An algebraic setting for the validity of Pavelka style completeness for some natural expansions of Łukasiewicz logic by new connectives and rational constants is given. This algebraic approach is based on the fact that the standard MV-algebra on the real segment $[0, 1]$ is an injective MV-algebra. In particular the logics associated with MV-algebras with product and with divisible MV-algebras are considered.
We consider the explicit fragment of the basic justification stit logic introduced in earlier publications. We define a Hilbert-style axiomatic system for this logic and show that this system is strongly complete relative to the intended semantics.
We present a completeness result for a logical system which combines stit logic and justification logic in order to represent proving activity of the agents. This logic is interpreted over the semantics introduced in earlier publications. We define a Hilbert-style axiomatic system for this logic and show that this system is strongly complete relative to the intended semantics.
A Henkin-style proof of completeness of first-order classical logic is given with respect to a very small set (notably missing cut rule) of Genzten deduction rules for intuitionistic sequents. Insisting on sparing on derivation rules, satisfiability theorem is seen to need weaker assumptions than completeness theorem, the missing request being exactly the rule ~ p --> p, which gives a hint of intuitionism's motivations from a classical point of view. A bare treatment of standard, basic first-order syntax somehow more algebraic-flavoured than usual is also given.
The paper studies the containment companion of a logic $\vdash$. This consists of the consequence relation $\vdash^{r}$ which satisfies all the inferences of $\vdash$, where the variables of the conclusion are \emph{contained} into those of the (set of) premises. In accordance with our previous work on logics of left variable inclusion, we show that a different generalization of the Płonka sum construction, adapted from algebras to logical matrices, allows us to provide a matrix-based semantics for containment logics. In particular, we provide an appropriate completeness theorem for a wide fam…
We formulate a Hilbert-style axiomatic system for STIT logic of imagination recently proposed by H. Wansing and prove its completeness by the method of canonical models.
We prove the following completeness result about classical realizability: given any Boolean algebra with at least two elements, there exists a Krivine-style classical realizability model whose characteristic Boolean algebra is elementarily equivalent to it. This is done by controlling precisely which combinations of so-called "angelic" (or "may") and "demonic" (or "must") nondeterminism exist in the underlying model of computation.
We study the first-order primal infon logic. It is the core of the policy language DKAL. We provide Gentzen-style calculi for two versions of this logic that are not equivalent. For both versions we investigate the semantics: one of them is a generalization of the so-called quasi-boolean semantics, the other one is a Krypke-style semantics. We prove the completeness results and the disjunction property for both logics.
We study propositional and first-order Gödel logics over infinitary languages which are motivated semantically by corresponding interpretations into the unit interval [0,1]. We provide infinitary Hilbert-style calculi for the particular (propositional and first-order) cases with con-/disjunctions of countable length and prove corresponding completeness theorems by extending the usual Lindenbaum-Tarski construction to the infinitary case for a respective algebraic semantics via complete linear Heyting algebras. We provide infinitary hypersequent calculi and prove corresponding cut-elimination t…
In this paper we enrich the orthomodular structure by adding a modal operator, following a physical motivation. A logical system is developed, obtaining algebraic completeness and completeness with respect to a Kripke-style semantic founded on Baer *-semigroups as in [20].
In 1933, Gödel considered two modal approaches to describing provability. One captured formal provability and resulted in the logic GL and Solovay's Completeness Theorem. The other was based on the modal logic S4 and led to Artemov's Logic of Proofs LP. In this paper, we study introduced by the author logic GLA, which is a fusion of GL and LP in the union of their languages. GLA is supplied with a Kripke-style semantics and the corresponding completeness theorem. Soundness and completeness of GLA with respect to the arithmetical provability semantics is established.
We determine the modal logic of fixed-point models of truth and their axiomatizations by Solomon Feferman via Solovay-style completeness results. Given a fixed-point model $\mathcal{M}$, or an axiomatization $S$ thereof, we find a modal logic $M$ such that a modal sentence $φ$ is a theorem of $M$ if and only if the sentence $φ^*$ obtained by translating the modal operator with the truth predicate is true in $\mathcal{M}$ or a theorem of $S$ under all such translations. To this end, we introduce a novel version of possible worlds semantics featuring both classical and nonclassical worlds and es…
Bi-Intuitionistic Stable Tense Logics (BIST Logics) are tense logics with a Kripke semantics where worlds in a frame are equipped with a pre-order as well as with an accessibility relation which is 'stable' with respect to this pre-order. BIST logics are extensions of a logic, BiSKt, which arose in the semantic context of hypergraphs, since a special case of the pre-order can represent the incidence structure of a hypergraph. In this paper we provide, for the first time, a Hilbert-style axiomatisation of BISKt and prove the strong completeness of BiSKt. We go on to prove strong completeness of…
Duality for Constructive Modal Logics: from Sahqlvist to Goldblatt-Thomason
Mathematical Institute, University of Bern（伯尔尼大学数学研究所）；School of Computer Science, University of Birimingham（伯明翰大学计算机学院）；School of Computing, The Australian National University（澳大利亚国立大学计算学院）
Game logic was introduced by Rohit Parikh in the 1980s as a generalisation of propositional dynamic logic (PDL) for reasoning about outcomes that players can force in determined 2-player games. Semantically, the generalisation from programs to games is mirrored by moving from Kripke models to monotone neighbourhood models. Parikh proposed a natural PDL-style Hilbert system which was easily proved to be sound, but its completeness has thus far remained an open problem. In this paper, we introduce a cut-free sequent calculus for game logic, and two cut-free sequent calculi that manipulate annota…
It is well-known that a Hilbert-style deduction system for first-order classical logic is sound and complete for a model theory built using all Boolean algebras as truth-value algebras if and only if it is sound and complete for a model theory utilizing only irreducible Boolean algebras (which are all isomorphic to the two-element Boolean algebra). In this paper, we prove an analogous result for any first-order logic with an algebraic semantics satisfying certain minimal assumptions, and we then apply our result to first-order quantum logic.
There has been a significant interest in extending various modal logics with intersection, the most prominent examples being epistemic and doxastic logics with distributed knowledge. Completeness proofs for such logics tend to be complicated, in particular on model classes such as S5 like in standard epistemic logic, mainly due to the undefinability of intersection of modalities in standard modal logics. A standard proof method for the S5 case was outlined in [8] and later explicated in more detail in [13], using an "unraveling-folding method" case to achieve a treelike model to deal with the …
A class of models is presented, in the form of continuation monads polymorphic for first-order individuals, that is sound and complete for minimal intuitionistic predicate logic. The proofs of soundness and completeness are constructive and the computational content of their composition is, in particular, a $β$-normalisation-by-evaluation program for simply typed lambda calculus with sum types. Although the inspiration comes from Danvy's type-directed partial evaluator for the same lambda calculus, the there essential use of delimited control operators (i.e. computational effects) is avoided. …
Given a regular cardinal $κ$ such that $κ^{<κ}=κ$ (e.g., if the Generalized Continuum Hypothesis holds), we develop a proof system for classical infinitary logic that includes heterogeneous quantification (i.e., infinite alternate sequences of quantifiers) within the language $\mathcal{L}_{κ^+, κ}$, where there are conjunctions and disjunctions of at most $κ$ any formulas and quantification (including the heterogeneous one) is applied to less than $κ$ many variables. This type of quantification is interpreted in $\mathcal{S}et$ using the usual second-order formulation in terms of strategies fo…
This paper exhibits a general and uniform method to prove completeness for certain modal fixpoint logics. Given a set Γof modal formulas of the form γ(x, p1, . . ., pn), where x occurs only positively in γ, the language L\sharp (Γ) is obtained by adding to the language of polymodal logic a connective \sharp\_γfor each γε. The term \sharp\_γ(\varphi1, . . ., \varphin) is meant to be interpreted as the least fixed point of the functional interpretation of the term γ(x, φ1, . . ., φn). We consider the following problem: given Γ, construct an axiom system which is sound and complete with respect t…
Institute of Computer Science of the Czech Academy of Sciences（捷克科学院计算机研究所）；Department of Algebra, Faculty of Mathematics and Physics, Charles University（查理大学数学物理学院代数系）
Algebraic quantum field theory, or AQFT for short, is a rigorous analysis of the structure of relativistic quantum mechanics. It is formulated in terms of a net of operator algebras indexed by regions of a Lorentzian manifold. In several cases the mentioned net is represented by a family of von Neumann algebras, concretely, type III factors. Local quantum field logic arises as a logical system that captures the propositional structure encoded in the algebras of the net. In this framework, this work contributes to the solution of a family of open problems, emerged since the 30s, about the chara…
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…
