We introduce a relativized version of random Kripke's schema and show how it may be applied in the investigation of the expressive power of intuitionistic real algebra by interpreting second-order Heyting arithmetic in it.
We investigate the logical structure of intuitionistic Kripke-Platek set theory IKP, and show that the first-order logic of IKP is intuitionistic first-order logic IQC.
We introduce a realisability semantics for infinitary intuitionistic set theory that is based on Ordinal Turing Machines (OTMs). We show that our notion of OTM-realisability is sound with respect to certain systems of infinitary intuitionistic logic, and that all axioms of infinitary Kripke-Platek set theory are realised. Finally, we use a variant of our notion of realisability to show that the propositional admissible rules of (finitary) intuitionistic Kripke-Platek set theory are exactly the admissible rules of intuitionistic propositional logic.
It is known that intuitionistic Kripke semantics can be generalized so that it can treat arbitrary propositional connectives characterized by truth functions. We extend this generalized Kripke semantics to first-order logic, and study how the choice of connectives changes the relation between intuitionistic predicate logic and the logic of constant domains in terms of validity of sequents. Our main result gives a simple necessary and sufficient condition for the set of valid sequents in intuitionistic predicate logic to coincide with the set of valid sequents in the logic of constant domains.
In their seminal paper Artemov and Protopopescu provide Hilbert formal systems, Brower-Heyting-Kolmogorov and Kripke semantics for the logics of intuitionistic belief and knowledge. Subsequently Krupski has proved that the logic of intuitionistic knowledge is PSPACE-complete and Su and Sano have provided calculi enjoying the subformula property. This paper continues the investigations around to sequent calculi for Intuitionistic Epistemic Logics by providing sequent calculi that have the subformula property and that are terminating in linear depth. Our calculi allow us to design a procedure th…
Following G. Mints(Kluwer 2000 and draft 2013), we present terminating and bicomplete proof searches in multi-succedent sequent calculi for intuitionistic propositional logic, fragments of intuitionistic predicate logic and full intuitionistic predicate logic in the spirit of Schuette's schema.
Quantified propositional intuitionistic logic is obtained from propositional intuitionistic logic by adding quantifiers \forall p, \exists p over propositions. In the context of Kripke semantics, a proposition is a subset of the worlds in a model structure which is upward closed. Kremer (1997) has shown that the quantified propositional intuitionistic logic Hπ+ based on the class of all partial orders is recursively isomorphic to full second-order logic. He raised the question of whether the logic resulting from restriction to trees is axiomatizable. It is shown that it is, in fact, decidable.…
Until the 1970s, proof theoretic investigations were mainly concerned with theories of inductive definitions, subsystems of analysis and finite type systems. With the pioneering work of Gerhard Jaeger in the late 1970s and early 1980s, the focus switched to set theories, furnishing ordinal-theoretic proof theory with a uniform and elegant framework. More recently it was shown that these tools can even sometimes be adapted to the context of strong axioms such as the powerset axiom, where one does not attain complete cut elimination but can nevertheless extract witnessing information and charact…
The basics of Intuitionistic Kripke-Platek set theory are developed, and some independence results among related classically equivalent theories are shown using Kripke models.
Kripke's Schema (better the Brouwer-Kripke Schema) and the Kreisel-Troelstra Theory of the Creating Subject were introduced around the same time for the same purpose, that of analysing Brouwer's 'Creating Subject arguments'; other applications have been found since. I first look in detail at a representative choice of Brouwer's arguments. Then I discuss the original use of the Schema and the Theory, their justification from a Brouwerian perspective, and instances of the Schema that can in fact be found in Brouwer's own writings. Finally, I defend the Schema and the Theory against a number of o…
The paper treats 4 different fragments of first-order logic induced by their respective versions of Kripke style semantics for modal intuitionistic logic. In order to capture these fragments, the notion of asimulation is modified and extended to yield Van Benthem type of semantic characterization of their respective expressive powers. It is shown, further, that this characterization can be easily carried over to arbitrary first-order definable subclasses of classical first-order models.
We prove that, over Kripke-Platek set theory with infinity (KP), transfinite induction along the ordinal $ε_{Ω+1}$ is equivalent to the schema asserting the soundness of KP, where $Ω$ denotes the supremum of all ordinals in the universe; this is analogous to the result that, over Peano arithmetic (PA), transfinite induction along $ε_0$ is equivalent to the schema asserting the soundness of PA. In the proof we need to code infinitary proofs within KP, and it is done by using partial recursive set functions. This result can be generalised to KP + $Γ$-separation + $Γ$-collection where $Γ$ is any …
Intuitionistic Propositional Logic is proved to be an infinitely many valued logic by Kurt Gödel (1932), and it is proved by Stanisław Jaśkowski (1936) to be a countably many valued logic. In this paper, we provide alternative proofs for these theorems by using models of Saul Kripke (1959). Gödel's proof gave rise to an intermediate propositional logic (between intuitionistic and classical), that is known nowadays as Gödel or the Gödel-Dummet Logic, and is studied by fuzzy logicians as well. We also provide some results on the inter-definablility of propositional connectives in this logic.
We study different representation theorems for various reducts of Heyting polyadic algebras. Superamalgamation is proved for several (natural reducts) and our results are compared to the finitizability problem in classical algebraic logic dealing with cylindric and polyadic (Boolean algebras). We also prove several new neat embedding theorems, and obtain that the class of representable algebras based on (a generalized) Kripke semantics coincide with the class of algebras having the neat embedding property, that is those algebras that are subneat reducts of algebras having $ω$ extra dimensions.
Universidad de los Andes（德洛斯安第斯大学）；Universidad de Buenos Aires（布宜诺斯艾利斯大学）
We consider the Gödel bi-modal logic determined by fuzzy Kripke models where both the propositions and the accessibility relation are infinitely valued over the standard Gödel algebra [0,1] and prove strong completeness of Fischer Servi intuitionistic modal logic IK plus the prelinearity axiom with respect to this semantics. We axiomatize also the bi-modal analogues of $T,$ $S4,$ and $S5$ obtained by restricting to models over frames satisfying the [0,1]-valued versions of the structural properties which characterize these logics. As application of the completeness theorems we obtain a represe…
Subresiduated lattices were introduced during the decade of 1970 by Epstein and Horn as an algebraic counterpart of some logics with strong implication previously studied by Lewy and Hacking. These logics are examples of subuintuitionistic logics, i.e., logics in the language of intuitionistic logic that are defined semantically by using Kripke models, in the same way as intuitionistic logic is defined, but without requiring of the models some of the properties required in the intuitionistic case. Also in relation with the study of subintuitionistic logics, Celani and Jansana get these algebra…
The multi-valued logic of Łukasiewicz is a substructural logic that has been widely studied and has many interesting properties. It is classical, in the sense that it admits the axiom schema of double negation, [DNE]. However, our understanding of Łukasiewicz logic can be improved by separating its classical and intuitionistic aspects. The intuitionistic aspect of Łukasiewicz logic is captured in an axiom schema, [CWC], which asserts the commutativity of a weak form of conjunction. This is equivalent to a very restricted form of contraction. We show how Łukasiewicz Logic can be viewed both as …
In 1938, Tarski proved that a formula is not intuitionistically valid if, and only if, it has a counter-model in the Heyting algebra of open sets of some topological space. In fact, Tarski showed that any Euclidean space R^n with n >= 1 suffices, as does e.g. the Cantor space. In particular, intuitionistic logic cannot detect topological dimension in the frame of all open sets of a Euclidean space. By contrast, we consider the lattice of open subpolyhedra of a given compact polyhedron P \subseteq R^n, prove that it is a locally finite Heyting subalgebra of the (non-locally-finite) algebra of a…
