We develop a second-order extension of intuitionistic modal logic, allowing quantification over propositions, both syntactically and semantically. A key feature of second-order logic is its capacity to define positive connectives from the negative fragment. Duly we are able to recover the diamond (and its associated theory) using only boxes, as long as we include both forward and backward modalities (`tense' modalities).
We propose axiomatic, proof theoretic and model theoretic definitions of `second-order intuitionistic tense logic', and ultimately prove that they all coincide. In particular we establish completeness of a labelled sequent calculus via a proof search argument, yielding at the same time a cut-admissibility result. Our methodology also applies to the classical version of second-order tense logic, which we develop in tandem with the intuitionistic case.
We generalize intuitionistic tense logics to the multi-modal case by placing grammar logics on an intuitionistic footing. We provide axiomatizations for a class of base intuitionistic grammar logics as well as provide axiomatizations for extensions with combinations of seriality axioms and what we call "intuitionistic path axioms". We show that each axiomatization is sound and complete with completeness being shown via a typical canonical model construction.
We introduce translations between display calculus proofs and labeled calculus proofs in the context of tense logics. First, we show that every derivation in the display calculus for the minimal tense logic Kt extended with general path axioms can be effectively transformed into a derivation in the corresponding labeled calculus. Concerning the converse translation, we show that for Kt extended with path axioms, every derivation in the corresponding labeled calculus can be put into a special form that is translatable to a derivation in the associated display calculus. A key insight in this con…
Most Properties are Undecidable for Transitive Tense Logics
The Tsinghua-UvA JRC for Logic Department of Philosophy Tsinghua University（清华大学哲学系清华-阿姆斯特丹逻辑联合研究中心）；Institute for Logic, Language and Computation University of Amsterdam（阿姆斯特丹大学逻辑、语言与计算研究所）
We prove a Goldblatt-Thomason theorem for dialgebraic intuitionistic logics, and instantiate it to Goldblatt-Thomason theorems for a wide variety of modal intuitionistic logics from the literature.
The logic of constant domains is intuitionistic logic extended with the so-called forall-shift axiom, a classically valid statement which implies the excluded middle over decidable formulas. Surprisingly, this logic is constructive and so far this has been proved by cut-elimination for ad-hoc sequent calculi. Here we use the methods of natural deduction and Curry-Howard correspondence to provide a simple computational interpretation of the logic.
We define a family of intuitionistic non-normal modal logics; they can bee seen as intuitionistic counterparts of classical ones. We first consider monomodal logics, which contain only one between Necessity and Possibility. We then consider the more important case of bimodal logics, which contain both modal operators. In this case we define several interactions between Necessity and Possibility of increasing strength, although weaker than duality. For all logics we provide both a Hilbert axiomatisation and a cut-free sequent calculus, on its basis we also prove their decidability. We then give…
We introduce bisimulations for the logic $ITL^e$ with `next', `until' and `release', an intuitionistic temporal logic based on structures equipped with a partial order used to interpret intuitionistic implication and a monotone function used to interpret the temporal modalities. Our main results are that `eventually', which is definable in terms of `until', cannot be defined in terms of `next' and `henceforth', and similarly that `henceforth', definable in terms of `release', cannot be defined in terms of `next' and `until', even over the smaller class of here-and-there models.
This paper studies the relationship between labelled and nested calculi for propositional intuitionistic logic, first-order intuitionistic logic with non-constant domains and first-order intuitionistic logic with constant domains. It is shown that Fitting's nested calculi naturally arise from their corresponding labelled calculi--for each of the aforementioned logics--via the elimination of structural rules in labelled derivations. The translational correspondence between the two types of systems is leveraged to show that the nested calculi inherit proof-theoretic properties from their associa…
This paper employs the linear nested sequent framework to design a new cut-free calculus LNIF for intuitionistic fuzzy logic--the first-order Gödel logic characterized by linear relational frames with constant domains. Linear nested sequents--which are nested sequents restricted to linear structures--prove to be a well-suited proof-theoretic formalism for intuitionistic fuzzy logic. We show that the calculus LNIF possesses highly desirable proof-theoretic properties such as invertibility of all rules, admissibility of structural rules, and syntactic cut-elimination.
We propose four axiomatic systems for intuitionistic linear temporal logic and show that each of these systems is sound for a class of structures based either on Kripke frames or on dynamic topological systems. Our topological semantics features a new interpretation for the `henceforth' modality that is a natural intuitionistic variant of the classical one. Using the soundness results, we show that the four logics obtained from the axiomatic systems are distinct. Finally, we show that when the language is restricted to the `henceforth'-free fragment, the set of valid formulas for the relationa…
In this paper we show that the intuitionistic monotone modal logic $\mathsf{iM}$ has the uniform Lyndon interpolation property (ULIP). The logic $\mathsf{iM}$ is a non-normal modal logic on an intuitionistic basis, and the property ULIP is a strengthening of interpolation in which the interpolant depends only on the premise or the conclusion of an implication, respecting the polarities of the propositional variables. Our method to prove ULIP yields explicit uniform interpolants and makes use of a terminating sequent calculus for $\mathsf{iM}$ that we have developed for this purpose. As far as …
In this paper we present simple example of propositional logic which has one modal operator and is based on intuitionistic core. This system is very weak in modal sense - e.g. rules of regularity or monotonicity do not hold. It has complete semantics composed of possible worlds equipped with neighborhoods and pre-order relation. We discuss certain restrictions imposed on those structures. Also, we present characterization of axiom 4 known from logic S4.
This paper shows how to derive nested calculi from labelled calculi for propositional intuitionistic logic and first-order intuitionistic logic with constant domains, thus connecting the general results for labelled calculi with the more refined formalism of nested sequents. The extraction of nested calculi from labelled calculi obtains via considerations pertaining to the elimination of structural rules in labelled derivations. Each aspect of the extraction process is motivated and detailed, showing that each nested calculus inherits favorable proof-theoretic properties from its associated la…
A cyclic proof system gives us another way of representing inductive definitions and efficient proof search. In 2011 Brotherston and Simpson conjectured the equivalence between the provability of the classical cyclic proof system and that of the classical system of Martin-Lof's inductive definitions. This paper studies the conjecture for intuitionistic logic. This paper first points out that the countermodel of FOSSACS 2017 paper by the same authors shows the conjecture for intuitionistic logic is false in general. Then this paper shows the conjecture for intuitionistic logic is true under ari…
The purpose of this paper is to introduce a bi-intuitionistic sequent calculus and to give proofs of admissibility for its structural rules. The calculus I will present, called SC2Int, is a sequent calculus for the bi-intuitionistic logic 2Int. Calculi for this logic represent a kind of bilateralist reasoning, since they do not only internalize processes of verification or provability but also the dual processes in terms of falsification or what is called dual provability. A normal form theorem for a natural deduction calculus of 2Int has already been stated, in this paper I want to prove a cu…
A modal logic amalgam of classical and intuitionistic propositional logic
由哥德尔在1932年提出并由麦肯锡和塔斯基在1948年证明的一个著名的结果是：$φ$是直觉命题逻辑IPC的一个定理当且仅当它的哥德尔翻译$φ'$是模态逻辑S4的一个定理。在本文中，我们扩展了在我们的前一篇文章（S. Lewitzka，Algebraic Semantics for a Modal Logic Close to S1，J. Logic and Comp.，doi：10.1093/logcom/exu 067），并证明了：命题公式$φ$是IPC的一个定理当且仅当$\squareφ$是L的一个定理（实际上，我们证明了：$Φ\vdash_{IPC}φ$当且仅当$\squareΦ\vdash_L\squareφ$，对于命题$Φ,φ$）.因此，映射$φ\mapsto\squareφ$是IPC到L中的嵌入，即L包含IPC的副本。此外，L是经典命题逻辑CPC的保守扩展。在这个意义上，L是CPC和IPC的混合物。证明了L是完备的。一类具有（非正规）模态算子的特殊Heyting代数。
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…
This paper studies a formalisation of intuitionistic logic by Negri and von Plato which has general introduction and elimination rules. The philosophical importance of the system is expounded. Definitions of `maximal formula', `segment' and `maximal segment' suitable to the system are formulated and corresponding reduction procedures for maximal formulas and permutative reduction procedures for maximal segments given. Alternatives to the main method used are also considered. It is shown that deductions in the system convert into normal form and that deductions in normal form have the subformul…
We give a linear nested sequent calculus for the basic normal tense logic Kt. We show that the calculus enables backwards proof-search, counter-model construction and syntactic cut-elimination. Linear nested sequents thus provide the minimal amount of nesting necessary to provide an adequate proof-theory for modal logics containing converse. As a bonus, this yields a cut-free calculus for symmetric modal logic KB.
In this paper, we introduce a general family of sequent-style calculi over the modal language and its fragments to capture the essence of all constructively acceptable systems. Calling these calculi \emph{constructive}, we show that any strong enough constructive sequent calculus, satisfying a mild technical condition, feasibly admits all Visser's rules, i.e., there is a polynomial time algorithm that reads a proof of the premise of a Visser's rule and provides a proof for its conclusion. As a positive application, we show the feasible admissibility of Visser's rules in several sequent calculi…
