Weak Logic Research Articles

Meeting strength in substructural logics

This paper contributes to the theory of hybrid substructural logics, i.e. weak logics given by a Gentzen-style proof theory in which there is only alimited possibility to use structural rules. Following the literture, we use an operator to mark formulas to which the extra structural rules may be applied. New in our approach is that we do not see this ∇ as a modality, but rather as themeet of the marked formula with a special typeQ. In this way we can make the specific structural behaviour of marked formulas more explicit. The main motivation for our approach is that we can provide a nice, intuitive semantics for hybrid substructural logics. Soundness and completeness for this semantics are proved; besides this we consider some proof-theoretical aspects like cut-elimination and embeddings of the ‘strong’ system in the hybrid one.

Weak logic theory

Weak logic is a partial logic for program verification and system specification. In this article a sequent calculus for weak logic is first defined. Then the completeness theorem, a cut elimination theorem and some interpolation theorems are proved. In the proofs of the theorems, strategies from first order logic are used with some additions to adapt them to weak logic. One of the interpolation theorems is used for a deeper study of one of the main criterias for the definition of weak logic. In order to find effective strategies for doing proofs within weak logic, a resolution principle for weak logic is given. The relationship between provability in weak logic and provability in first order logic is also studied to make it possible to use all known first order logic proof methods more directly.

Jerome, Alcuin and Vergil's ‘Old entellus’

In 798, Alcuin quarreled with Charlemagne's court scholars over the calculation of the lunar saltus. In Epistola 145, an important source for this argument, Alcuin called his rivals the ‘Egyptian boys’, because they wanted to follow Alexandrian custom, which began the Easter cycle on 1 September instead of the Roman 1 January. Alcuin expressed his anger with the ‘boys’ over their violation of Roman custom by quoting aptly from Jerome's letters. Jerome's warning to Augustine in Epistola 102.2, to remember Vergil's Dares and Entellus, suggested to Alcuin that his scholarly debate with the ‘boys’ should become a boxing match. Accordingly, Alcuin became ‘old Entellus’, whom Charlemagne's young scholars had hit unfairly with their criticism. In Epistola 145, Alcuin jabbed at the weak logic of the ‘Egyptians’ and satirized their ignorance. Besides revealing Alcuin's literary techniques, Epistola 145 is important for illuminating the tensions in Alcuin's relationship with Charlemagne.

Sets as singularities in the intensional universe

This paper is motivated by the search for a natural and deductively powerful extension of classical set theory. A theory of properties U is developed, based on a system of relevant logic related to RQ. In U the set {a, b, c,...} is identified with the property [x: x=a ∨ x=b ∨ x=c...]. The universe of all sets V, is identified with the property of being a hereditary set. The main result is that relevant implication → collapses to material implication ⊃ for sentences with quantifiers restricted to V. This demonstrates the naturalness of the system. However, an aparent lack of deductive power leads to the conclusion that the best extension of classical set theory is to be found in intensional theories with the unrestricted comprehension schema based on weak relevant logics. The author has obtained similar collapses of → to ⊃ for these systems.

Reasoning about knowledge and belief: a survey

We examine a number of logics of knowledge and belief from the perspective of knowledge‐based systems. We are concerned with the beliefs of a knowledge‐based system, including both the system's base set of beliefs–those garnered directly from the world–and beliefs that follow from the base set. Three things to consider with such logics are the expressive power of the language of the logic, the correctness and completeness of the inferences sanctioned, and the speed with which it is possible to determine whether a given sentence is believed. The influential possible worlds approach to representing belief has the property of logical omniscience, which makes for inferences that are unacceptable in the context of belief and may take too much time to make. We examine a number of weak logics which attempt to deal with these problems. These logics divide into three categories: those that admit incomplete or inconsistent situations into their semantics, those that posit a number of distinct states for a believer which correspond roughly to frames of mind, and those that incorporate axioms or other syntactic entities directly into the semantics. As to expressive power, we consider whether belief should be represented by a predicate or a sentential operator and examine the boundary between self‐referential and inconsistent systems. Finally, we consider logics of believing only, which add the assumption that a system's base set of beliefs are, in a certain sense, all that it believes.

On a theory of weak implications

The purpose of this paper is to study logical implications which are much weaker than the implication of intuitionistic logic.In §1 we define the system SI (system of Simple Implication) which is obtained from intuitionistic logic by restricting the inference rules of intuitionistic implication. The implication of the system SI is called the “simple implication” and denoted by ⊃, where the simple implication ⊃ has the following properties:(1) The simple implication ⊃ is much weaker than the usual intuitionistic implication.(2) The simple implication ⊃ can be interpreted by the notion of provability, i.e., we have a very simple semantics for SI so that a sentence A ⊃ B is interpreted as “there exists a proof of B from A”.(3) The full-strength intuitionistic implication ⇒ is definable in a weak second order extension of SI; in other words, it is definable by help of a variant of the weak comprehension schema and the simple implication ⊃. Therefore, though SI is much weaker than the intuitionistic logic, the second order extension of SI is equivalent to the second order extension of the intuitionistic logic.(4) The simple implication is definable in a weak modal logic MI by the use of the modal operator and the intuitionistic implication ⇒ with full strength. More precisely, A ⊃ B is defined as the strict implication of the form ◽(A ⇒ B).In §1, we show (3) and (4). (2) is shown in §2 in a more general setting.Semantics by introduction rules of logical connectives has been studied from various points of view by many authors (e.g. Gentzen [4], Lorentzen [5], Dummett [1], [2], Prawitz [8]. Martin-Löf [7], Maehara [6]). Among them Gentzen (in §§10 and 11 of [4]) introduced such a semantics in order to justify logical inferences and the mathematical induction rule. He observed that all of the inference rules of intuitionistic arithmetic, except for those on implication and negation, are justified by means of his semantics, but justification of the inference rules on implication and negation contains a circular argument for the interpretation by introduction rules, where the natural interpretation of A ⊃ B by ⊃-introduction rule is “there exists a proof of B from A ” (cf. §11 of Gentzen [4]).

A weak intuitionistic propositional logic with purely constructive implication

We introduce subsystems WLJ and SI of the intuitionistic propositional logic LJ, by weakening the intuitionistic implication. These systems are justifiable by purely constructive semantics. Then the intuitionistic implication with full strength is definable in the second order versions of these systems. We give a relationship between SI and a weak modal system WM. In Appendix the Kripke-type model theory for WM is given.

Weak Logics with Strict Implication

Weak Logics with Strict Implication

Polynomial rings and weak second-order logic

This article is a rewriting of my Ph.D. Thesis, supervised by Professor G. Sabbagh, and incorporates a suggestion from Professor B. Poizat. My main result can be crudely summarized (but see below for detailed statements) by the equality: first-order theory of F[Xi]i∈I = weak second-order theory of F.§I.1. Conventions. The letter F will always denote a commutative field, and I a nonempty set. A field or a ring (A; +, ·) will often be written A for short. We shall use symbols which are definable in all our models, and in the structure of natural numbers (N; +, ·):— the constant 0, defined by the formula Z(x): ∀y (x + y = y);— the constant 1, defined by the formula U(x): ∀y (x · y = y);— the operation ∹ x − y = z ↔ x = y + z;— the relation of division: x ∣ y ↔ ∃ z(x · z = y).A domain is a commutative ring with unity and without any zero divisor.By “… → …” we mean “… is definable in …, uniformly in any model M of L”.All our constructions will be uniform, unless otherwise mentioned.§I.2. Weak second-order models and languages. First of all, we have to define the models Pf(M), Sf(M), Sf′(M) and HF(M) associated to a model M = {A; ℐ) of a first-order language L [CK, pp. 18–20]. Let L1 be the extension of L obtained by adjunction of a second list of variables (denoted by capital letters), and of a membership symbol ∈. Pf(M) is the model (A, Pf(A); ℐ, ∈) of L1, (where Pf(A) is the set of finite subsets of A. Let L2 be the extension of L obtained by adjunction of a second list of variables, a membership symbol ∈, and a concatenation symbol ◠.

Algorithmic Logic with Stacks and Its Model-Theoretical Properties

In this paper we propose to transform the Algorithmic Theory of Stacks (cf. Salwicki [30]) into a logic for expressing and proving properties of programs with stacks. We compare this logic to the Weak Second Order Logic (cf. [11, 15]) and prove theorems concerning axiomatizability without quantifiers (an analogon of Łoś-Tarski theorem) and χ 0 - categoricity (an analogon of Ryll-Nardzewski’s theorem).

Conjunctive normal forms and weak modal logics without the axiom of necessity.

On definit les logiques modales faibles. Definitions et caracterisations des L-tautologie. Demonstrations de completude. Applications

Tree Rewriting Systems, Combinatory Weak Reduction Systems and Type Free Languages1

In this paper we consider combinators as tree transducers: this approach is based on the one-to-one correspondence between terms of Combinatory Logic and trees, and on the fact that combinators may be considered as transformers of terms. Since combinators are terms themselves, we will deal with trees as objects to be transformed and tree transformers as well. Methods for defining and studying tree rewriting systems inside Combinatory Weak Reduction Systems and Weak Combinatory Logic are also analyzed and particular attention is devoted to the problem of finiteness and infinity of the generated tree languages (here defined). This implies the study of the termination of the rewriting process (i.e. reduction) for combinators.

Posibilidades de generalización de las lógicas cuánticas

Resumen

Welding Semantics For Weak Strict Modal Logics into the General Framework of Modal Logic Semantics

Welding Semantics For Weak Strict Modal Logics into the General Framework of Modal Logic Semantics

Boolean algebras with a category theory in a weak second order logic

Boolean algebras with a category theory in a weak second order logic

Introduction of a Basic Theory of Objects

In constructing various kind of mathematical theories on the basis of a common basic theory, it has been very usual to take up the set theory as the common basic theory. This approach has been already successful to a certain extent and looks like successfully developable in the future not only in constructing mathematical theories standing on the classical logic but also in constructing formal theories standing on weaker logics. In constructing mathematical theories standing on the classical logic, it has been successful in most cases only by interpreting mathematical notions in the set theory without defining any special interpretation of logical notions. In constructing any mathematical theory standing on weaker logics such as the intuitionistic logic, however, we have to give a special interpretation for logical notions, too.