Provable Formulas Research Articles

Intuitionistic Propositional Calculus in the Extended Framework with Modal Operator. Part II

Summary This paper is a continuation of Inoué [5]. As already mentioned in the paper, a number of intuitionistic provable formulas are given with a Hilbert-style proof. For that, we make use of a family of intuitionistic deduction theorems, which are also presented in this paper by means of Mizar system [2], [1]. Our axiom system of intuitionistic propositional logic IPC is based on the propositional subsystem of H1-IQC in Troelstra and van Dalen [6, p. 68]. We also owe Heyting [4] and van Dalen [7]. Our treatment of a set-theoretic intuitionistic deduction theorem is due to Agata Darmochwał’s Mizar article “Calculus of Quantifiers. Deduction Theorem” [3].

EPSILON THEOREMS IN INTERMEDIATE LOGICS

AbstractAny intermediate propositional logic (i.e., a logic including intuitionistic logic and contained in classical logic) can be extended to a calculus with epsilon- and tau-operators and critical formulas. For classical logic, this results in Hilbert’s $\varepsilon $ -calculus. The first and second $\varepsilon $ -theorems for classical logic establish conservativity of the $\varepsilon $ -calculus over its classical base logic. It is well known that the second $\varepsilon $ -theorem fails for the intuitionistic $\varepsilon $ -calculus, as prenexation is impossible. The paper investigates the effect of adding critical $\varepsilon $ - and $\tau $ -formulas and using the translation of quantifiers into $\varepsilon $ - and $\tau $ -terms to intermediate logics. It is shown that conservativity over the propositional base logic also holds for such intermediate ${\varepsilon \tau }$ -calculi. The “extended” first $\varepsilon $ -theorem holds if the base logic is finite-valued Gödel–Dummett logic, and fails otherwise, but holds for certain provable formulas in infinite-valued Gödel logic. The second $\varepsilon $ -theorem also holds for finite-valued first-order Gödel logics. The methods used to prove the extended first $\varepsilon $ -theorem for infinite-valued Gödel logic suggest applications to theories of arithmetic.

A Deep Inference System for Differential Linear Logic

Differential linear logic (DiLL) provides a fine analysis of resource consumption in cut-elimination. We investigate the subsystem of DiLL without promotion in a deep inference formalism, where cuts are at an atomic level. In our system every provable formula admits a derivation in normal form, via a normalization procedure that commutes with the translation from sequent calculus to deep inference.

The New Normal: We Cannot Eliminate Cuts in Coinductive Calculi, But We Can Explore Them

AbstractIn sequent calculi, cut elimination is a property that guarantees that any provable formula can be proven analytically. For example, Gentzen’s classical and intuitionistic calculi LK and LJ enjoy cut elimination. The property is less studied in coinductive extensions of sequent calculi. In this paper, we use coinductive Horn clause theories to show that cut is not eliminable in a coinductive extension of LJ, a system we call CLJ. We derive two further practical results from this study. We show that CoLP by Gupta et al. gives rise to cut-free proofs in CLJ with fixpoint terms, and we formulate and implement a novel method of coinductive theory exploration that provides several heuristics for discovery of cut formulae in CLJ.

Deriving Theorems in Implicational Linear Logic, Declaratively

The problem we want to solve is how to generate all theorems of a given size in the implicational fragment of propositional intuitionistic linear logic. We start by filtering for linearity the proof terms associated by our Prolog-based theorem prover for Implicational Intuitionistic Logic. This works, but using for each formula a PSPACE-complete algorithm limits it to very small formulas. We take a few walks back and forth over the bridge between proof terms and theorems, provided by the Curry-Howard isomorphism, and derive step-by-step an efficient algorithm requiring a low polynomial effort per generated theorem. The resulting Prolog program runs in O(N) space for terms of size N and generates in a few hours 7,566,084,686 theorems in the implicational fragment of Linear Intuitionistic Logic together with their proof terms in normal form. As applications, we generate datasets for correctness and scalability testing of linear logic theorem provers and training data for neural networks working on theorem proving challenges. The results in the paper, organized as a literate Prolog program, are fully replicable. Keywords: combinatorial generation of provable formulas of a given size, intuitionistic and linear logic theorem provers, theorems of the implicational fragment of propositional linear intuitionistic logic, Curry-Howard isomorphism, efficient generation of linear lambda terms in normal form, Prolog programs for lambda term generation and theorem proving.

Intuitionistic ancestral logic

Abstract In this article we define pure intuitionistic Ancestral Logic ( iAL ), extending pure intuitionistic First-Order Logic ( iFOL ). This logic is a dependently typed abstract programming language with computational functionality beyond iFOL given by its realizer for the transitive closure, TC . We derive this operator from the natural type theoretic definition of TC using intersection. We show that provable formulas in iAL are uniformly realizable, thus iAL is sound with respect to constructive type theory. We further show that iAL subsumes Kleene Algebras with tests and thus serves as a natural programming logic for proving properties of program schemes. We also extract schemes from proofs that iAL specifications are solvable.

Monadic pseudo BCI-algebras and corresponding logics

We introduce the notion of monadic pseudo BCI-algebras and study some related properties. Then, we introduce monadic filters and monadic congruences of monadic pseudo BCI-algebras and discuss the relations between them. We proved that there is a one-to-one correspondence between the set of closed m-congruence relations and the set of normal closed m-filters in a monadic pseudo BCI-algebra. Moreover, we introduce a notion of strong residuated mappings and study the relation between monadic operators and strong residuated mappings in pseudo BCI-algebras. Let A be a pseudo BCI-algebra and \(f:A\rightarrow A\) be a mapping, we obtain that \((f, f^+)\) is a monadic operator on A if and only if f is a strong residuated mapping on A where \(f^+\) is the residual of f. Also we exhibit an axiom system of monadic pseudo BCI-logic, which enrich the language of pseudo BCI-logics. Based on the monadic pseudo BCI-algebras, we prove the completeness and soundness of the monadic pseudo BCI-logic propositional system. Finally, using provable formula set, normal subset and monadic subset in the set of all formulas of a monadic pseudo BCI-logic \(\mathcal {L}\), we characterize filters, normal filters and monadic normal filters in a monadic pseudo BCI-algebra, respectively.

When the lexicographic product of two po-groups has the Riesz decomposition property

We study conditions when a certain type of the Riesz Decomposition Property (RDP for short) holds in the lexicographic product of two po-groups. Defining two important properties of po-groups, we extend known situations showing that the lexicographic product satisfies RDP or even $${{\rm RDP}_1}$$ , a stronger type of RDP. We recall that a very strong type of RDP, $${{\rm RDP}_2}$$ , entails that the group is lattice ordered. RDP's of the lexicographic products are important for the study of lexicographic pseudo effect algebras, or perfect types of pseudo MV-algebras and pseudo effect algebras, where infinitesimal elements play an important role both for algebras as well as for the first order logic of valid but not provable formulas.

Riesz decomposition properties and the lexicographic product of po-groups

We establish conditions when a certain type of the Riesz decomposition property (RDP) holds in the lexicographic product of two po-groups. It is well known that the resulting product is an $$\ell $$l-group if and only if the first one is linearly ordered and the second one is an $$\ell $$l-group. This can be equivalently studied as po-groups with a special type of the RDP. In the paper we study three different types of RDPs. RDPs of the lexicographic products are important for the study of pseudo effect algebras where infinitesimal elements play an important role both for algebras as well as for the first-order logic of valid but not provable formulas.

Generalized intuitionistic fuzzy filters on residuated lattices

Filters play an important role in studying fuzzy logics. From logical point of view, filters correspond to sets of provable formulae. The aim of this paper is to solve two problems appeared in α, β...

On lifting quasi-filters and strong lifting quasi-filters in MV-algebras

A lifting inference rule is a many-valued inference rule that manipulates with evaluated formulas, it has played an important role in Pavelka-style fuzzy propositional logic. In this paper, we consider the algebraic characterization of Pavelka-style fuzzy logic deductive system based on lifting rul es in MV-algebras from the algebraic semantical point of view, which can be a tool to analyze the provability in fuzzy logic with evaluated syntax based on the lifting inference rules. We introduce the notions of lifting quasi-filter and strong lifting quasi-filter in MV-algebras, they are the algebraic abstractions of the sets of provable formulae which are closed with respect to lifting inference rules in the corresponding formal logic system, and some important properties are presented. We also characterize two kinds of extended lifting quasi-filters and several kinds of generated lifting quasi-filters, and prove that the class F α (L) of all lifting quasi-filters with respect to α is an MV-algebra, the class Fs α (L) of all strong lifting quasi-filters with respect to α is a complete Heyting algebra.

Generalized intuitionistic fuzzy filters on residuated lattices

Filters play an important role in studying fuzzy logics. From logical point of view, filters correspond to sets of provable formulae. The aim of this paper is to solve two problems appeared in α, β]-intuitionistic fuzzy filters of residuated lattices. Some characteristics of α, β]-intuitionistic fuzzy filters α, β]-intuitionistic fuzzy P-filters are obtained. The relations among α, β]-intuitionistic fuzzy filters, cut sets and s-upper cut sets are established. The “Extension property” and “Triple of equivalent characteristics” of α, β]-intuitionistic fuzzy P-filters are also presented.

ゲンツェンを読む : 三つの無矛盾性証明の統一的解釈

Gentzen proved the consistency of elementary arithmetic (i.e. first-order Peano arithmetic) in 1936 before his most famous and influential proof in 1938. The consistency proof in 1936 contains some ambiguous parts and seems to be quite different from his consistency proof in 1938. The aim of the consistency proof in 1936 is "to give finitist sense" to provable formula. In this paper, we give an exact reconstruction of the consistency proof in 1936 and claim that "to give finitist sense" is a uniform idea behind Gentzen's three consistency proofs including the proof in 1938. First we explain Gentzen's basic ideas of the proof in 1936 in detail. In particular, the idea of finitist interpretation and the main structure of the proof are explained. Secondly, we define a reduction step via the modern method of proof theory called "finite notation for infinitary derivations" due to Mints-Buchholz. It is shown that the reduction essentially coincides with Gentzen's reduction in 1936. Especially we give a definition of "normalization tree" describing Gentzen's reduction step. Moreover, the well-foundedness of this tree is proved. The well-foundedness of the normalization tree implies the consistency of elementary arithmetic. Together with Buchholz's analysis of Gentzen's 1938 consistency proof, this shows that the proof in 1938 is just a special case of the proof in 1936. Thirdly, we clarify what the normalization tree is. According to Gentzen, the normalization tree makes us possible to see the "correctness" of a provable formula in elementary arithmetic. Then we propose a uniform reading of three consistency proofs as based on the same spirit. Finally we discuss some relationship between Gentzen's idea, the method of "finite notation for infinitary derivations", and Gödel's idea of his famous Dialectica interpretation. According to our analysis, Gentzen's idea and the method of "finite notation for infinitary derivations" can be explained in the same way as "carrying out finite proof as program". Moreover, we suggest that Gödel's interpretation (no-counterexample interpretation) should be obtained by describing the normalization tree as functionals.

Classes of fuzzy filters of residuated lattice ordered monoids

The logical foundations of processes handling uncertainty in information use some classes of algebras as algebraic semantics. Bounded residuated lattice ordered monoids (\rl monoids) are common generalizations of $\text{\rm BL}$-algebras, i.e., algebras of the propositional basic fuzzy logic, and Heyting algebras, i.e., algebras of the propositional intuitionistic logic. From the point of view of uncertain information, sets of provable formulas in inference systems could be described by fuzzy filters of the corresponding algebras. In the paper we investigate implicative, positive implicative, Boolean and fantastic fuzzy filters of bounded $\text{\rm Rl}$-monoids.

Fuzzy filters and fuzzy prime filters of bounded [formula omitted]-monoids and pseudo BL-algebras

The logical foundations of processes handling uncertainty in information use some classes of algebras as algebraic semantics. The sets of provable formulas in corresponding inference systems from the point of view of uncertain information can be described by fuzzy filters of those algebraic semantics. Bounded residuated lattice ordered monoids ( R ℓ -monoids) are a common generalization of pseudo BL-algebras (and consequently of pseudo MV-algebras) and Heyting algebras, i.e. algebras behind fuzzy and intuitionistic reasoning. In the paper, we introduce and investigate fuzzy filters of bounded R ℓ -monoids and fuzzy prime filters of pseudo BL-algebras.

A Weak Constructive Second-Order Arithmetic with Extraction of Algorithms Computable in Polynomial Time

In this paper, a family of weak constructive theories, containing arithmetic and a theory of natural-valued functions of natural arguments, is suggested. The functions are polynomially bounded and computable in time bounded by polynomials in their arguments. The languages of these theories contain functional constants for addition and multiplication and the equality predicate. Other functional constants are also allowed, provided that the corresponding functions satisfy the above conditions of polynomial boundedness. From the proofs of the theories considered witness functions for provable formulas, computable in polynomial time, can algorithmically be extracted. If an argument of a witness is a function, then the latter is used in the witness algorithm as an oracle. Bibliography: 2 titles.

A strong induction scheme that leads to polynomially computable realizations

Formal intuitionistic theories are built with a relatively strong induction scheme, for which any provable formula of the form ∀x∃yB(x,y) is realizable by an algorithm f that is computable in polynomial time: ∀xB(x,f(x)).

On semilattice relevant logics

AbstractThe semilattice relevant logics ∪R, ∪T, ∪RW, and ∪TW (slightly different from the orthodox relevant logics R, T, RW, and TW) are defined by semilattice models in which conjunction and disjunction are interpreted in a natural way. For each of them, there is a cut‐free labelled sequent calculus with plural succedents (like LK). We prove that these systems are equivalent, with respect to provable formulas, to the restricted systems with single succedents (like LJ). Moreover, using this equivalence, we give a new Hilbert‐style axiomatizations for ∪R and ∪T and prove equivalence between two semantics (commutative monoid and distributive semilattice) for the contractionless logics ∪RW and ∪TW.

A note on the notion of truth in fuzzy logic

In fuzzy predicate logic, assignment of truth values may be partial, i.e. the truth value of a formula in an interpretation may be undefined (due to lack of some infinite suprema or infima in the underlying structure of truth values). A logic is supersound if each provable formula ϕ is true (has truth value 1) in each interpretation in which the truth value of ϕ is defined. It is shown that among the logics given by continuous t-norms, Gödel logic is the only one that is supersound; all others are (sound but) not supersound. This supports the view that the usual restriction of semantics to safe interpretations (in which the truth assignments is total) is very natural.

An elementary definability theorem for first order logic

In this paper, we will present a definability theorem for first order logic. This theorem is very easy to state, and its proof only uses elementary tools. To explain the theorem, let us first observe that if M is a model of a theory T in a language , then, clearly, any definable subset S ⊂ M (i.e., a subset S = {a ∣ M ⊨ φ(a)} defined by some formula φ) is invariant under all automorphisms of M. The same is of course true for subsets of Mn defined by formulas with n free variables.Our theorem states that, if one allows Boolean valued models, the converse holds. More precisely, for any theory T we will construct a Boolean valued model M, in which precisely the T -provable formulas hold, and in which every (Boolean valued) subset which is invariant under all automorphisms of M is definable by a formula .Our presentation is entirely selfcontained, and only requires familiarity with the most elementary properties of model theory. In particular, we have added a first section in which we review the basic definitions concerning Boolean valued models.