using the notions of exact truth and exact falsity, one can give 16 distinct definitions of classical consequence. This paper studies the class of relations that results from these definitions in settings that are paracomplete, paraconsistent or both and that are governed by the Strong Kleene schema. Besides familiar logics such as Strong Kleene logic, the Logic of Paradox and First Degree Entailment, the resulting class of all Strong Kleene generalizations of classical logic also contains a (...) host of unfamiliar logics. We first study the members of our class semantically, after which we present a uniform sequent calculus that is sound and complete with respect to all of them. Two further sequent calculi and \ calculus) will be considered, which serve the same purpose and which are obtained by applying general methods to construct sequent calculi for many-valued logics. Rules and proofs in the SK calculus are much simpler and shorter than those of the \ and the \ calculus, which is one of the reasons to prefer the SK calculus over the latter two. Besides favourably comparing the SK calculus to both the \ and the \ calculus, we also hint at its philosophical significance. (shrink)

There is a familiar derivation of G¨ odel’s Theorem from the proof by diagonalization of the unsolvability of the Halting Problem. That proof, though, still involves a kind of self-referential trick, as we in effect construct a sentence that says ‘the algorithm searching for a proof of me doesn’t halt’. It is worth showing, then, that some core results in the theory of partial recursive functions directly entail G¨ odel’s First Incompleteness Theorem without any further self-referential trick.

This paper presents a semantical analysis of the Weak Kleene Logics Kw3 and PWK from the tradition of Bochvar and Halldén. These are three-valued logics in which a formula takes the third value if at least one of its components does. The paper establishes two main results: a characterisation result for the relation of logical con- sequence in PWK – that is, we individuate necessary and sufficient conditions for a set.

We consider various classes of algebras obtained by expanding idempotent semirings with meet, residuals and Kleene-*. An investigation of congruence properties (e-permutability, e-regularity, congruence distributivity) is followed by a section on algebraic Gentzen systems for proving inequalities in idempotent semirings, in residuated lattices, and in (residuated) Kleene lattices (with cut). Finally we define (one-sorted) residuated Kleene lattices with tests to complement two-sorted Kleene algebras with tests.

The Hilbert–Bernays Theorem establishes that for any satisfiable first-order quantificational schema S, one can write out linguistic expressions that are guaranteed to yield a true sentence of elementary arithmetic when they are substituted for the predicate letters in S. The theorem implies that if L is a consistent, fully interpreted language rich enough to express elementary arithmetic, then a schema S is valid if and only if every sentence of L that can be obtained by substituting predicates of L for (...) predicate letters in S is true. The theorem therefore licenses us to define validity substitutionally in languages rich enough to express arithmetic. The heart of the theorem is an arithmetization of Gödel's completeness proof for first-order predicate logic. Hilbert and Bernays were the first to prove that there is such an arithmetization. Kleene established a strengthened version of it, and Kreisel, Mostowski, and Putnam refined Kleene's result. Despite the later refinements, Kleene's presentation of th.. (shrink)

Kleene's realizability interpretation for first-order arithmetic was shown by Hyland to fit into the internal logic of an elementary topos, the “Effective topos” . In this paper it is shown, that there is an internal realizability definition in , i.e. a syntactical translation of the internal language of into itself of form “n realizes ” , which extends Kleene's definition, and such that for sentences , the equivalence [harr]n is true in . The internal realizability definition depends on (...) finding separated covers for non-separated objects of . However, for the objects arising in higher-order arithmetic, canonical covers are available, which are definable in higher-order arithmetic. These canonical covers yield “covering axioms”. It is shown that these covering axioms, together with uniformity principles and Extended Church's Thesis, axiomatize a formalized extension of Kleene realizability to a constructive system of higher-order arithmetic. The details are worked out for second and third-order arithmetic, but the method can be extended to any order. As an application, it is shown in the final section that a certain completeness property of the category of “modest sets” can be derived in third-order arithmetic from the realizability axioms. (shrink)

A possible world semantics for standard modal languages is presented, where the valuation functions are allowed to be partial, the truth--functional connectives are interpreted according to weak Kleene matrices, and the necessity operator is given a "weak" interpretation. Completeness and incompleteness results for some (axiomatic) systems are then established. Extensions of these modal logics in which figure "statability" operators are also examined.

It is well known that the following features hold of AR + T under the strong Kleene scheme, regardless of the way the language is Gödel numbered: 1. There exist sentences that are neither paradoxical nor grounded. 2. There are 2ℵ0 fixed points. 3. In the minimal fixed point the weakly definable sets (i.e., sets definable as {n∣ A(n) is true in the minimal fixed point where A(x) is a formula of AR + T) are precisely the Π1 1 (...) sets. 4. In the minimal fixed point the totally defined sets (sets weakly defined by formulae all of whose instances are true or false) are precisely the ▵1 1 sets. 5. The closure ordinal for Kripke's construction of the minimal fixed point is ωCK 1. In contrast, we show that under the weak Kleene scheme, depending on the way the Gödel numbering is chosen: 1. There may or may not exist nonparadoxical, ungrounded sentences. 2. The number of fixed points may be any positive finite number, ℵ0, or 2ℵ0 . 3. In the minimal fixed point, the sets that are weakly definable may range from a subclass of the sets 1-1 reducible to the truth set of AR to the Π1 1 sets, including intermediate cases. 4. Similarly, the totally definable sets in the minimal fixed point range from precisely the arithmetical sets up to precisely the ▵1 1 sets. 5. The closure ordinal for the construction of the minimal fixed point may be ω, ωCK 1, or any successor limit ordinal in between. In addition we suggest how one may supplement AR + T with a function symbol interpreted by a certain primitive recursive function so that, irrespective of the choice of the Godel numbering, the resulting language based on the weak Kleene scheme has the five features noted above for the strong Kleene language. (shrink)

Barrio et al. (_Journal of Philosophical Logic_, _49_(1), 93–120, 2020 ) and Pailos (_Review of Symbolic Logic_, _2020_(2), 249–268, 2020 ) develop an approach to define various metainferential hierarchies on strong Kleene models by transferring the idea of distinct standards for premises and conclusions from inferences to metainferences. In particular, they focus on a hierarchy named the \(\mathbb {S}\mathbb {T}\) -hierarchy where the inferential logic at the bottom of the hierarchy is the non-transitive logic ST but where each subsequent (...) metainferential logic ‘says’ about the former logic that it is transitive. While Barrio et al. ( 2020 ) suggests that this hierarchy is such that each subsequent level ‘in some intuitive sense, more classical than’ the previous level, Pailos ( 2020 ) proposes an extension of the hierarchy through which a ‘fully classical’ metainferential logic can be defined. Both Barrio et al. ( 2020 ) and Pailos ( 2020 ) explore the hierarchy in terms of semantic definitions and every proof proceeds by a rather cumbersome reasoning about those semantic definitions. The aim of this paper is to present and illustrate the virtues of a proof-theoretic tool for reasoning about the \(\mathbb {S}\mathbb {T}\) -hierarchy and the other metainferential hierarchies definable on strong Kleene models. Using the tool, this paper argues that each level in the \(\mathbb {S}\mathbb {T}\) -hierarchy is non-classical to an equal extent and that the ‘fully classical’ metainferential logic is actually just the original non-transitive logic ST ‘in disguise’. The paper concludes with some remarks about how the various results about the \(\mathbb {S}\mathbb {T}\) -hierarchy could be seen as a guide to help us imagine what a non-transitive metalogic for ST would tell us about ST. In particular, it teaches us that ST is from the perspective of ST as metatheory not only non-transitive but also transitive. (shrink)

Paraconsistent weak Kleene logic is the $3$-valued logic based on the weak Kleene matrices and with two designated values. In this paper, we investigate the poset of prevarieties of generalized involutive bisemilattices, focussing in particular on the order ideal generated by Α$\textrm{lg} $. Applying to this poset a general result by Alexej Pynko, we prove that, exactly like Priest’s logic of paradox, $\textrm{PWK}$ has only one proper nontrivial extension apart from classical logic: $\textrm{PWK}_{\textrm{E}}\textrm{,}$ PWK logic plus explosion. This (...) $6$-valued logic, unlike $\textrm{PWK} $, fails to be paraconsistent. We describe its consequence relation via a variable inclusion criterion and identify its Suszko-reduced models. (shrink)

A Kleene lattice is a distributive lattice equipped with an antitone involution and satisfying the so-called normality condition. These lattices were introduced by J. A. Kalman. We extended this concept also for posets with an antitone involution. In our recent paper (Chajda, Länger and Paseka, in: Proceeding of 2022 IEEE 52th International Symposium on Multiple-Valued Logic, Springer, 2022), we showed how to construct such Kleene lattices or Kleene posets from a given distributive lattice or poset and a (...) fixed element of this lattice or poset by using the so-called twist product construction, respectively. We extend this construction of Kleene lattices and Kleene posets by considering a fixed subset instead of a fixed element. Moreover, we show that in some cases, this generating poset can be embedded into the resulting Kleene poset. We investigate the question when a Kleene poset can be represented by a Kleene poset obtained by the mentioned construction. We show that a direct product of representable Kleene posets is again representable and hence a direct product of finite chains is representable. This does not hold in general for subdirect products, but we show some examples where it holds. We present large classes of representable and non-representable Kleene posets. Finally, we investigate two kinds of extensions of a distributive poset $${\textbf{A}}$$ A, namely its Dedekind-MacNeille completion $${{\,\mathrm{\textbf{DM}}\,}}({\textbf{A}})$$ DM ( A ) and a completion $$G({\textbf{A}})$$ G ( A ) which coincides with $${{\,\mathrm{\textbf{DM}}\,}}({\textbf{A}})$$ DM ( A ) provided $${\textbf{A}}$$ A is finite. In particular we prove that if $${\textbf{A}}$$ A is a Kleene poset then its extension $$G({\textbf{A}})$$ G ( A ) is also a Kleene lattice. If the subset X of principal order ideals of $${\textbf{A}}$$ A is involution-closed and doubly dense in $$G({\textbf{A}})$$ G ( A ) then it generates $$G({\textbf{A}})$$ G ( A ) and it is isomorphic to $${\textbf{A}}$$ A itself. (shrink)

In the literature, Weak Kleene logics are usually taken as three-valued logics. However, Suszko has challenged the main idea of many-valued logic claiming that every logic can be presented in a two-valued fashion. In this paper, we provide two-valued semantics for the Weak Kleene logics and for a number of four-valued subsystems of them. We do the same for the so-called Logics of Nonsense, which are extensions of the Weak Kleene logics with unary operators that allow looking (...) at them as Logics of Formal Inconsistency and Logics of Formal Underterminedness. Our aim with this work, rather than arguing for Suszko’s thesis, is to show that two-valued presentations of these peculiar logics enlighten the non-standard behavior of their logical connectives. More specifically, the two-valued presentations of paraconsistent logics illustrate and clarify the disjunctive flavor of the conjunction, and dually, the two-valued presentations of paracomplete subsystems of Weak Kleene logics reveal the conjunctive flavor of the disjunction. (shrink)

Let A and B be subsets of the reals. Say that A κ ≥ B, if there is a real a such that the relation "x ∈ B" is uniformly Δ 1 (a, A) in L[ ω x,a,A 1 , x,a,A]. This reducibility induces an equivalence relation $\equiv_\kappa$ on the sets of reals; the $\equiv_\kappa$ -equivalence class of a set is called its Kleene degree. Let K be the structure that consists of the Kleene degrees and the induced (...) partial order K ≥. A substructure of K that is of interest is P, the Kleene degrees of the Π 1 1 sets of reals. If sharps exist, then there is not much to P, as Steel [9] has shown that the existence of sharps implies that P has only two elements: the degree of the empty set and the degree of the complete Π 1 1 set. Legrand [4] used the hypothesis that there is a real whose sharp does not exist to show that there are incomparable elements in P; in the context of V = L, Hrbacek has shown that P is dense and has no minimal pairs. The Hrbacek results led Simpson [6] to make the following conjecture: if V = L, then p forms a universal homogeneous upper semilattice with 0 and 1. Simpson's conjecture is shown to be false by showing that if V = L, then Godel's maximal thin Π 1 1 set is the infimum of two strictly larger elements of P. The second main result deals with the notion of jump in K. Let A' be the complete Kleene enumerable set relative to A. Say that A is low-n if A (n) has the same degree as $\varnothing^{(n)}$ , and A is high-n if A (n) has the same degree as $\varnothing^{(n + 1)}$ . Simpson and Weitkamp [7] have shown that there is a high (high-1) incomplete Π 1 1 set in L. They have also shown that various other Π 1 1 sets are neither high nor low in L. Legrand [5] extended their results by showing that, if there is a real x such that x # does not exist, then there is an element of P that, for all n, is neither low-n nor high-n. In § 2, ZFC is used to show that, for all n, if A is Π 1 1 and low-n then A is Borel. The proof uses a strengthened version of Jensen's theorem on sequences of admissible ordinals that appears in [7, Simpson-Weitkamp]. (shrink)

Concerning Post's problem for Kleene degrees and its relativization, Hrbacek showed in [1] and [2] that if V = L, then Kleene degrees of coanalytic sets are dense, and then for all K ⊆ωω, there are N1 sets which are Kleene semirecursive in K and not Kleene recursive in each other and K. But the density of Kleene semirecursive in K Kleene degrees is not obtained from these theorems. In this note, we extend these (...) theorems by showing that if V = L, then for all K ⊆ ωω, Kleene semirecursive in K Kleene degrees are dense above K. (shrink)

In this paper several assumptions concerning omniscience and future contingents on the one side, and omniscience and self-reference on the other, areexamined with respect to a classical and a three-valued semantic setting (the latter pertains especially to Łukasiewicz’s, Kleene’s and Blau’s three-valued logics).Interesting features of both settings are highlighted and their basic assumptions concerning omniscience are explored. To generate a context in which the notion of omniscience does not deviate from some basic intuitions, two special futurity operators are introduced (...) in this article: one for what will definitely take place and another one for what is indeterminate as to whether it will take place. Once these operators are introduced, some puzzles about omniscience in combination with future contingents are removed. An analogous solution to some puzzles concerning omniscience and selfreferentiality is also provided. (shrink)

In this paper we study the subdirectly irreducible algebras in the variety of pseudocomplemented De Morgan algebras by means of their De Morgan p‐spaces. We introduce the notion of the body of an algebra and determine when is subdirectly irreducible. As a consequence of this, in the case of pseudocomplemented Kleene algebras, two special subvarieties arise naturally, for which we give explicit identities that characterise them. We also introduce a subvariety of, namely the variety of bundle pseudocomplemented Kleene (...) algebras, fully describe its subvariety lattice and find explicit equational bases for each subvariety. In addition, we study the subvariety of generated by the simple members of, determine the structure of the free algebra over a finite set in this variety and their finite weakly projective algebras. (shrink)

A proof is given that 0′ is definable in the structure of the degrees of unsolvability. This answers a long-standing question of Kleene and Post, and has a number of corollaries including the definability of the jump operator.

We consider the Lambek calculus, or noncommutative multiplicative intuitionistic linear logic, extended with iteration, or Kleene star, axiomatised by means of an$\omega $-rule, and prove that the derivability problem in this calculus is$\Pi _1^0$-hard. This solves a problem left open by Buszkowski (2007), who obtained the same complexity bound for infinitary action logic, which additionally includes additive conjunction and disjunction. As a by-product, we prove that any context-free language without the empty word can be generated by a Lambek grammar (...) with unique type assignment, without Lambek’s nonemptiness restriction imposed (cf. Safiullin, 2007). (shrink)

A possible world semantics for standard modal languages is presented, where the valuation functions are allowed to be partial, the truth–functional connectives are interpreted according to weak Kleene matrices, and the necessity operator is given a “weak” interpretation. Completeness and incompleteness results for some (axiomatic) systems are then established. Extensions of these modal logics in which figure “statability” operators are also examined.

This paper extends Fitting's epistemic interpretation of some Kleene logics, to also account for Paraconsistent Weak Kleene logic. To achieve this goal, a dualization of Fitting's "cut-down" operator is discussed, rendering a "track-down" operator later used to represent the idea that no consistent opinion can arise from a set including an inconsistent opinion. It is shown that, if some reasonable assumptions are made, the truth-functions of Paraconsistent Weak Kleene coincide with certain operations defined in this track-down fashion. (...) Finally, further reflections on conjunction and disjunction in the weak Kleene logics accompany this paper, particularly concerning their relation with containment logics. These considerations motivate a special approach to defining sound and complete Gentzen-style sequent calculi for some of their four-valued generalizations. (shrink)