The provability logic of a theory $T$ is the set of modal formulas, which under any arithmetical realization are provable in $T$. We slightly modify this notion by requiring the arithmetical realizations to come from a specified set $\Gamma$. We make an analogous modification for interpretability logics. We first study provability logics with restricted realizations and show that for various natural candidates of $T$ and restriction set $\Gamma$, the result is the logic of linear frames. However, (...) for the theory Primitive Recursive Arithmetic (PRA), we define a fragment that gives rise to a more interesting provability logic by capitalizing on the well-studied relationship between PRA and I$\Sigma_1$. We then study interpretability logics, obtaining upper bounds for IL(PRA), whose characterization remains a major open question in interpretability logic. Again this upper bound is closely related to linear frames. The technique is also applied to yield the nontrivial result that IL(PRA) $\subset$ ILM. (shrink)

We consider fragments of the relational logic RL(1) obtained by posing various constraints on the relational terms involving the operator of composition of relations. These fragments allow to express several non classical logics including modal and description logics. We show how relational dual tableaux can be employed to provide decision procedures for each of them.

In this paper, a suitable notion of bisimulation is proposed for the bimodal logic with contingency and accident. We obtain several van Benthem Characterization Theorems, and axiomatize the bimodal logic over the class of Eulidean frames and over some more restricted classes, showing their strong completeness via a novel strategy, thereby answering two open questions raised in the literature. With the new bisimulation notion, we also correct an error in the expressivity results in the literature.

We investigate some logics with Henkin quantifiers. For a given logic L, we consider questions of the form: what is the degree of the set of L–tautologies in a poor vocabulary (monadic or empty)? We prove that the set of tautologies of the logic with all Henkin quantifiers in empty vocabulary L*∅ is of degree 0’. We show that the same holds also for some weaker logics like L ∅(Hω) and L ∅(Eω). We show that each (...) logic of the form L ∅ (k)(Q), with the number of variables restricted to k, is decidable. Nevertheless – following the argument of M. Mostowski from [Mos89] – for each reasonable set theory no concrete algorithm can provably decide L (k) (Q), for some (Q). We improve also some results related to undecidability and expressibility for logics L(H4) and L(F2) of Krynicki and M. Mostowski from [KM92]. (shrink)

We introduce structural rules mingle, and investigatetheorem-equivalence, cut- eliminability, decidability, interpolabilityand variable sharing property for sequent calculi having the mingle.These results include new cut-elimination results for the extendedlogics: FLm (full Lambek logic with the mingle), GLm(Girard's linear logic with the mingle) and Lm (Lambek calculuswith restricted mingle).

We develop intuitionistic public announcement logic over intuitionistic \({\textbf{K}}\), \({{\textbf{K}}}{{\textbf{T}}}\), \({{\textbf{K}}}{{\textbf{4}}}\), and \({{\textbf{S}}}{{\textbf{4}}}\) with distributed knowledge. We reveal that a recursion axiom for the distributed knowledge is _not_ valid for a frame class discussed in [ 12 ] but valid for the restricted frame class introduced in [ 20, 26 ]. The semantic completeness of the static logics for this restricted frame class is established via the concept of pseudo-model.

We introduce the logics GLP Λ, a generalization of Japaridze’s polymodal provability logic GLP ω where Λ is any linearly ordered set representing a hierarchy of provability operators of increasing strength. We shall provide a reduction of these logics to GLP ω yielding among other things a finitary proof of the normal form theorem for the variable-free fragment of GLP Λ and the decidability of GLP Λ for recursive orderings Λ. Further, we give a restricted axiomatization of the (...) variable-free fragment of GLP Λ. (shrink)

If the language is extended by new individual variables, in classical first order logic, then the deduction system obtained is a conservative extension of the original one. This fails to be true for the logics with infinitary predicates. But it is shown that restricting the commutativity of quantifiers and the equality axioms in the extended system and supposing the merry-go-round property in the original system, the foregoing extension is already conservative. It is shown that these restrictions are crucial (...) for an extension to be conservative. The origin of the results is algebraic logic. (shrink)

Fluted logic is the restriction of pure predicate logic to formulas in which variables play no essential role. Although fluted logic is significantly weaker than pure predicate logic, it is of interest because it seems closely to parallel natural logic, the logic that is conducted in natural language. It has been known since 1969 that if conjunction in fluted formulas is restricted to subformulas of equal arity, satisfiability is decidable. However, the decidability of (...) sublogics lying between this restricted (homogeneous) fluted logic and full predicate logic remained unknown. In 1994 it was shown that the satisfiability of fluted formulas without restriction is decidable, thus reducing the unknown region significantly. This paper further reduces the unknown region. It shows that fluted logic with the logical identity is decidable. Since the reflection functor can be defined in fluted logic with identity, it follows that fluted logic with the reflection functor also lies within the region of decidability. Relevance to natural logic is increased since the identity permits definition of singular predicates, which can represent anaphoric pronouns. (shrink)

The least common subsumer of a set of concept descriptions is the most specific concept description that subsumes all of the concept descriptions in the given set. By computing the lcs, commonalities between concept descriptions can be made explicit. This is an important inference task useful in several applications, including, for instance, the bottom-up construction of description logic knowledge bases. Previous work on the lcs has concentrated on description logics that either allow for number restrictions or for existential restrictions. (...) Many applications, however, require to combine these constructors. In this work, we present an algorithm for computing the lcs in the description logic ALEN which comprises both constructors—number and existential restrictions—as well as concept conjunction, primitive negation, and value restrictions. To prove correctness of our lcs algorithm, we develop a structural characterization of subsumption in ALEN. (shrink)

In this paper, we present a prenex form theorem for a version of Independence Friendly logic, a logic with imperfect information. Lifting classical results to such logics turns out not to be straightforward, because independence conditions make the formulas sensitive to signalling phenomena. In particular, nested quantification over the same variable is shown to cause problems. For instance, renaming of bound variables may change the interpretations of a formula, there are only restricted quantifier extraction theorems, and (...) slashed connectives cannot be so easily removed. Thus we correct some claims from Hintikka [8], Caicedo & Krynicki [3] and Hodges [11]. We refine definitions, in particular the notion of equivalence, and sharpen preconditions, allowing us to restore those claims, including the prenex form theorem of Caicedo & Krynicki [3], and, as a side result, we obtain an application to Skolem forms of classical formulas. It is a known fact that a complete calculus for IF-logic is impossible, but with our results we establish several quantifier rules that form a partial calculus of equivalence for a general version of IF-logic reflecting general properties of information flow in games. (shrink)

This thesis presents a proof theoretical investigation of some constructive set theories with restricted set induction. The set theories considered are various systems of Constructive Zermelo Fraenkel set theory, CZF ([1]), in which the schema of $\in$ - Induction is either removed or weakened. We shall examine the theories $CZF^\Sigma_\omega$ and $CZF_\omega$, in which the $\in$ - Induction scheme is replaced by a scheme of induction on the natural numbers (only for  formulas in the case of the (...) first theory, for arbitrary formulas in the second theory). Of particular interest is the system CZF− + INAC, which is obtained by removing $\in$ - Induction and by adding an axiom, INAC, for inaccessible sets. The main outcome is that CZF− + INAC is a predicative set theory with inaccessible sets, its proof theoretic ordinal being $\Gamma_0$. The proof theoretic strength of these theories, has been determined by means of proof theoretical reductions. The upper bounds have been obtained by interpreting the set theories in classical theories of (iterated) inductive definitions with ordinals. A significant feature of these interpretations is that they are realizability interpretations, which make an indirect use of Aczel’s interpretation of CZF in Martin Löf type theory, and, in the case of CZF− + INAC, also employ a maximum bisimulation relation to represent equality between sets. The lower bounds are obtained by interpreting appropriate subsystems of intuitionistic second order arithmetic in the set theories. At the conclusion of the thesis we present work unrelated to the previous chapters. It is a preliminary study of the constructible hierarchy in CZF. We begin by formalizing the notion of definability in constructive set theory and then introduce Gödel’s L in CZF. We prove that intuitionistic L is a model of a subsystem of CZF strong enough to include intuitionistic Kripke-Platek set theory, and also that L is a model of the axiom of constructibility, provably in CZF. (shrink)

We consider countable so‐called rich subsemigroups of ; each such semigroup T gives a variety CPEAT that is axiomatizable by a finite schema of equations taken in a countable subsignature of that of ω‐dimensional cylindric‐polyadic algebras with equality where substitutions are restricted to maps in T. It is shown that for any such T, if and only if is representable as a concrete set algebra of ω‐ary relations. The operations in the signature are set‐theoretically interpreted like in polyadic (...) equality set algebras, but such operations are relativized to a union of cartesian spaces that are not necessarily disjoint. This is a form of guarding semantics. We show that CPEAT is canonical and atom‐canonical. Imposing an extra condition on T, we prove that atomic algebras in CPEAT are completely representable and that CPEAT has the super amalgamation property. If T is rich and finitely represented, it is shown that CPEAT is term definitionally equivalent to a finitely axiomatizable Sahlqvist variety. Such semigroups exist. This can be regarded as a solution to the central finitizability problem in algebraic logic for first order logic with equality if we do not insist on full fledged commutativity of quantifiers. The finite dimensional case is approached from the view point of guarded and clique guarded (relativized) semantics of fragments of first order logic using finitely many variables. Both positive and negative results are presented. (shrink)

This paper aims at being a systematic investigation of different completeness properties of first-order predicate logics with truth-constants based on a large class of left-continuous t-norms . We consider standard semantics over the real unit interval but also we explore alternative semantics based on the rational unit interval and on finite chains. We prove that expansions with truth-constants are conservative and we study their real, rational and finite chain completeness properties. Particularly interesting is the case of considering canonical (...) real and rational semantics provided by the algebras where the truth-constants are interpreted as the numbers they actually name. Finally, we study completeness properties restricted to evaluated formulae of the kind , where φ has no additional truth-constants. (shrink)

In the present paper, we continue the research in Zhao (2021, Logic J. IGPL) to develop the Sahlqvist completeness theory for hybrid logic with satisfaction operators and downarrow binders |$\mathcal {L}( @, {\downarrow })$|⁠. We define the class of restricted Sahlqvist formulas for |$\mathcal {L}( @, {\downarrow })$| following the ideas in Conradie and Robinson (2017, J. Logic Comput., 27, 867–900), but we follow a different proof strategy which is purely proof-theoretic, namely showing that for (...) every restricted Sahlqvist formula |$\varphi $| and its hybrid pure correspondence |$\pi $|⁠, |$\textbf {K}_{\mathcal {H}( @, {\downarrow })}+\varphi $| proves |$\pi $|⁠; therefore, |$\textbf {K}_{\mathcal {H}( @, {\downarrow })}+\varphi $| is complete with respect to the class of frames defined by |$\pi $|⁠, using a modified version |$\textsf {ALBA}^{{\downarrow }}_{\textsf {Modified}}$| of the algorithm |$\textsf {ALBA}^{{\downarrow }}$| defined in Zhao (2021, Logic J. IGPL). (shrink)

In this paper we present BTC, which is a complete logic for branchingtime whose modal operator quantifies over histories and whose temporal operators involve a restricted quantification over histories in a given possible choice. This is a technical novelty, since the operators of the usual logics for branching-time such as CTL express an unrestricted quantification over histories and moments. The value of the apparatus we introduce is connected to those logics of agency that are interpreted on branching-time, as (...) for instance Stit Logics. (shrink)

A common misconception among logicians is to think that intuitionism is necessarily tied-up with single conclusion calculi. Single conclusion calculi can be used to model intuitionism and they are convenient, but by no means are they necessary. This has been shown by such influential textbook authors as Kleene, Takeuti and Dummett, to cite only three. If single conclusions are not necessary, how do we guarantee that only intuitionistic derivations are allowed? Traditionally one insists on restrictions on particular rules: implication (...) right, negation right and universal quantification right are required to be single conclusion rules. In this note we show that instead of a cardinality restriction such as one-conclusion-only, we can use a notion of dependency between formulae to enforce the constructive character of derivations. The system we obtain, called FIL for full intuitionistic logic, satisfies basic properties such as soundness, completeness and cut elimination. We present two motivating applications of FIL and discuss some future work. (shrink)

We investigate mathematical structures that provide natural semantics for families of (quantified) non-classical logics featuring special unary connectives, known as recovery operators, that allow us to ‘recover’ the properties of classical logic in a controlled manner. These structures are known as topological Boolean algebras, which are Boolean algebras extended with additional operations subject to specific conditions of a topological nature. In this study, we focus on the paradigmatic case of negation. We demonstrate how these algebras are well-suited to (...) provide a semantics for some families of paraconsistent Logics of Formal Inconsistency and paracomplete Logics of Formal Undeterminedness. These logics feature recovery operators used to earmark propositions that behave ‘classically’ when interacting with non-classical negations. Unlike traditional semantical investigations, which are carried out in natural language (extended with mathematical shorthand), our formal meta-language is a system of higher-order logic (HOL) for which automated reasoning tools exist. In our approach, topological Boolean algebras are encoded as algebras of sets via their Stone-type representation. We use our higher-order meta-logic to define and interrelate several transformations on unary set operations, which naturally give rise to a topological cube of opposition. Additionally, our approach enables a uniform characterization of propositional, first-order and higher-order quantification, including restrictions to constant and varying domains. With this work, we aim to make a case for the utilization of automated theorem proving technology for conducting computer-supported research in non-classical logics. All the results presented in this paper have been formally verified, and in many cases obtained, using the Isabelle/HOL proof assistant. (shrink)

Harold Hodes in [1] introduces an extension of first-order modal logic featuring a backtracking operator, and provides a possible worlds semantics, according to which the operator is a kind of device for ‘world travel’; he does not provide a proof theory. In this paper, I provide a natural deduction system for modal logic featuring this operator, and argue that the system can be motivated in terms of a reading of the backtracking operator whereby it serves to indicate modal (...) scope. I prove soundness and completeness theorems with respect to Hodes’ semantics, as well as semantics with fewer restrictions on the accessibility relation. (shrink)

The decision problem for provability logic remains PSPACE-complete even if the number of propositional atoms is restricted to one.

In this paper we prove the equivalence between the Gentzen system G LJ*\c , obtained by deleting the contraction rule from the sequent calculus LJ* (which is a redundant version of LJ), the deductive system IPC*\c and the equational system associated with the variety RL of residuated lattices. This means that the variety RL is the equivalent algebraic semantics for both systems G LJ*\c in the sense of [18] and [4], respectively. The equivalence between G LJ*\c and IPC*\c is (...) a strengthening of a result obtained by H. Ono and Y. Komori [14, Corollary 2.8.1] and the equivalence between G LJ*\c and the equational system associated with the variety RL of residuated lattices is a strengthening of a result obtained by P.M. Idziak [13, Theorem 1].An axiomatization of the restriction of IPC*\c to the formulas whose main connective is the implication connective is obtained by using an interpretation of G LJ*\c in IPC*\c. (shrink)

LCo with the Converse Ackermann Property is defined as the result of restricting Contraction in LC. Intuitionistic and Superintuitionistic Negation is shown to be compatible with the CAP.

We show that the spectrum of a sentence ϕ in Counting Monadic Second Order Logic (CMSOL) using one binary relation symbol and finitely many unary relation symbols, is ultimately periodic, provided all the models of ϕ are of clique width at most k, for some fixed k. We prove a similar statement for arbitrary finite relational vocabularies τ and a variant of clique width for τ-structures. This includes the cases where the models of ϕ are of tree width at (...) most k. For the case of bounded tree-width, the ultimate periodicity is even proved for Guarded Second Order Logic GSOL. We also generalize this result to many-sorted spectra, which can be viewed as an analogue of Parikh's Theorem on context-free languages, and its analogues for context-free graph grammars due to Habel and Courcelle. Our work was inspired by Gurevich and Shelah (2003), who showed ultimate periodicity of the spectrum for sentences of Monadic Second Order Logic where only finitely many unary predicates and one unary function are allowed. This restriction implies that the models are all of tree width at most 2, and hence it follows from our result. (shrink)

We prove that the existence of atomic models for countable atomic theories does not hold for Ln the first order logic restricted to n variables for finite n > 2. Our proof is algebraic, via polyadic algebras. We note that Lnhas been studied in recent times as a multi-modal logic with applications in computer science. 2000 MATHEMATICS SUBJECT CLASSIFICATION. 03C07, 03G15.

In this paper we analyze some aspects of the question of using methods from model theory to study structures of functional analysis.By a well known result of P. Lindström, one cannot extend the expressive power of first order logic and yet preserve its most outstanding model theoretic characteristics (e.g., compactness and the Löwenheim-Skolem theorem). However, one may consider extending the scope of first order in a different sense, specifically, by expanding the class of structures that are regarded as models (...) (e.g., including Banach algebras or other structures of functional analysis), and ask whether the resulting extensions of first order model theory preserve some of its desirable characteristics.A formal framework for the study of structures based on Banach spaces from the perspective of model theory was first introduced by C. W. Henson in [8] and [6]. Notions of syntax and semantics for these structures were defined, and it was shown that using them one obtains a model theoretic apparatus that satisfies many of the fundamental properties of first order model theory. For instance, one has compactness, Löwenheim-Skolem, and omitting types theorems. Further aspects of the theory, namely, the fundamentals of stability and forking, were first introduced in [10] and [9].The classes of mathematical structures formally encompassed by this framework are normed linear spaces, possibly expanded with additional structure,e.g., operations, real-valued relations, and constants. This notion subsumes wide classes of structures from functional analysis. However, the restriction that the universe of a structure be a normed space is not necessary. (This restriction has a historical, rather than technical origin; specifically, the development of the theory was originally motivated by questions in Banach space geometry.) Analogous techniques can be applied if the universe is a metric space. Now, when the underlying metric topology is discrete, the resulting model theory coincides with first order model theory, so this logic extends first order in the sense described above. Furthermore, without any cost in the mathematical complexity, one can also work in multi-sorted contexts, so, for instance, one sort could be an operator algebra while another is. say, a metric space. (shrink)

The paper studies a cluster of systems for fully disquotational truth based on the restriction of initial sequents. Unlike well-known alternative approaches, such systems display both a simple and intuitive model theory and remarkable proof-theoretic properties. We start by showing that, due to a strong form of invertibility of the truth rules, cut is eliminable in the systems via a standard strategy supplemented by a suitable measure of the number of applications of truth rules to formulas in derivations. Next, we (...) notice that cut remains eliminable when suitable arithmetical axioms are added to the system. Finally, we establish a direct link between cut-free derivability in infinitary formulations of the systems considered and fixed-point semantics. Noticeably, unlike what happens with other background logics, such links are established without imposing any restriction to the premisses of the truth rules. (shrink)

This paper proves normalisation theorems for intuitionist and classical negative free logic, without and with the $\invertediota$ operator for definite descriptions. Rules specific to free logic give rise to new kinds of maximal formulas additional to those familiar from standard intuitionist and classical logic. When $\invertediota$ is added it must be ensured that reduction procedures involving replacements of parameters by terms do not introduce new maximal formulas of higher degree than the ones removed. The problem is (...) solved by a rule that permits restricting these terms in the rules for $\forall$, $\exists$ and $\invertediota$ to parameters or constants. A restricted subformula property for deductions in systems without $\invertediota$ is considered. It is improved upon by an alternative formalisation of free logic building on an idea of Ja\'skowski's. In the classical system the rules for $\invertediota$ require treatment known from normalisation for classical logic with $\lor$ or $\exists$. The philosophical significance of the results is also indicated. (shrink)

We prove a local normal form theorem of the Gaifman type for the infinitary logic L∞ωω whose formulas involve arbitrary unary quantifiers but finite quantifier rank. We use a local Ehrenfeucht-Fraïssé type game similar to the one in [9]. A consequence is that every sentence of L∞ωω of quantifier rank n is equivalent to an infinite Boolean combination of sentences of the form ψ, where ψ has counting quantifiers restricted to the -neighborhood of y.

This is the second part of a two-part series on the logic of hyperlogic, a formal system for regimenting metalogical claims in the object language (even within embedded environments). Part A provided a minimal logic for hyperlogic that is sound and complete over the class of all models. In this part, we extend these completeness results to stronger logics that are sound and complete over restricted classes of models. We also investigate the logic of hyperlogic when (...) the language is enriched with hyperintensional operators such as counterfactual conditionals and belief operators. (shrink)

This paper is devoted to the completeness issue of RMLCI — the relative modal logic with composition and intersection— a restriction of the propositional dynamic logic with intersection. The trouble with RMLCI is that the operation of intersection is not modally definable. Using the notion of mosaics, we give a new proof of a theorem considered in a previous paper “Complete axiomatization of a relative modal logic with composition and intersection”. The theorem asserts (...) that the proof theory of RMLCI is complete for the standard Kripke semantics of RMLCI. (shrink)

We prove a local normal form theorem of the Gaifman type for the infinitary logic L∞ωω whose formulas involve arbitrary unary quantifiers but finite quantifier rank. We use a local Ehrenfeucht-Fraïssé type game similar to the one in [9]. A consequence is that every sentence of L∞ωω of quantifier rank n is equivalent to an infinite Boolean combination of sentences of the form ψ, where ψ has counting quantifiers restricted to the -neighborhood of y.

A fascinating recent turn in epistemology focuses on inquiring attitudes like wondering and being curious. Many have argued that these attitudes are governed by norms similar to those that govern our doxastic attitudes. Yet, to date, this work has only considered norms that might prohibit having certain inquiring attitudes (“norms of restriction”), while ignoring those that might require having them (“norms of expansion”). We aim to address that omission by offering a framework that generates norms of expansion for inquiring attitudes. (...) The framework draws on inferential erotetic logic, which we explain and augment with some theorems. We explore several of the norms that it yields—some sympathetically, others unsympathetically. (shrink)

We introduce a restricted second-order logic $\textrm{SO}^{\textit{plog}}$ for finite structures where second-order quantification ranges over relations of size at most poly-logarithmic in the size of the structure. We demonstrate the relevance of this logic and complexity class by several problems in database theory. We then prove a Fagin’s style theorem showing that the Boolean queries which can be expressed in the existential fragment of $\textrm{SO}^{\textit{plog}}$ correspond exactly to the class of decision problems that can be computed by (...) a non-deterministic Turing machine with random access to the input in time $O^k)$ for some $k \ge 0$, i.e. to the class of problems computable in non-deterministic poly-logarithmic time. It should be noted that unlike Fagin’s theorem which proves that the existential fragment of second-order logic captures NP over arbitrary finite structures, our result only holds over ordered finite structures, since $\textrm{SO}^{\textit{plog}}$ is too weak as to define a total order of the domain. Nevertheless, $\textrm{SO}^{\textit{plog}}$ provides natural levels of expressibility within poly-logarithmic space in a way which is closely related to how second-order logic provides natural levels of expressibility within polynomial space. Indeed, we show an exact correspondence between the quantifier prefix classes of $\textrm{SO}^{\textit{plog}}$ and the levels of the non-deterministic poly-logarithmic time hierarchy, analogous to the correspondence between the quantifier prefix classes of second-order logic and the polynomial-time hierarchy. Our work closely relates to the constant depth quasipolynomial size AND/OR circuits and corresponding restricted second-order logic defined by David A. Mix Barrington in 1992. We explore this relationship in detail. (shrink)

A fascinating recent turn in epistemology focuses on inquiring attitudes like wondering and being curious. Many have argued that these attitudes are governed by norms similar to those that govern our doxastic attitudes. Yet, to date, this work has only considered norms that might *prohibit* having certain inquiring attitudes (``norms of restriction''), while ignoring those that might *require* having them (``norms of expansion''). We aim to address that omission by offering a framework that generates norms of expansion for inquiring attitudes. (...) The framework draws on inferential erotetic logic, which we explain and augment with some theorems. We explore several of the norms that it yields - some sympathetically, others unsympathetically. (shrink)

This article concerns the metatheory of a class of modal logics whose language includes propositional constants of various kinds. The main novelties are the use of general frames with specific restrictions and the definition of the strict range of a formula. Many examples from the literature are treated within the framework provided and some traditional model-theoretic issues such as preservation results concerning the validity of formulas and definability results concerning frame properties are addressed.

As a result of these restrictions, an implementation of L [subscript lambda] does not need to implement full higher-order unification. Instead, an extension to first-order unification that respects bound variable names and scopes is all that is required. Such unification problems are shown to be decidable and to possess most general unifiers when unifiers exist. A unification algorithm and logic programming interpreter are described and proved correct. Several examples of using L[subscript lambda] as a meta-programming language are presented.

Recently, Coquand and Palmgren considered systems of intuitionistic arithmetic in a finite types together with various forms of the axiom of choice and a numerical omniscience schema which implies classical logic for arithmetical formulas. Feferman subsequently observed that the proof theoretic strength of such systems can be determined by functional interpretation based on a non-constructive μ-operator and his well-known results on the strength of this operator from the 70's. In this note we consider a weaker form LNOS of (...) NOS which suffices to derive the strong form of binary König's lemma studied by Coquand/Palmgren and gives rise to a new and mathematically strong semi-classical system which, nevertheless, can proof theoretically be reduced to primitive recursive arithmetic PRA. The proof of this fact relies on functional interpretation and a majorization technique developed in a previous paper. (shrink)

Despite being fairly powerful, finite non-deterministic matrices are unable to characterize some logics of formal inconsistency, such as those found between mbCcl and Cila. In order to overcome this limitation, we propose here restricted non-deterministic matrices (in short, RNmatrices), which are non-deterministic algebras together with a subset of the set of valuations. This allows us to characterize not only mbCcl and Cila (which is equivalent, up to language, to da Costa's logic C_1) but the whole hierarchy of (...) da Costa's calculi C_n. This produces a novel decision procedure for these logics. Moreover, we show that the RNmatrix semantics proposed here induces naturally a labelled tableau system for each C_n, which constitutes another decision procedure for these logics. This new semantics allows us to conceive da Costa's hierarchy of C-systems as a family of (non deterministically) (n+2)-valued logics, where n is the number of "inconsistently true" truth-values and 2 is the number of "classical" or "consistent" truth-values, for every C_n. (shrink)

We study the expressive power in the finite of the logic Fixed-Point+Counting, the extension of first-order logic which is obtained through adding both the fixed-point constructor and the ability to count. To this end an isomorphism preserving (`generic') model of computation is introduced whose PTime restriction exactly corresponds to this level of expressive power, while its PSpace restriction corresponds to While+Counting. From this model we obtain a normal form which shows a rather clear separation of the relational vs. (...) the arithmetical side of the algorithms involved. In parallel, we study the relations of Fixed-Point+Counting with the infinitary logics L r ∞ω (∃ ≥ m ) m∈ω and the corresponding pebble games. The main result, however, involves the concept of an arithmetical invariant. By this we mean a functor taking every finite relational structure to an expansion of (an initial segment of) the standard arithmetical structure. In particular its values are linearly ordered structures. We establish the existence of a family of arithmetical invariants (J r ) r≥ 1 with the following properties: $\bullet$ The invariants themselves can be evaluated in polynomial time. $\bullet$ A class of finite relational structures is definable in Fixed-Point+Counting if and only if membership can be decided in polynomial time on the basis of the values of one of the invariants. $\bullet$ The invariant J r classifies all finite relational structures exactly up to equivalence with respect to the logic L r ∞ω (∃ ≥ m ) m∈ω . We also give a characterization of Fixed-Point+Counting in terms of sequences of formulae in the L r ωω (∃ ≥ m ) m∈ω : It corresponds exactly to the polynomial time computable families (φ n ) n∈ω in these logics. Towards a positive assessment of the expressive power of Fixed-Point+Counting, it is shown that the natural extension of fixed-point logic by Lindström quantifiers, which capture all the PTime computable properties of cardinalities of definable predicates, is strictly weaker than what we get here. This implies in particular that every extension of fixed-point logic by means of monadic Lindstrom quantifiers, which stays within PTime, must be strictly contained in Fixed-Point+Counting. (shrink)

The logic of common belief does not always reflect that of individual beliefs. In particular, even when the individual belief operators satisfy the KD45 logic, the common belief operator may fail to satisfy axiom 5. That is, it can happen that neither is A commonly believed nor is it common belief that A is not commonly believed. We identify the intersubjective restrictions on individual beliefs that are incorporated in axiom 5 for common belief.

Temporal logic as a modern discipline is separate from classical logic; it is seen as an addition or expansion of the more basic propositional and predicate logics. This approach is in contrast with logic in the Middle Ages, which was primarily intended as a tool for the analysis of natural language. Because all natural language sentences have tensed verbs, medieval logic is inherently a temporal logic. This fact is most clearly exemplified in medieval theories (...) of supposition. As a case study, we look at the supposition theory of Lambert of Lagny, extracting from it a temporal logic and providing a formalization of that logic. (shrink)

This book presents an up-to-date, unified treatment of research in bounded arithmetic and complexity of propositional logic, with emphasis on independence proofs and lower bound proofs. The author discusses the deep connections between logic and complexity theory and lists a number of intriguing open problems. An introduction to the basics of logic and complexity theory is followed by discussion of important results in propositional proof systems and systems of bounded arithmetic. More advanced topics are then treated, (...) including polynomial simulations and conservativity results, various witnessing theorems, the translation of bounded formulas (and their proofs) into propositional ones, the method of random partial restrictions and its applications, direct independence proofs, complete systems of partial relations, lower bounds to the size of constant-depth propositional proofs, the method of Boolean valuations, the issue of hard tautologies and optimal proof systems, combinatorics and complexity theory within bounded arithmetic, and relations to complexity issues of predicate calculus. Students and researchers in mathematical logic and complexity theory will find this comprehensive treatment an excellent guide to this expanding interdisciplinary area. (shrink)

A propositional logic with modal operators for logical necessity and possibility is formulated as a formal ontology for logical atomism (with negative facts). It is shown that such modal operators represent purely formal, Internal 'properties' of propositions if and only if the notion of 'all possible worlds' has its standard and not the secondary interpretation which it is usually given (as, E.G., In kripke model-Structures). Allowing arbitrary restrictions on the notion of 'all possible worlds', At least in (...) such a framework as logical atomism, Generates internal 'properties' of propositions with material instead of purely formal content. (shrink)

A restriction of R-Mingle with the variable-sharing property and the Ackermann properties is defined. From an intuitive semantical point of view, this restriction is an alternative to Anderson and Belnap’s logic of entailment E.

Lattice logic, bilattice logic, and paraconsistent quantum logic are investigated based on monosequent systems. Paraconsistent quantum logic is an extension of lattice logic, and bilattice logic is an extension of paraconsistent quantum logic. Monosequent system is a sequent calculus based on the restricted sequent that contains exactly one formula in both the antecedent and succedent. It is known that a completeness theorem with respect to a lattice-valued semantics holds for a monosequent (...) system for lattice logic. A completeness theorem with respect to a lattice-valued semantics is proved for paraconsistent quantum logic, and a completeness theorem with respect to a bilattice-valued semantics is proved for bilattice logic. Some syntactical properties, including cut-elimination and duality, are also investigated for the monosequent systems for these logics. (shrink)

Partly motivated by description logics, poor man's logics have been proposed as an interesting fragment of modal logics. A poor man's logic is a propositional modal logic where only literals and the connectives ∧, □, and ◊ are allowed. It is known that the complexity of the satisfiability problem may drop dramatically when going from a full modal logic to the corresponding poor man's logic, e.g., in the case of modal logic K one goes from (...) PSPACE-complete to coNP-complete. We prove that it is sometimes possible to extend poor man's logics with restricted disjunctions (i.e., clauses) without increasing the computational complexity. For Horn and Krom clauses, we provide necessary and sufficient conditions for when the resulting logic is polynomial-time. Such extensions correspond to allowing a restricted use of the union operator in description logics. (shrink)

In many natural languages, there are clear syntactic and/or intonational differences between declarative sentences, which are primarily used to provide information, and interrogative sentences, which are primarily used to request information. Most logical frameworks restrict their attention to the former. Those that are concerned with both usually assume a logical language that makes a clear syntactic distinction between declaratives and interrogatives, and usually assign different types of semantic values to these two types of sentences. A different approach has been (...) taken in recent work on inquisitive semantics. This approach does not take the basic syntactic distinction between declaratives and interrogatives as its starting point, but rather a new notion of meaning that captures both informative and inquisitive content in an integrated way. The standard way to treat the logical connectives in this approach is to associate them with the basic algebraic operations on these new types of meanings. For instance, conjunction and disjunction are treated as meet and join operators, just as in classical logic. This gives rise to a hybrid system, where sentences can be both informative and inquisitive at the same time, and there is no clearcut division between declaratives and interrogatives. It may seem that these two general approaches in the existing literature are quite incompatible. The main aim of this paper is to show that this is not the case. We develop an inquisitive semantics for a logical language that has a clearcut division between declaratives and interrogatives. We show that this language coincides in expressive power with the hybrid language that is standardly assumed in inquisitive semantics, we establish a sound and complete axiomatization for the associated logic, and we consider a natural enrichment of the system with presuppositional interrogatives. (shrink)