We investigate properties of monadic purely negational fragment of Intuitionistic Control Logic (ICL). This logic arises from Intuitionistic Propositional Logic (IPL) by extending language of IPL by additional new constant for falsum. Having two different falsum constants enables to define two forms of negation. We analyse implicational relations between negational monadic formulae and present a poset of non equivalent formulae of this fragment of ICL.
In 1932, Gödel proved that there is no finite semantics for intuitionistic logic. We consider all fragments of intuitionistic logic and check in each case whether a finite semantics exists. We may fulfill a didactic goal, as little logic and algebra are presupposed.
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…
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 study implicational formulas in the context of proof complexity of intuitionistic propositional logic (IPC). On the one hand, we give an efficient transformation of tautologies to implicational tautologies that preserves the lengths of intuitionistic extended Frege (EF) or substitution Frege (SF) proofs up to a polynomial. On the other hand, EF proofs in the implicational fragment of IPC polynomially simulate full intuitionistic logic for implicational tautologies. The results also apply to other fragments of other superintuitionistic logics under certain conditions. In particular, the expo…
In this paper we construct a Beth model for intuitionistic functionals of high types and use it to create a relatively strong theory SLP containg intuitionistic principles for functionals, in particular, the theory of the "creating subject", axioms for lawless functionals and some versions of choice axioms. We prove that the intuitionistic theory SLP is equiconsistent with a classical typed set theory TI, where the comprehension axiom for sets of type n is restricted to formulas with no parameters of types > n. We show that each fragment of SLP with types <= s is equiconsistent with the corres…
In this paper we investigate the fragment of intuitionistic logic which only uses conjunction (meet) and implication, using finite duality for distributive lattices and universal models. We give a description of the finitely generated universal models of this fragment and give a complete characterization of the up-sets of Kripke models of intuitionistic logic which can be defined by meet-implication-formulas. We use these results to derive a new version of subframe formulas for intuitionistic logic and to show that the uniform interpolants of meet-implication-formulas are not necessarily unifo…
We study the correspondence theory of intuitionistic modal logic in modal Fairtlough-Mendler semantics (modal FM semantics) \cite{FaMe97}, which is the intuitionistic modal version of possibility semantics \cite{Ho16}. We identify the fragment of inductive formulas \cite{GorankoV06} in this language and give the algorithm $\mathsf{ALBA}$ \cite{CoPa12} in this semantic setting. There are two major features in the paper: one is that in the expanded modal language, the nominal variables, which are interpreted as atoms in perfect Boolean algebras, complete join-prime elements in perfect distributi…
Boudou and the authors have recently introduced the intuitionistic temporal logic $\sf ITL^e$ and shown it to be decidable. In this article we show that the `henceforth'-free fragment of this logic is complete for the class of non-deterministic quasimodels introduced by Fernández-Duque. From this and results of Boudou, Romero and the authors, we conclude that this fragment is also complete for the standard semantics of $\sf ITL^e$ over the class of expanding posets.
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…
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.
We prove that the positive fragment of first-order intuitionistic logic in the language with two variables and a single monadic predicate letter, without constants and equality, is undecidable. This holds true regardless of whether we consider semantics with expanding or constant domains. We then generalise this result to intervals [QBL, QKC] and [QBL, QFL], where QKC is the logic of the weak law of the excluded middle and QBL and QFL are first-order counterparts of Visser's basic and formal logics, respectively. We also show that, for most "natural" first-order modal logics, the two-variable …
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…
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…
In a recent paper we showed that intuitionistic quantifiers admit the following temporal interpretation: "always in the future" (for $\forall$) and "sometime in the past" (for $\exists$). In this paper we study this interpretation for the monadic fragment $\sf MIPC$ of the intuitionistic predicate logic. It is well known that $\sf MIPC$ is translated fully and faithfully into the monadic fragment $\sf MS4$ of the predicate $\sf S4$ (Gödel translation). We introduce a new tense extension of $\sf S4$, denoted by $\sf TS4$, and provide an alternative full and faithful translation of $\sf MIPC$ in…
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…
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 ? …
We present a reading of the traditional syllogistics in a fragment of the propositional intuitionistic multiplicative linear logic and prove that with respect to a diagrammatic logical calculus that we introduced in a previous paper, a syllogism is provable in such a fragment if and only if it is diagrammatically provable. We extend this result to syllogistics with complemented terms à la De Morgan, with respect to a suitable extension of the diagrammatic reasoning system for the traditional case and a corresponding reading of such De Morgan style syllogistics in the previously referred to fra…
Computability logic (CL) is a systematic formal theory of computational tasks and resources, which, in a sense, can be seen as a semantics-based alternative to (the syntactically introduced) linear logic. With its expressive and flexible language, where formulas represent computational problems and "truth" is understood as algorithmic solvability, CL potentially offers a comprehensive logical basis for constructive applied theories and computing systems inherently requiring constructive and computationally meaningful underlying logics. Among the best known constructivistic logics is Heyting's …
The paper gives a soundness and completeness proof for the implicative fragment of intuitionistic calculus with respect to the semantics of computability logic, which understands intuitionistic implication as interactive algorithmic reduction. This concept -- more precisely, the associated concept of reducibility -- is a generalization of Turing reducibility from the traditional, input/output sorts of problems to computational tasks of arbitrary degrees of interactivity. See http://www.cis.upenn.edu/~giorgi/cl.html for a comprehensive online source on computability logic.
We investigate four well-known negative translations of classical logic into intuitionistic logic within a substructural setting. We find that in affine logic the translation schemes due to Kolmogorov and Gödel both satisfy Troelstra's criteria for a negative translation. On the other hand, the schemes of Glivenko and Gentzen both fail for affine logic, but for different reasons: one can extend affine logic to make Glivenko work and Gentzen fail and vice versa. By contrast, in the setting of Lukasiewicz logic, we can prove a general result asserting that a wide class of formula translations in…
The one-variable fragment of any first-order logic may be considered as a modal logic, where the universal and existential quantifiers are replaced by a box and diamond modality, respectively. In several cases, axiomatizations of algebraic semantics for these logics have been obtained: most notably, for the modal counterparts S5 and MIPC of the one-variable fragments of first-order classical logic and intuitionistic logic, respectively. Outside the setting of first-order intermediate logics, however, a general approach is lacking. This paper provides the basis for such an approach in the setti…
In this paper we study some fragments without implications of the (Hilbert) full Lambek logic $\mathbf{HFL}$ and also some fragments without implications of some of the substructural extensions of that logic. To do this, we perform an algebraic analysis of the Gentzen systems defined by the substructural calculi $\FL_σ$. Such systems are extensions of the full Lambek calculus $\FL$ with the rules codified by a subsequence, $σ$, of the sequence $e w_l w_r c$; where $e$ stands for \emph{exchange}, $w_l$ for \emph{left weakening}, $w_r$ for \emph{right weakening}, and $c$ for \emph{contraction}. …
In this paper we intend to study implications in their most general form, generalizing different classes of implications including the Heyting implication, sub-structural implications and weak strict implications. Following the topological interpretation of the intuitionistic logic, we will introduce non-commutative spacetimes to provide a more dynamic and subjective interpretation of an intuitionistic proposition. These combinations of space and time are natural sources for well-behaved implications and we will show that their spatio-temporal implications represent any other reasonable abstra…
Faculty of Technical Sciences, University of Novi Sad（诺维萨德大学技术科学学院）；Mathematical Institute of the Serbian Academy of Sciences and Arts（塞尔维亚科学院数学研究所）；School of Computer Science, University of Birmingham（伯明翰大学计算机学院）
