We find a translation with particularly nice properties from intuitionistic propositional logic in countably many variables to intuitionistic propositional logic in two variables. In addition, the existence of a possibly-not-as-nice translation from any countable logic into intuitionistic propositional logic in two variables is shown. The nonexistence of a translation from classical logic into intuitionistic propositional logic which preserves ``and'' and ``or'' but not necessarily ``true'' is proven. These results about translations follow from additional results about embeddings into free Heyting algebras.
In this paper we show that an intuitionistic theory for fixed points is conservative over the Heyting arithmetic with respect to a certain class of formulas. This extends partly the result of mine. The proof is inspired by the quick cut-elimination due to G. Mints.
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.
In this paper intuitionistic topological system and its properties have been introduced. Categorical interrelationships among Heyting algebra, Gödel algebra, Esakia space and proposed intuitionistic topological systems have also been studied in details.
The paper is devoted to an algebraic interpretation of Kuznetsov's theorem which establishes the assertoric equipollence of intuitionistic and proof-intuitionistic propositional calculi. Given a Heyting algebra, we define an enrichable Heyting algebra, in which the former algebra is embedded. Moreover, we show that both algebras generate one and the same variety of Heyting algebras. This algebraic result is equivalent to the Kuznetsov theorem. The proposed construction of the enrichable `counterpart' of a given Heyting algebra allows one to observe that some properties of the original algebra …
It is shown that propositional intuitionistic logic is the maximal (with respect to expressive power) abstract logic satisfying a certain topological property reminiscent of compactness, the Tarski union property and preservation under asimulations.
It is known that the set of tautologies of second order intuitionistic propositional logic, $\mathrm{IPC} 2$, is undecidable. Here, we prove that the sets of formulas of $\mathrm{IPC} 2$ which are true in the algebra of open subsets of reals or rationals are decidable.
Notions of k-asimulation and asimulation are introduced as asymmetric counterparts to k-bisimulation and bisimulation, respectively. It is proved that a first-order formula is equivalent to a standard translation of an intuitionistic propositional formula iff it is invariant with respect to k-asimulations for some k, and then that a first-order formula is equivalent to a standard translation of an intuitionistic propositional formula iff it is invariant with respect to asimulations. Finally, it is proved that a first-order formula is intuitionistically equivalent to a standard translation of a…
In the style of Lindström's theorem for classical first-order logic, this article characterizes propositional bi-intuitionistic logic as the maximal (with respect to expressive power) abstract logic satisfying a certain form of compactness, the Tarski union property and preservation under bi-asimulations. Since bi-intuitionistic logic introduces new complexities in the intuitionistic setting by adding the analogue of a backwards looking modality, the present paper constitutes a non-trivial modification of previous work done by the authors for intuitionistic logic in: G. Badia and G. Olkhovikov…
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.…
We show that intuitionistic logic is deductively equivalent to Connexive Heyting Logic (CHL), hereby introduced as an example of a strong connexive logic with intuitive semantics. We use the reverse algebraisation paradigm: CHL is presented as the assertional logic of a point regular variety (whose structure theory is examined in detail) that turns out to be term equivalent to the variety of Heyting algebras. We provide Hilbert-style and Gentzen-style proof systems for CHL; moreover, we suggest a possible computational interpretation of its connexive conditional, and we revisit Kapsner's idea …
Both propositional dependence logic and inquisitive logic are expressively complete. As a consequence, every formula with intuitionistic disjunction or intuitionistic implication can be translated equivalently into a formula in the language of propositional dependence logic without these two connectives. We show that although such a (non-compositional) translation exists, neither intuitionistic disjunction nor intuitionistic implication is uniformly definable in propositional dependence logic.
We show that intuitionistic propositional logic is \emph{Carnap categorical}: the only interpretation of the connectives consistent with the intuitionistic consequence relation is the standard interpretation. This holds relative to the most well-known semantics with respect to which intuitionistic logic is sound and complete; among them Kripke semantics, Beth semantics, Dragalin semantics, and topological semantics. It also holds for algebraic semantics, although categoricity in that case is different in kind from categoricity relative to possible worlds style semantics.
In this paper, we will study Heyting algebras endowed with tense negative operators, which we call tense H-algebras and we proof that these algebras are the algebraic semantics of the Intuitionistic Propositional Logic with Galois Negations. Finally, we will develop a Priestley-style duality for H-algebras.
This paper presents a novel proof of the conservativity of the intuitionistic theory of strictly positive fixpoints, $\widehat{\mathrm{ID}}{}_{1}^{\mathrm{i}}$, over Heyting arithmetic (HA), originally proved in full generality by Arai (2011). The proof embeds $\widehat{\mathrm{ID}}{}_{1}^{\mathrm{i}}$ into the corresponding theory over Beeson's logic of partial terms and then uses two consecutive interpretations, a realizability interpretation of this theory into the subtheory generated by almost negative fixpoints, and a direct interpretation into Heyting arithmetic with partial terms using …
We show that several reducts of Heyting polyadic algebras of infinite dimension, with and without equality enjoy various amalgamation properties. In the equality free case we obtain superamalgamation, but when we have equality we obtain a weaker interpolation property, for the corresponding infinitary intuitionistic logic with equality.
We improve the answer to the question: what set of excluded middles for propositional variables in a formula suffices to prove the formula in intuitionistic propositional logic whenever it is provable in classical propositional logic.
We prove that the propositional logic of intuitionistic set theory IZF is intuitionistic propositional logic IPC. More generally, we show that IZF has the de Jongh property with respect to every intermediate logic that is complete with respect to a class of finite trees. The same results follow for CZF.
In this paper, we consider the complexity of propositional proofs of classical and intuitionistic tautologies. In fact, we describe a nondeterministic polynomial-time decision procedure for intuitionistic implicational tautologies. For this purpose, we reduce a decision problem for intuitionistic implicational tautologies to a decision problem for deductive proof diagrams which are a short form for representing of proofs in the intuitionistic implicational calculus. Next, we transform deductive proof diagrams to a special form in which any proof has the size bounded by a polynomial in the leng…
Intuitionistic logic extended with decidable propositional atoms combines classical properties in its propositional part and intuitionistic properties for derivable formulas not containing propositional symbols. Sequent calculus is used as a framework for investigating this extension. Admissibility of cut is retained. Constrained Kripke structures are introduced for modeling intuitionistic logic with decidable propositional atoms. The extent of the disjunction and existence properties is investigated. The latest information about this research can be found at http://sakharov.net/median.html
We extend the polynomial time algorithms due to Buss and Mints(APAL 1999) and Ferrari, Fiorentini and Fiorino(LPAR 2002) to yield a polynomial time complete disjunction property in intuitionistic propositional logic.
We prove that the first-order logic of CZF is intuitionistic first-order logic. To do so, we introduce a new model of transfinite computation (Set Register Machines) and combine the resulting notion of realisability with Beth semantics. On the way, we also show that the propositional admissible rules of CZF are exactly those of intuitionistic propositional logic.
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.
Gödel-McKinsey-Tarski and (not quite) Blok-Esakia for Heyting-Lewis Implication
School of Computing The Australian National University（澳大利亚国立大学计算机学院）；Chair For Theoretical Computer Science FAU Erlangen-Nuremberg（埃尔朗根-纽伦堡大学理论计算机科学教席）
In 1936, Stanislaw Jaśkowski gave a construction of an interesting sequence of what he called "matrices", which we would today call "finite Heyting Algebras". He then gave a very brief sketch of a proof that if a propositional formula holds in each of these algebras then it is provable in intuitionistic propositional logic (IPL). The sketch just describes a certain normal form for propositional formulas and gives a very terse outline of an inductive argument showing that an unprovable formula in the normal form can be refuted in one of the algebras. Unfortunately, it is far from clear how to r…
In this paper, we present a propositional logic (called mixed logic) containing disjoint copies of minimal, intuitionistic and classical logics. We prove a completeness theorem for this logic with respect to a Kripke semantics. We establish some relations between mixed logic and minimal, intuitionistic and classical logics. We present at the end a sequent calculus version for this logic.
We prove that there is a factor of the Muchnik lattice that captures intuitionistic propositional logic. This complements a now classic result of Skvortsova for the Medvedev lattice.
In this article, we continue the study of tense symmetric Heyting algebras (or TSH-algebras). These algebras constitute a generalization of tense algebras. In particular, we describe a discrete duality for TSHalgebras bearing in mind the results indicated by E. Or lowska and I. Rewitzky in [E. Or lowska and I. Rewitzky, Discrete Dualities for Heyting Algebras with Operators, Fund. Inform. 81 (2007), no.1-3, 275-295.] for Heyting algebras. In addition, we introduce a propositional calculus and prove this calculus has TSH-algebras as algebraic counterpart. Finally, the duality mentioned above al…
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.
The inverse method is a saturation based theorem proving technique; it relies on a forward proof-search strategy and can be applied to cut-free calculi enjoying the subformula property. Here we apply this method to derive the unprovability of a goal formula G in Intuitionistic Propositional Logic. To this aim we design a forward calculus FRJ(G) for Intuitionistic unprovability which is prone to constructively ascertain the unprovability of a formula G by providing a concise countermodel for it; in particular we prove that the generated countermodels have minimal height. Moreover, we clarify th…
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…
In this paper we show that the intuitionistic theory for finitely many iterations of strictly positive operators is a conservative extension of the Heyting arithmetic. The proof is inspired by the quick cut-elimination due to G. Mints. This technique is also applied to fragments of Heyting arithmetic.
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…
A famous result, conjectured by Gödel in 1932 and proved by McKinsey and Tarski in 1948, says that $φ$ is a theorem of intuitionistic propositional logic IPC iff its Gödel-translation $φ'$ is a theorem of modal logic S4. In this paper, we extend an intuitionistic version of modal logic S1+SP, introduced in our previous paper (S. Lewitzka, Algebraic semantics for a modal logic close to S1, J. Logic and Comp., doi:10.1093/logcom/exu067) to a classical modal logic L and prove the following: a propositional formula $φ$ is a theorem of IPC iff $\squareφ$ is a theorem of L (actually, we show: $Φ\vda…
In this paper we will show that using implicative algebras one can produce models of intuitionistic set theory generalizing both realizability and Heyting-valued models. This has as consequence that if one assumes the inaccessible cardinal axiom, then every topos which is obtained from a Set-based tripos as the result of the tripos-to-topos construction hosts a model of intuitionistic set theory.
This paper presents a soundness and completeness proof for propositional intuitionistic calculus with respect to the semantics of computability logic. The latter interprets formulas as interactive computational problems, formalized as games between a machine and its environment. Intuitionistic implication is understood as algorithmic reduction in the weakest possible -- and hence most natural -- sense, disjunction and conjunction as deterministic-choice combinations of problems (disjunction = machine's choice, conjunction = environment's choice), and "absurd" as a computational problem of univ…
In this paper, we construct an undecidable 3-variable superintuitionistic propositional calculus, i.e., a finitely axiomatizable extension of the intuitionistic propositional calculus with axioms containing only 3 variables. Since there are no 2-variable superintuitionistic propositional calculi, this is the minimal possible number of variables.
In a 1985 commentary to his collected works, Kolmogorov remarked that his 1932 paper "was written in hope that with time, the logic of solution of problems [i.e., intuitionistic logic] will become a permanent part of a [standard] course of logic. A unified logical apparatus was intended to be created, which would deal with objects of two types - propositions and problems." We construct such a formal system, as well as its predicate version QHC, which is a conservative extension of both the intuitionistic predicate calculus QH and the classical predicate calculus QC. The only new connectives ? …
Following A. Kuznetsov's outline, we restore Kuznetsov's syntactic proof of the assertoric equipollence of the intuitionistic propositional calculus and the proof-intuitionistic calculus KM (Kuznetsov's Theorem). Then, we show that this property is true for a broad class of modal logics on an intuitionistic basis, which includes, e.g., the modalized Heyting calculus mHC. The last fact is one of two key properties necessary for the commutativity of a diagram involving the lattices of normal extensions of four well-known logics. Also, we give an algebraic interpretation of the assertoric equipol…
We extend the main result of (G. Badia and G. Olkhovikov. A Lindström theorem for intuitionistic propositional logic. Notre Dame Journal of Formal Logic, 61 (1): 11--30 (2020)) to the first-order intuitionistic logic (with and without equality), showing that it is the maximal (with respect to expressive power) abstract logic satisfying a certain form of compactness, the Tarski union property and preservation under asimulations. A similar result is also shown for the intuitionistic logic of constant domains.
In this article we develop a new version of the intuitionist existential graphs presented by Arnol Oostra [4]. The deductive rules presented in this article have the same meaning as those described in the work of Yuri Poveda [5], because the deductions according to the parity of the cuts are eliminated and are replaced by a finite set of recursive rules. This way, $ Alfa_I $ the existential graphs system for intuitional propositional logic follows the course of the deductive rules of the system $ Alfa_0 $ described by Poveda [5], and is equivalent to the intuitionistic propositional calculus. …
In 1933, Gödel introduced a provability interpretation of the propositional intuitionistic logic to establish a formalization for the BHK interpretation. He used the modal system, $\mathbf{S4}$, as a formalization of the intuitive concept of provability and then translated $\mathbf{IPC}$ to $\mathbf{S4}$. His work suggested the problem to find a concrete provability interpretation of the modal logic $\mathbf{S4}$. In this paper, we will try to answer this problem. In fact, we will generalize Solovay's provability interpretation of the modal logic $\mathbf{GL}$ to capture other modal logics suc…
It is a consequence of existing literature that least and greatest fixed-points of monotone polynomials on Heyting algebras-that is, the alge- braic models of the Intuitionistic Propositional Calculus-always exist, even when these algebras are not complete as lattices. The reason is that these extremal fixed-points are definable by formulas of the IPC. Consequently, the $μ$-calculus based on intuitionistic logic is trivial, every $μ$-formula being equiv- alent to a fixed-point free formula. We give in this paper an axiomatization of least and greatest fixed-points of formulas, and an algorithm…
We outline an intuitionistic view of knowledge which maintains the original Brou\-wer-Heyting-Kolmogorov semantics for intuitionism and is consistent with the well-known approach that intuitionistic knowledge be regarded as the result of verification. We argue that on this view co-reflection $A \rightarrow {\bf{K}} A$ is valid and the factivity of knowledge holds in the form ${\bf{K}} A \rightarrow \neg\neg A$ `known propositions cannot be false'. We show that the traditional form of factivity ${\bf{K}} A \rightarrow A$ is a distinctly classical principle which, like {\it tertium non datur} $A…
In this paper we explore the following question: how weak can a logic be for Rosser's essential undecidability result to be provable for a weak arithmetical theory? It is well known that Robinson's Q is essentially undecidable in intuitionistic logic, and P. Hajek proved it in the fuzzy logic BL for Grzegorczyk's variant of Q which interprets the arithmetic operations as non-total non-functional relations. We present a proof of essential undecidability in a much weaker substructural logic and for a much weaker arithmetic theory, a version of Robinson's R (with arithmetic operations also interp…
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.
Proof-theoretic semantics (P-tS) is the paradigm of semantics in which meaning in logic is based on proof (as opposed to truth). A particular instance of P-tS for intuitionistic propositional logic (IPL) is its base-extension semantics (B-eS). This semantics is given by a relation called support, explaining the meaning of the logical constants, which is parameterized by systems of rules called bases that provide the semantics of atomic propositions. In this paper, we interpret bases as collections of definite formulae and use the operational view of the latter as provided by uniform proof-sear…
