We extend to the predicate frame a previous characterization of the maximal intermediate propositional constructive logics. This provides a technique to get maximal intermediate predicate constructive logics starting from suitable sets of classically valid predicate formulae we call maximal nonstandard predicate constructive logics. As an example of this technique, we exhibit two maximal intermediate predicate constructive logics, yet leaving open the problem of stating whether the two logics are distinct. Further properties of these logics will (...) be also investigated. (shrink)

Suppose that fn is a sequence of nonnegative functions with compact support on a locally compact metric space, that T is a nonnegative linear functional, and that equation imageT fn < T f0. A result of Bishop, foundational to a constructive theory of functional analysis, asserts the existence of a point x such that equation imagefn < f0. This paper extends this result to arbitrary Hausdorff spaces, and gives short proofs using nonstandard analysis. While such arguments used are (...) not themselves constructive, they can illuminate where the difficulty lies in finding the point x. An algorithm for constructing x is then given, with a nonstandard proof that the algorithm converges to a correct value. (shrink)

In the present paper we introduce a constructive theory of nonstandard arithmetic in higher types. The theory is intended as a framework for developing elementary nonstandard analysis constructively. More specifically, the theory introduced is a conservative extension of HAω + AC. A predicate for distinguishing standard objects is added as in Nelson's internal set theory. Weak transfer and idealisation principles are proved from the axioms. Finally, the use of the theory is illustrated by extending Bishop's constructive (...) analysis with infinitesimals. (shrink)

Currently the two popular ways to practice Robinson’s nonstandard analysis are the model-theoretic approach and the axiomatic/syntactic approach. It is sometimes claimed that the internal axiomatic approach is unable to handle constructions relying on external sets. We show that internal frameworks provide successful accounts of nonstandard hulls and Loeb measures. The basic fact this work relies on is that the ultrapower of the standard universe by a standard ultrafilter is naturally isomorphic to a subuniverse of the internal universe.

We develop a constructive version of nonstandard analysis, extending Bishop's constructive analysis with infinitesimal methods. A full transfer principle and a strong idealisation principle are obtained by using a sheaf-theoretic construction due to I. Moerdijk. The construction is, in a precise sense, a reduced power with variable filter structure. We avoid the nonconstructive standard part map by the use of nonstandard hulls. This leads to an infinitesimal analysis which includes nonconstructive theorems such as the Heine-Borel theorem, (...) the Cauchy-Peano existence theorem for ordinary differential equations and the exact intermediate-value theorem, while it at the same time provides constructive results for concrete statements. A nonstandard measure theory which is considerably simpler than that of Bishop and Cheng is developed within this context. (shrink)

Recently, the second author, Briseid, and Safarik introduced nonstandard Dialectica, a functional interpretation capable of eliminating instances of familiar principles of nonstandard arithmetic—including overspill, underspill, and generalizations to higher types—from proofs. We show that the properties of this interpretation are mirrored by first-order logic in a constructive sheaf model of nonstandard arithmetic due to Moerdijk, later developed by Palmgren, and draw some new connections between nonstandard principles and principles that are rejected by strict constructivism. (...) Furthermore, we introduce a variant of the Diller–Nahm interpretation with two different kinds of quantifiers, similar to Hernest’s light Dialectica interpretation, and show that one can obtain nonstandard Dialectica by weakening the computational content of the existential quantifiers—a process called herbrandization. We also define a constructive sheaf model mirroring this new functional interpretation, and show that the process of herbrandization has a clear meaning in terms of these sheaf models. (shrink)

A general definition of consequence relation is given, and a criterion for compactness based on a nonstandard construction is demonstrated.

This lecture, given at Beijing University in 1984, presents a remarkable (previously unpublished) proof of the Gödel Incompleteness Theorem due to Kripke. Today we know purely algebraic techniques that can be used to give direct proofs of the existence of nonstandard models in a style with which ordinary mathematicians feel perfectly comfortable--techniques that do not even require knowledge of the Completeness Theorem or even require that logic itself be axiomatized. Kripke used these techniques to establish incompleteness by means (...) that could, in principle, have been understood by nineteenth-century mathematicians. The proof exhibits a statement of number theory--one which is not at all "self referring"--and constructs two models, in one of which it is true and in the other of which it is false, thereby establishing "undecidability" (independence). (shrink)

A definition of nonstandard universe which gets over the limitation to the finite levels of the cumulative hierarchy is proposed. Though necessarily nonwellfounded, nonstandard universes are arranged in strata in the likeness of superstructures and allow a rank function taking linearly ordered values. Nonstandard universes are also constructed which model the whole ZFC theory without regularity and satisfy the κ-saturation property.

This article establishes the existence of a definable , countably saturated nonstandard enlargement of the superstructure over the reals. This nonstandard universe is obtained as the union of an inductive chain of bounded ultrapowers . The underlying ultrafilter is the one constructed by Kanovei and Shelah [10].

We introduce constructive and classical systems for nonstandard arithmetic and show how variants of the functional interpretations due to Gödel and Shoenfield can be used to rewrite proofs performed in these systems into standard ones. These functional interpretations show in particular that our nonstandard systems are conservative extensions of E-HAω and E-PAω, strengthening earlier results by Moerdijk and Palmgren, and Avigad and Helzner. We will also indicate how our rewriting algorithm can be used for term extraction purposes. (...) To conclude the paper, we will point out some open problems and directions for future research, including some initial results on saturation principles. (shrink)

Following [3], we build higher-order models of analysis resembling the frameworks of nonstandard analysis. The models are entirely canonical, constructed without Choice. Weak transfer principles are developed and the models are applied to topology, graph theory, and measure theory. A Loeb-like measure is constructed.

A nonstandard universe is constructed from a superstructure in a Boolean-valued model of set theory. This provides a new framework of nonstandard analysis with which methods of forcing are incorporated naturally. Various new principles in this framework are provided together with the following applications: An example of an 1-saturated Boolean ultrapower of the real number field which is not Scott complete is constructed. Infinitesimal analysis based on the generic extension of the hyperreal numbers is provided, and the hull (...) completeness theorem and the Loeb measure construction are extended to objects in the generic extension of the internal universe. The reduction theory of the Boolean-valued complex numbers are developed as a prototype of the applications to the topological reduction theory of Boolean sheaves or operator algebras. (shrink)

Following [3], we build higher-order models of analysis resembling the frameworks of nonstandard analysis. The models are entirely canonical, constructed without Choice. Weak transfer principles are developed and the models are applied to topology, graph theory, and measure theory. A Loeb-like measure is constructed.

A definition of nonstandard universe which gets over the limitation to the finite levels of the cumulative hierarchy is proposed. Though necessarily nonwellfounded, nonstandard universes are arranged in strata in the likeness of superstructures and allow a rank function taking linearly ordered values. Nonstandard universes are also constructed which model the whole ZFC theory without regularity and satisfy the $\kappa$-saturation property.

We investigate how nonstandard reals can be established constructively as arbitrary infinite sequences of rationals, following the classical approach due to Schmieden and Laugwitz. In particular, a total standard part map into Richman's generalised Dedekind reals is constructed without countable choice.

We provide results allowing to state, by the simple inspection of suitable classes of posets , that the corresponding intermediate propositional logics are maximal among the ones which satisfy the disjunction property. Starting from these results, we directly exhibit, without using the axiom of choice, the Kripke frames semantics of 2No maximal intermediate propositional logics with the disjunction property. This improves previous evaluations, giving rise to the same conclusion but made with an essential use of the axiom of choice, of (...) the cardinality of the set of the maximal intermediate propositional logics with the disjunction property. (shrink)

Derndinger [2] and Krupa [5] defined the F-product of a semigroup and presented some applications . Wolff investigated some kind of nonstandard analogon and applied it to spectral theory of group representations. The question arises in which way these constructions are related. In this paper we show that the classical and the nonstandard F-product are isomorphic . We also prove a little “classical” corollary.

Assuming the existence of an inaccessible cardinal, transitive full models of the whole set theory, equipped with a linearly valued rank function, are constructed. Such models provide a global framework for nonstandard mathematics.

A bounded ultrasheaf is a nonstandard universe constructed from a superstructure in a Boolean valued model of set theory. We consider the bounded elementary embeddings between bounded ultrasheaves. Then the standardization principle is true if and only if the ultrafilters are comparable by the Rudin-Frolik order. The base concept is that the bounded elementary embeddings correspond to the complete Boolean homomorphisms. We represent this by the Rudin-Keisler order of ultrafilters of Boolean algebras.

A new foundation for constructive nonstandard analysis is presented. It is based on an extension of a sheaf-theoretic model of nonstandard arithmetic due to I. Moerdijk. The model consists of representable sheaves over a site of filter bases. Nonstandard characterisations of various notions from analysis are obtained: modes of convergence, uniform continuity and differentiability, and some topological notions. We also obtain some additional results about the model. As in the classical case, the order type of the (...) nonstandard natural numbers is a dense set of copies of the integers. Every standard set has a hyperfinite enumeration of its standard elements in the model. All arguments are carried out within a constructive and predicative metatheory: Martin-Löf's type theory. (shrink)

Using a slight generalization, due to Palmgren, of sheaf semantics, we present a term-model construction that assigns a model to any first-order intuitionistic theory. A modification of this construction then assigns a nonstandard model to any theory of arithmetic, enabling us to reproduce conservation results of Moerdijk and Palmgren for nonstandard Heyting arithmetic. Internalizing the construction allows us to strengthen these results with additional transfer rules; we then show that even trivial transfer axioms or minor strengthenings of these (...) rules destroy conservativity over HA. The analysis also shows that nonstandard HA has neither the disjunction property nor the explicit definability property. Finally, careful attention to the complexity of our definitions allows us to show that a certain weak fragment of intuitionistic nonstandard arithmetic is conservative over primitive recursive arithmetic. (shrink)

Using a slight generalization, due to Palmgren, of sheaf semantics, we present a term-model construction that assigns a model to any first-order intuitionistic theory. A modification of this construction then assigns a nonstandard model to any theory of arithmetic, enabling us to reproduce conservation results of Moerdijk and Palmgren for nonstandard Heyting arithmetic. Internalizing the construction allows us to strengthen these results with additional transfer rules; we then show that even trivial transfer axioms or minor strengthenings of these (...) rules destroy conservativity over HA. The analysis also shows that nonstandard HA has neither the disjunction property nor the explicit definability property. Finally, careful attention to the complexity of our definitions allows us to show that a certain weak fragment of intuitionistic nonstandard arithmetic is conservative over primitive recursive arithmetic. (shrink)

We give an axiomatic characterization for complete elementary extensions, that is, elementary extensions of the first-order structure consisting of all finitary relations and functions on the underlying set. Such axiom systems have been studied using various types of primitive notions . Our system uses the notion of partial functions as primitive. Properties of nonstandard extensions are derived from five axioms in a rather algebraic way, without the use of metamathematical notions such as formulas or satisfaction. For example, when applied (...) to the real number system, it provides a complete framework for developing nonstandard analysis based on hyperreals without having to construct them and without any use of logic. This has possible pedagogical and expository applications as presented in, e.g., [5], [6]. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim. (shrink)

In this paper, we investigate definable models of Peano Arithmetic PA in a model of PA. For any definable model N without parameters in a model M, we show that N is isomorphic to M if M is elementary extension of the standard model and N is elementarily equivalent to M. On the other hand, we show that there is a model M and a definable model N with parameters in M such that N is elementarily equivalent to M but (...) N is not isomorphic to M. We also show that there is a model M and a definable model N with parameters in M such that N is elementarily equivalent to M, and N is isomorphic to M, but N is not definably isomorphic to M. And also, we give a generalization of Tennenbaum's theorem. At the end, we give a new method to construct a definable model by a refinement of Kotlarski's method. (shrink)

This discussion explores the possibility of distinguishing a tighter notion of contrariety evident in the Square of Opposition, especially in its modal incarnations, than as that binary relation holding statements that cannot both be true, with or without the added rider ‘though can both be false’. More than one theorist has voiced the intuition that the paradigmatic contraries of the traditional Square are related in some such tighter way—involving the specific role played by negation in contrasting them—that distinguishes them from (...) other pairs of incompatible statements constructed from the same conceptual materials. Prominent among examples, these other nonstandard pairs are the ‘new contraries’ presented by Robert Blanché’s hexagon(s) of opposition. With special, though not exclusive, attention to these cases, we investigate whether contrariety in the distinguished sense can be captured by adding to the incompatibility condition the further demand that the pair of statements concerned can be represented as the results of applying some sentence operator to the content in its scope, for one of the pair, and, for the other, the application of that same operator to the negation of that content. For one of the two cases, a Blanché case, of nonstandard contrariety singled out for attention, the question of whether such a representation is available is settled at the end of Section 4, and then in a more satisfying way in Section 5, though for the other case, noticed by Peter Simons, the question remains open, after some tentative discussion in one subsection, 6.2, of an Appendix (Section 6). (shrink)

The author's motivation for constructing the calculi of this paper\nis so that time and tense can be "discussed together in the same\nlanguage" (p. 282). Two types of enriched propositional caluli for\ntense logic are considered, both containing ordinary propositional\nvariables for which any proposition may be substituted. One type\nalso contains "clock-propositional" variables, a,b,c, etc., for\nwhich only clock-propositional variables may be substituted and that\ncorrespond to instants or moments in the semantics. The other type\nalso contains "history-propositional" variables, u,v,w, etc., for\nwhich only history-propositional variables (...) may be substituted and\nthat correspond to maximally linearly ordered subsets of instants.\nThis second type is for tense logics in which time is not linear\nand where at any moment there may exist alternative future courses\nof events. Quantifiers are included in both types but only for clock-\nand history-propositional variables. However, these quantifiers are\ngiven an essentially substitutional interpretation. In addition,\nbesides each clock-propositional variable being true at exactly one\ninstant, the semantics requires that for each instant there be at\nleast one clock-propositional variable true at that instant, which\nin effect amounts to having each instant of the model "denoted" in\nthe formalism by some clock-propositional variable. Thus it is not\nsurprising that quantification over clock-propositional variables\nturns out to have a "well behaved first-order semantics" (p. 284).\n{One axiom relating history- with clock-propositions seems to need\nrevision: CKTauTbuAUabUba. Since the clock-propositional variables\nmight be true at the same instant the consequent should include an\nalternative, namely, LCab.} Besides the standard truth-functional\nand tense operators G,H,F,P, the author includes L for "it is\nalways the case that". The model-theoretic earlier-than relation\nordering instants and the semantic truth-at (an instant) relation\nare formalized by Uab and Taα (for a wff α) which\nare defined respectively as LCbPa (whenever the clock-proposition\nb is the case the clock-proposition a was the case) and LCaα\n(α is the case whenever the clock-proposition a is the case).\nBecause the future tense operators are definable in terms of U\nand T and the latter are definable in terms of L and P, the\ndual of H, the author notes that his completeness proofs can be\nrestricted to L and H as primitive and with the future tense\noperators used for other interpretations, especially Prior, A.'s\nindeterminist "it will be the case that", where time is not linear.\nThe author concludes with some rather technical observations showing\nthat "in ordinary tense logics the instants are nonstandard elements\nof the models. This provides", according to the author, "a semantic\npressure to build the instants into tense logic, so that they will\nappear as standard elements in the models" (p. 283). (shrink)

In continuation of our study of HST, Hrbaek set theory (a nonstandard set theory which includes, in particular, the ZFC Replacement and Separation schemata in the st--language, and Saturation for well-orderable families of internal sets), we consider the problem of existence of elementary extensions of inner "external" subclasses of the HST universe.We show that, given a standard cardinal , any set R * generates an "internal" class S(R) of all sets standard relatively to elements of R, and an "external" (...) class L[S(R)] of all sets constructible (in a sense close to the Gödel constructibility) from sets in S(R). We prove that under some mild saturation-like requirements for R the class L[S(R)] models a certain -version of HST including the principle of +-saturation; moreover, in this case L[S(R)] is an elementary extension of L[S(R)] in the st--language whenever sets R R satisfy the requirements. (shrink)

It is and has been widely assumed, e.g., in Hintikka and Hintikka, that the logical theory available from Wittgenstein’s Tractatus Logico-Philosophicus affords a foundation for the conventional logic represented in standard formulations of classical propositional, first-order predicate, and perhaps higher-order formal systems. The present article is a detailed attempt at a mathematical demonstration, or as much demonstration as the sources will allow, that this assumption is false by contemporary lights and according to a preferred account of argument validity. When (...) Wittgenstein’s description of the forms of propositions or Sätze in the 5-numbered remarks and Remark 6 is given a close reconstruction, one sees that no Tractarian proposition is logically equivalent to a simple universally or existentially quantified formula of first-order predicate logic. Therefore, although Wittgenstein employs the sign ∀—or, in his notation—occasionally in explanations and illustrations, e.g., 4.0411 and 5.1311, when it comes to logic, the sign ∀ should receive in Tractatus a semantical treatment that is nonstandard. En route to that result, we show that the hierarchy of variables–and, hence, of propositions–defined at 5.501 incorporates the expressive power of finitary classical propositional logic. Also, when constructed over a first-order language for arithmetic, formulae corresponding to the specification of 5.501 pick out all and only arithmetic sets of numbers. Consequently, Wittgenstein’s hierarchy of iterated N-propositions–as described in vide Remark 6–does not collapse: at any level k, one finds propositions at k + 1 or above that are not logically equivalent to any proposition formed at k or below. (shrink)

Using an alternative to Tarskian semantics for first-order logic known as possibility semantics, I introduce an approach to nonstandard analysis that remains within the bounds of semiconstructive mathematics, i.e., does not assume any fragment of the Axiom of Choice beyond the Axiom of Dependent Choices. I define the Fr´echet hyperreal line †R as a possibility structure and show that it shares many fundamental properties of the classical hyperreal line, such as a Transfer Principle and a Saturation Principle. I (...) discuss the technical advantages of †R over some other alternative approaches to nonstandard analysis and argue that it is well-suited to address some of the philosophical and methodological concerns that have been raised against the application of nonstandard methods to ordinary mathematics. (shrink)

We develop a set-theoretic semantics for Cocchiarella's second-order logical system . Such a semantics is a modification of the nonstandard sort of second-order semantics described, firstly, by Simms and later extended by Cocchiarella. We formulate a new second order logical system and prove its relative consistency. We call such a system and construct its set-theoretic semantics. Finally, we prove completeness theorems for proper normal extensions of the two systems with respect to certain notions of validity provided by the semantics.

In this paper, we argue that the usual approach to modelling knowledge and belief with the necessity modality |$\Box $| does not produce intuitive outcomes in the framework of the Belnap–Dunn logic (⁠|$\textsf{BD}$|⁠, alias |$\textbf{FDE}$|—first-degree entailment). We then motivate and introduce a nonstandard modality |$\blacksquare $| that formalizes knowledge and belief in |$\textsf{BD}$| and use |$\blacksquare $| to define |$\bullet $| and |$\blacktriangledown $| that formalize the unknown truth and ignorance as not knowing whether, respectively. Moreover, we introduce (...) another modality |$\textbf{I}$| that stands for factive ignorance and show its connection with |$\blacksquare $|⁠. We equip these modalities with Kripke-frame-based semantics and construct a sound and complete analytic cut system for |$\textsf{BD}^{\blacksquare }$| and |$\textsf{BD}^{\textbf{I}}$|—the expansions of |$\textsf{BD}$| with |$\blacksquare $| and |$\textbf{I}$|⁠. In addition, we show that |$\Box $| as it is customarily defined in |$\textsf{BD}$| cannot define any of the introduced modalities, nor, conversely, neither |$\blacksquare $| nor |$\textbf{I}$| can define |$\Box $|⁠. We also demonstrate that |$\blacksquare $| and |$\textbf{I}$| are not interdefinable and establish the definability of several important classes of frames using |$\blacksquare $|⁠. (shrink)

The main theorem of this article is that every countable model of set theory 〈M, ∈M〉, including every well-founded model, is isomorphic to a submodel of its own constructible universe 〈LM, ∈M〉 by means of an embedding j : M → LM. It follows from the proof that the countable models of set theory are linearly pre-ordered by embeddability: if 〈M, ∈M〉 and 〈N, ∈N〉 are countable models of set theory, then either M is isomorphic to a submodel of N (...) or conversely. Indeed, these models are pre-well-ordered by embeddability in order-type exactly ω1+ 1. Specifically, the countable well-founded models are ordered under embeddability exactly in accordance with the heights of their ordinals; every shorter model embeds into every taller model; every model of set theory M is universal for all countable well-founded binary relations of rank at most OrdM; and every ill-founded model of set theory is universal for all countable acyclic binary relations. Finally, strengthening a classical theorem of Ressayre, the proof method shows that if M is any nonstandard model of PA, then every countable model of set theory — in particular, every model of ZFC plus large cardinals — is isomorphic to a submodel of the hereditarily finite sets 〈 HFM, ∈M〉 of M. Indeed, 〈 HFM, ∈M〉 is universal for all countable acyclic binary relations. (shrink)

In the fifth version of our response-paper [26] to Imamura’s criticism, we recall that NonStandard Neutrosophic Logic was never used by neutrosophic community in no application, that the quarter of century old neutrosophic operators (1995-1998) criticized by Imamura were never utilized since they were improved shortly after but he omits to tell their development, and that in real world applications we need to convert/approximate the NonStandard Analysis hyperreals, monads and binads to tiny intervals with the desired accuracy (...) – otherwise they would be inapplicable. We point out several errors and false statements by Imamura [21] with respect to the inf/sup of nonstandard subsets, also Imamura’s “rigorous definition of neutrosophic logic” is wrong and the same for his definition of nonstandard unit interval, and we prove that there is not a total order on the set of hyperreals (because of the newly introduced Neutrosophic Hyperreals that are indeterminate), whence the Transfer Principle from R to R* is questionable. After his criticism, several response publications on theoretical nonstandard neutrosophics followed in the period 2018-2022. As such, I extended the NonStandard Analysis by adding the left monad closed to the right, right monad closed to the left, pierced binad (we introduced in 1998), and unpierced binad - all these in order to close the newly extended nonstandard space (R*) under nonstandard addition, nonstandard subtraction, nonstandard multiplication, nonstandard division, and nonstandard power operations [23, 24]. Improved definitions of NonStandard Unit Interval and NonStandard Neutrosophic Logic, together with NonStandard Neutrosophic Operators are presented. (shrink)

It is widely accepted that there is a clear sense in which the first-order paraconsistent constructive logic with strong negation of Almukdad and Nelson, QN4, is more constructive than intuitionistic first-order logic, QInt. While QInt and QN4 both possess the disjunction property and the existence property as characteristics of constructiveness (or constructivity), QInt lacks certain features of constructiveness enjoyed by QN4, namely the constructible falsity property and the dual of the existence property. This paper deals with (...) the constructiveness of the contra-classical, connexive, paraconsistent, and contradictory non-trivial first-order logic QC, which is a connexive variant of QN4. It is shown that there is a sense in which QC is even more constructive than QN4. The argument focuses on a problem that is mirror-inverted to Raymond Smullyan’s drinker paradox, namely the invalidity of what will be called the drinker truism and its dual in QN4 (and QInt), and on a version of the Brouwer-Heyting-Kolmogorov interpretation of the logical operations that treats proofs and disproofs on a par. The validity of the drinker truism and its dual together with the greater constructiveness of QC in comparison to QN4 may serve as further motivation for the study of connexive logics and suggests that constructive logic is connexive and contradictory (the latter understood as being negation inconsistent). (shrink)

In this paper we investigate total correctness in Nonstandard Dynamic Logic . Here we show that despite of the celebrated Kfoury-Park [5] result, termination is a rst order notion if approached properly.

The goal of this two-part series of papers is to show that constructive logic with strong negation N is definitionally equivalent to a certain axiomatic extension NFL ew of the substructural logic FL ew . In this paper, it is shown that the equivalent variety semantics of N (namely, the variety of Nelson algebras) and the equivalent variety semantics of NFL ew (namely, a certain variety of FL ew -algebras) are term equivalent. This answers a longstanding question (...) of Nelson [30]. Extensive use is made of the automated theorem-prover Prover9 in order to establish the result. The main result of this paper is exploited in Part II of this series [40] to show that the deductive systems N and NFL ew are definitionally equivalent, and hence that constructive logic with strong negation is a substructural logic over FL ew. (shrink)

We study the connection between factors of the Medvedev lattice and constructive logic. The algebraic properties of these factors determine logics lying in between intuitionistic propositional logic and the logic of the weak law of the excluded middle (also known as De Morgan, or Jankov, logic). We discuss the relation between the weak law of the excluded middle and the algebraic notion of join-reducibility. Finally we discuss autoreducible degrees.

The goal of this two-part series of papers is to show that constructive logic with strong negation N is definitionally equivalent to a certain axiomatic extension NFL ew of the substructural logic FL ew. The main result of Part I of this series [41] shows that the equivalent variety semantics of N and the equivalent variety semantics of NFL ew are term equivalent. In this paper, the term equivalence result of Part I [41] is lifted to the (...) setting of deductive systems to establish the definitional equivalence of the logics N and NFL ew. It follows from the definitional equivalence of these systems that constructive logic with strong negation is a substructural logic. (shrink)

The aim of the paper is to show that topoi are useful in the categorial analysis of the constructive logic with strong negation. In any topos ϵ we can distinguish an object Λ and its truth-arrows such that sets ϵ have a Nelson algebra structure. The object Λ is defined by the categorial counterpart of the algebraic FIDEL-VAKARELOV construction. Then it is possible to define the universal quantifier morphism which permits us to make the first order predicate calculus. (...) The completeness theorem is proved using the Kripke-type semantic defined by THOMASON. (shrink)

This paper considers a topos-theoretic structure for the interpretation of co-constructive logic for proofs and refutations following Trafford :22–40, 2015). It is notoriously tricky to define a proof-theoretic semantics for logics that adequately represent constructivity over proofs and refutations. By developing abstractions of elementary topoi, we consider an elementary topos as structure for proofs, and complement topos as structure for refutation. In doing so, it is possible to consider a dialogue structure between these topoi, and also control their (...) relation such that classical logic is simulated where proofs and refutations are conclusive. (shrink)

In this paper, consistency is understood as absolute consistency (i.e. non-triviality). The basic constructive logic BKc6, which is adequate to this sense of consistency in the ternary relational semantics without a set of designated points, is defined. Then, it is shown how to define a series of logics by extending BKc6 up to contractionless intuitionistic logic. All logics defined in this paper are paraconsistent logics.

Shapiro and Taschek have argued that simply using intuitionistic logic and its Heyting semantics, one can show that there are no gaps in warranted assertability. That is, given that a discourse is faithfully modeled using Heyting's semantics for the logical constants, then if a statement _S is not warrantedly assertable, its negation (superscript box) _S is. Tennant has argued for this conclusion on similar grounds. I show that these arguments fail, albeit in illuminating ways. An appeal to constructive (...) logic does not commit one to this strong epistemological thesis, but appeals to semantics of intuitionistic logic none the less do give us certain conclusions about the connections between warranted assertability and truth. (edited). (shrink)

This paper considers logics which are formally dual to intuitionistic logic in order to investigate a co-constructive logic for proofs and refutations. This is philosophically motivated by a set of problems regarding the nature of constructive truth, and its relation to falsity. It is well known both that intuitionism can not deal constructively with negative information, and that defining falsity by means of intuitionistic negation leads, under widely-held assumptions, to a justification of bivalence. For example, we (...) do not want to equate falsity with the non-existence of a proof since this would render a statement such as “pi is transcendental” false prior to 1882. In addition, the intuitionist account of negation as shorthand for the derivation of absurdity is inadequate, particularly outside of purely mathematical contexts. To deal with these issues, I investigate the dual of intuitionistic logic, co-intuitionistic logic, as a logic of refutation, alongside intuitionistic logic of proofs. Direct proof and refutation are dual to each other, and are constructive, whilst there also exist syntactic, weak, negations within both logics. In this respect, the logic of refutation is weakly paraconsistent in the sense that it allows for statements for which, neither they, nor their negation, are refuted. I provide a proof theory for the co-constructive logic, a formal dualizing map between the logics, and a Kripke-style semantics. This is given an intuitive philosophical rendering in a re-interpretation of Kolmogorov’s logic of problems. (shrink)

We study axiomatic extensions of the propositional constructive logic with strong negation having the disjunction property in terms of corresponding to them varieties of Nelson algebras. Any such varietyV is characterized by the property: (PQWC) ifA,B V, thenA×B is a homomorphic image of some well-connected algebra ofV.We prove:• each varietyV of Nelson algebras with PQWC lies in the fibre –1(W) for some varietyW of Heyting algebras having PQWC, • for any varietyW of Heyting algebras with PQWC the least (...) and the greatest varieties in –1(W) have PQWC, • there exist varietiesW of Heyting algebras having PQWC such that –1(W) contains infinitely many varieties (of Nelson algebras) with PQWC. (shrink)

In this paper, consistency is understood as the absence of the negation of a theorem, and not, in general, as the absence of any contradiction. We define the basic constructive logic BKc1 adequate to this sense of consistency in the ternary relational semantics without a set of designated points. Then we show how to define a series of logics extending BKc1 within the spectrum delimited by contractionless minimal intuitionistic logic. All logics defined in the paper are paraconsistent (...) logics. (shrink)