In this paper, we will provide a method to compute the density of tautologies among the set of well-formed formulae consisting of $m$ variables, the negation symbol and the implication symbol; which has a possibility to be applied for other logical systems. This paper contains computational numerical values of the density of tautologies for two, three, and four variable cases. Also, for certain quadratic systems, we will build a theory of the $s$-cut concept to make a memory-time trade-off when we compute the ratio by brute-force counting, and discover a fundamental relation between generating functions' values on the singularity point and ratios of coefficients, which can be understood as another interpretation of the Szegő lemma for such quadratic systems. With this relation, we will provide an asymptotic lower bound $m^{-1}-(7/4)m^{-3/2}+O(m^{-2})$ of the density of tautologies in the logic system with $m$ variables, the negation, and the implication, as $m$ goes to the infinity
Cieśliński asked whether compositional truth theory with the additional axiom that all propositional tautologies are true is conservative over Peano Arithmetic. We provide a partial answer to this question, showing that if we additionally assume that truth predicate agrees with arithmetical truth on quantifier-free sentences, the resulting theory is as strong as $Δ_0$-induction for the compositional truth predicate, hence non-conservative. On the other hand, it can be shown with a routine argument that the principle of quantifier-free correctness is itself conservative.
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…
Bernays introduced a method for proving underivability results in propositional calculi by truth tables. In general, this motivates an investigations of how to find, given a propositional logic, a finite-valued logic which has as few tautologies as possible, but which has all the valid formulas of the given logic as tautologies. It is investigated how far this method can be carried using (1) one or (2) an infinite sequence of finite-valued logics. It is shown that the best candidate matrices for (1) can be computed from a calculus, and how sequences for (2) can be found for certain classes of …
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.
As defined by Dunn, Moss, and Wang, an universal test set in an ortholattice $L$ is a subset $T$ such that each term takes value $1$, only, if it does so under all substitutions from $T$. Generalizing their result for ortholattices of subspaces of finite dimensional Hilbert spaces, we show that no infinite modular ortholattice of finite dimension admits a finite universal test set. On the other hand, answering a question of the same authors, we provide a countable universal test set for the ortholattice of projections of any type II$_1$ von Neumann algebra factor as well as for von Neumann's a…
We extend classical Propositional Logic (PL) by adding a new primitive binary connective $φ|ψ$, intended to represent the "superposition" of sentences $φ$ and $ψ$, an operation motivated by the corresponding notion of quantum mechanics, but not intended to capture all aspects of the latter as they appear in physics. To interpret the new connective, we extend the classical Boolean semantics by employing models of the form $\langle M,f\rangle$, where $M$ is an ordinary two-valued assignment for the sentences of PL and $f$ is a choice function for all pairs of classical sentences. In the new sema…
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…
The thesis of this paper is that truth-relevant logic is a better foundation for mathematics than classical logic. It is a system proposed by Richard Diaz in 1981. In a certain sense t-relevant logic is based on Kleene strong tables. These define a system with three values: true, false, unknown. It turns out that there exist tautologies with the following property: there exists a proper subset of propositional variables (t-relevant variables) such that for all combinations true/false the tautology will be true, that is, the rest of the variables (redundant variables) occurring in the tautology…
In this paper, we study several propositional team logics that are closed under unions, including propositional inclusion logic. We prove that all these logics are expressively complete, and we introduce sound and complete systems of natural deduction for these logics. We also discuss the locality property and its connection with interpolation in these logics.
We consider team semantics for propositional logic, continuing our previous work (Yang & Väänänen 2016). In team semantics the truth of a propositional formula is considered in a set of valuations, called a team, rather than in an individual valuation. This offers the possibility to give meaning to concepts such as dependence, independence and inclusion. We define an expressively maximal propositional team logic, called full propositional team logic. This requires going beyond the logical operations of classical propositional logic. We exhibit a hierarchy of logics between the smallest, viz. c…
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.
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 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 He…
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.
In this paper we prove that three of the main propositional logics of dependence (including propositional dependence logic and inquisitive logic), none of which is structural, are structurally complete with respect to a class of substitutions under which the logics are closed. We obtain an analogues result with respect to stable substitutions, for the negative variants of some well-known intermediate logics, which are intermediate theories that are closely related to inquisitive logic.
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.
In this paper, we study logics of dependence on the propositional level. We prove that several interesting propositional logics of dependence, including propositional dependence logic, propositional intuitionistic dependence logic as well as propositional inquisitive logic, are expressively complete and have disjunctive or conjunctive normal forms. We provide deduction systems and prove the completeness theorems for these logics.
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.…
In this paper, we show that the derivability problem for the primal propositional logic remains solvable in polynomial time upon adding a certain form of the principle of equivalent form substitution; and that, upon adding another form of this principle, it becomes co-NP-hard.
Propositional logics in general, considered as a set of sentences, can be undecidable even if they have "nice" representations, e.g., are given by a calculus. Even decidable propositional logics can be computationally complex (e.g., already intuitionistic logic is PSPACE-complete). On the other hand, finite-valued logics are computationally relatively simple - at worst NP. Moreover, finite-valued semantics are simple, and general methods for theorem proving exist. This raises the question to what extent and under what circumstances propositional logics represented in various ways can be approx…
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…
We introduce a multi-type display calculus for Propositional Dynamic Logic (PDL). This calculus is complete w.r.t. PDL, and enjoys Belnap-style cut-elimination and subformula property.
We give sound and complete Hilbert-style axiomatizations for propositional dependence logic (PD), modal dependence logic (MDL), and extended modal dependence logic (EMDL) by extending existing axiomatizations for propositional logic and modal logic. In addition, we give novel labeled tableau calculi for PD, MDL, and EMDL. We prove soundness, completeness and termination for each of the labeled calculi.
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…
We introduce and investigate the notion of uniform Lyndon interpolation property (ULIP) which is a strengthening of both uniform interpolation property and Lyndon interpolation property. We prove several propositional modal logics including ${\bf K}$, ${\bf KB}$, ${\bf GL}$ and ${\bf Grz}$ enjoy ULIP. Our proofs are modifications of Visser's proofs of uniform interpolation property using layered bisimulations. Also we give a new upper bound on the complexity of uniform interpolants for ${\bf GL}$ and ${\bf Grz}$.
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.
Isomorphism between formulae is defined with respect to categories formalizing equality of deductions in classical propositional logic and in the multiplicative fragment of classical linear propositional logic caught by proof nets. This equality is motivated by generality of deductions. Characterizations are given for pairs of isomorphic formulae, which lead to decision procedures for this isomorphism.
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
Truth values of Modus Ponens and Modus Tollens rules for propositions having linguistic truth value that may be represented by lattice(Fig1, Fig2) are computed in this paper. The results show that the truth values are not always absolutely true but have graded truth values in some cases.
It is shown that G-up, the quantified propositional Goedel-Dummett logic based on the truth-values set V-up = {1 - 1/n : n >= 1} u {1}, is decidable. This result is obtained by reduction to Buechi's theory S1S. An alternative proof based on elimination of quantifiers is also given, which yields both an axiomatization and a characterization of G-up as the intersection of all finite-valued quantified propositional Goedel logics.
We give a procedure for counting the number of different proofs of a formula in various sorts of propositional logic. This number is either an integer (that may be 0 if the formula is not provable) or infinite.
It is known that not only classical semantics but also intuitionistic Kripke semantics can be generalized so that it can treat arbitrary propositional connectives characterized by truth tables, or truth functions. In our previous work, it has been shown that the set of Kripke-valid propositional sequents and that of classically valid propositional sequents coincide if and only if all available propositional connectives are monotonic. The present paper extend this result to frst-order logic showing that, in the case of predicate logic, the condition that all available propositional connectives …
Cirquent calculus is a new proof-theoretic and semantic approach introduced for the needs of computability logic by G.Japaridze, who also showed that, through cirquent calculus, one can capture, refine and generalize independence-friendly (IF) logic. Specifically, the approach allows us to account for independence from propositional connectives in the same spirit as the traditional IF logic accounts for independence from quantifiers. Japaridze's treatment of IF logic, however, was purely semantical, and no deductive system was proposed. The present paper constructs a formal system sound and co…
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.
This paper investigates formal logics for reasoning about determinacy and independence. Propositional Dependence Logic D and Propositional Independence Logic I are recently developed logical systems, based on team semantics, that provide a framework for such reasoning tasks. We introduce two new logics L_D and L_I, based on Kripke semantics, and propose them as alternatives for D and I, respectively. We analyze the relative expressive powers of these four logics and discuss the way these systems relate to natural language. We argue that L_D and L_I naturally resolve a range of interpretational…
In this work we explore some applications of the notions of Institution and Π-Institution in the setting of propositional logics and establish a precise categorial relation between these notions, i.e., we provide a pair of functors that establishes an adjunction between the categories Inst and Π-Inst.
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…
A weakly complete finitary axiomatization for EQPL (exogenous quantum propositional logic) is presented. The proof is carried out using a non trivial extension of the Fagin-Halpern-Megiddo technique together with three Henkin style completions.
We classify the computational complexity of the satisfiability, validity and model-checking problems for propositional independence, inclusion, and team logic. Our main result shows that the satisfiability and validity problems for propositional team logic are complete for alternating exponential-time with polynomially many alternations.
The BHK interpretation interprets propositional statements as descriptions of the world of proofs; a world which is hierarchical in nature. It consists of different layers of the concept of proof; the proofs, the proofs about proofs and so on. To describe this hierarchical world, one approach is the Russellian approach in which we use a typed language to reflect this hierarchical nature in the syntax level. In this case, since the connective responsible for this hierarchical behavior is implication, we will use a typed language equipped with a hierarchy of implications, $\{\rightarrow_n\}_{n=0…
Lindenbaum method is named after the Polish logician Adolf Lindenbaum who prematurely and without a clear trace disappeared in the turmoil of the Second World War at the age of about 37. The method is based on the symbolic nature of formalized languages of deductive systems and opens a gate for applications of algebra to logic and, thereby, to Abstract algebraic logic.
We investigate the complexity of satisfiability for finite-variable fragments of propositional dynamic logics. We consider three formalisms belonging to three representative complexity classes, broadly understood,---regular PDL, which is EXPTIME-complete, PDL with intersection, which is 2EXPTIME-complete, and PDL with parallel composition, which is undecidable. We show that, for each of these logics, the complexity of satisfiability remains unchanged even if we only allow as inputs formulas built solely out of propositional constants, i.e. without propositional variables. Moreover, we show tha…
This paper develops a general methodology to connect propositional and first-order interpolation. In fact, the existence of suitable skolemizations and of Herbrand expansions together with a propositional interpolant suffice to construct a first-order interpolant. This methodology is realized for lattice-based finitely-valued logics, the top element representing true. It is shown that interpolation is decidable for these logics.
This paper presents a formal theory which describes propositional binary logic as a semantically closed formal language, and allows for syntactically and semantically well-formed formulae, formal proofs (demonstrability in Hilbertian acception), deduction (Gentzen's view of demonstrability), CNF-ization, and deconstruction to be expressed and tested in the same (computational) formal language, using the same data structure. It is also shown here that Cognitive Binary Logic is a self-described theory in which the Liar Paradox is deconstructed.
In this paper, we shall give another proof of the faithfulness of Blass translation (for short, $B$-translation) of the propositional fragment $\bf L_1$ of Leśniewski's ontology in the modal logic $\bf K$ \it by means of Hintikka formula\rm . And we extend the result to von Wright-type deontic logics, i.e., ten Smiley-Hanson systems of monadic deontic logic. As a result of observing the proofs we shall give general theorems on the faithfulness of $B$-translation with respect to normal modal logics complete to certain sets of well-known accessibility relations with a restriction that transitivi…
