Classical Logic

This paper investigates the proof-theoretic foundations of double negation introduction (DNI: A ⊢ ¬¬A) and double negation elimination (DNE: ¬¬A ⊢ A) in classical logic. 1 By examining both sequent calculus and natural deduction, it is shown that these rules originate in reductio ad absurdum (RAA): DNI results from deriving ¬¬A via ¬I by discharging [¬A], while DNE arises from deriving A through a reductio on [¬A]. 2 Their significance extends beyond semantic equivalence, for DNI and DNE embody the identity relation A ⟷ ¬¬A as a structural principle of classical logic. The paper demonstrates that both rules possess harmony, ensuring balance between introduction and elimination, and normalisation, which guarantees that derivations reduce to canonical form without detours. These features reveal double negation not as a redundancy, but as a mechanism of proof-theoretic stability, securing the disciplined integration of RAA into classical logic.

This paper is an overview of a variety of results, all centered around a common theme, namely embedding of non-classical logics into first order logic and resolution theorem proving. We present several classes of non-classical logics, many of which are of great practical relevance in knowledge representation, which can be translated into tractable and relatively simple fragments of classical logic. In this context, we show that refinements of resolution can often be used successfully for automated theorem proving, and in many interesting cases yield optimal decision procedures.

The $\mathbf{LMT^{\rightarrow}}$ sequent calculus was introduced in Santos (2016). This paper presents a Termination proof and a new (more direct) Completeness proof for it. $\mathbf{LMT^{\rightarrow}}$ is aimed to be used for proof search in Propositional Minimal Implicational Logic ($\mathbf{M^{\rightarrow}}$), in a bottom-up approach. Termination of the calculus is guaranteed by a rule application strategy that stresses all the possible combinations. For an initial formula $\alpha$, proofs in $\mathbf{LMT^{\rightarrow}}$ have an upper bound of $|\alpha| \times 2^{|\alpha| + 1 + 2 \times log_2|\alpha|}$, which together with the system strategy ensure decidability. $\mathbf{LMT^{\rightarrow}}$ has the property to allow extractability of counter-models from failed proof searches (bicompleteness), i.e., the attempt proof tree of an expanded branch produces a Kripke model that falsifies the initial formula.

We review the close relationship between abstract machines for (call-by-name or call-by-value) λ-calculi (extended with Felleisen's C) and sequent calculus, reintroducing on the way Curien-Herbelin's syntactic kit expressing the duality of computation. We use this kit to provide a term language for a presentation of LK (with conjunction, disjunction, and negation), and to transcribe cut elimination as (non confluent) rewriting. A key slogan here, which may appear here in print for the first time, is that commutative cut elimination rules are explicit substitution propagation rules. We then describe the focalised proof search discipline (in the classical setting), and narrow down the language and the rewriting rules to a confluent calculus (a variant of the second author's focalising system L). We then define a game of patterns and counterpatterns, leading us to a fully focalised finitary syntax for a synthetic presentation of classical logic, that provides a quotient on (focalised) proofs, abstracting out the order of decomposition of negative connectives. 1 A state of the machine is thus a pair M | E where M is "where the computation is currently active", and E is the stack of things that are waiting to be done in the future, or the continuation, or the evaluation context. In λ-calculus litterature, contexts are more traditionally presented as terms with a hole: with this tradition, M How can we type the components of this machine? We have three categories of terms and of typing judgements: Expressions Contexts Commands

X is an untyped language for describing circuits by composition of basic components. This language is well suited to describe structures which we call "circuits" and which are made of parts that are connected by wires. Moreover X gives an expressive platform on which algebraic objects and many different (applicative) programming paradigms can be mapped. In this paper we will present the syntax and reduction rules for X and some its potential uses. To demonstrate the expressive power of X , we will show how, even in an untyped setting, elaborate calculi can be embedded, like the naturals, the λ-calculus, Bloe and Rose's calculus of explicit substitutions λx, Parigot's λµ and Curien and Herbelin's λµμ.

The liar and kindred paradoxes show that we can derive contradictions when we reason in accordance with classical logic from the schema (T) about truth, S is true iff p, where 'p' is to be replaced with a sentence and 'S' with a name of that sentence. The paper presents two arguments to the effect that the blame lies not in (T) but in classical logic. The arguments derive contradictions using classical logic, but instead of appealing to (T), they invoke semantic claims that seem even harder to reject. The first argument relies on two standard semantic principles that are not disquotational and on the claim that if there is such a thing as the property of being true, then 'true' expresses that property. The second argument relies on a schema about meaning, S means that p, where 'S' and 'p' are to be replaced as before.

The expected utility hypothesis is one of the building blocks of classical economic theory and founded on Savage's Sure-Thing Principle. It has been put forward, e.g. by situations such as the Allais and Ellsberg paradoxes, that real-life situations can violate Savage's Sure-Thing Principle and hence also expected utility. We analyze how this violation is connected to the presence of the 'disjunction effect' of decision theory and use our earlier study of this effect in concept theory to put forward an explanation of the violation of Savage's Sure-Thing Principle, namely the presence of 'quantum conceptual thought' next to 'classical logical thought' within a double layer structure of human thought during the decision process. Quantum conceptual thought can be modeled mathematically by the quantum mechanical formalism, which we illustrate by modeling the Hawaii problem situation, a well-known example of the disjunction effect, and we show how the dynamics in the Hawaii problem situation is generated by the whole conceptual landscape surrounding the decision situation.

Türkçe Önermelerde Doğruluk Koşulları adlı bu çalışmada Türkçe önermelerde doğruluk koşullarını şekillendiren unsurlar, bu koşulları oluşturan ön ve temel etmenler incelenmiştir. Öncelikle bir önermenin doğru veya yanlış olarak değerlendirilebilmesi için dilsel ve mantıksal açıdan anlamlı olması, doğrulanabilir-yanlışlanabilir özellikler taşıması ve ana dil konuşurlarının paylaştığı bir bağlamda üretilmesi gerektiği vurgulanmıştır. Bu aşama, önerme değerini kazanan sözcelerin doğruluk sınamasına tabi tutulabilmesi bakımından kritik bir zemin hazırlamaktadır.
Ön koşullara eşlik eden temel koşullar uygunluk, nesnellik, kanıtlanabilirlik ve şüpheden uzaklık gibi unsurları kapsamaktadır. Özellikle gerçeklikle uyum gösteren, bilimsel ya da deneysel verilerle desteklenebilen önermeler daha yüksek bir doğruluk değeri kazanmaktadır. Aynı şekilde, herkesçe kabul görmüş ve tarafların ortak deneyim alanında paylaşılan bilgilere dayalı sözceler, şüphe barındırmadıkları ölçüde doğru olarak değerlendirilmektedir. Bunun yanı sıra, salt gerçeklik düzeyine yakın sayılabilecek içeriklerin kanıtlarla tutarlı, evrensel ve öznellikten arınmış olması beklenmektedir. Konuşur bakış açısı ile bağlam değişkenliği, özellikle tarihsel ve toplumsal konularda, doğruluk değerlendirmesine ek yorum katmanları kazandırmaktadır. Ancak duygusal tutum, üslup veya kişisel inanç gibi kipsel etmenler, önermelerin doğruluk koşullarını doğrudan etkilememekte, yalnızca değerlendirme sürecine görece öznellik eklemektedir. Böylelikle, doğruluk sınaması hem dilsel hem de toplumsal açılardan çok boyutlu bir süreç hâline gelmektedir.

It has been shown that the rules of logic for the principle 'Ex Contradictione quod libet' (ECQ) do not cause U8 to explode. This is because the antecedent is non-designated and modus ponens blocks detachment for the conclusion. Therefore, ECQ is not a theorem of U8, however, ECQ is acknowledged to be a hypothetical theorem. This leaves the axioms disjunctive introduction and the disjunctive syllogism intact for U8, unlike other paraconsistent logics. Using traffic light signals, an example showing paraconsistent behaviour is given for the U8 AND operation. It can be concluded U8 is a paraconsistent null logic system.

Bir düşünme şekli olarak eleştirel düşünme akıl yürütme yapısı olarak da adlandırılan argümanlar aracılığıyla ifade edilmiş önermeler sisteminin desteklediği bir düşünme şeklidir. Düşünceler denetlemeye tabi tutulduklarında, tercih veya beğeni gibi öznel bir değerlendirme unsuru barındırmazlar. Düşüncelerin denetlenmesi mantık aracılığıyla yapılır. Mantık, önermeler arasındaki ilişkilerin nasıl olması gerektiğine dair kuralları verir. Bu kurallar aracılığıyla önermeler arası ilişkilerdeki karşıtlık, çelişki ve kıyas yapısı içinde geçerli argümanların nasıl olması gerektiği belirlenebilir. Bir önermenin çelişkisi ile çelişiği arasındaki fark doğruluk değeri ile birlikte yapısal karşılaştırmayı gerektirir. Akıl yürütme hataları her ne kadar kategorik kıyas yapısı içerisinde ele alınıyor olsa da genelleştirilerek düşünmenin önünde duran ve geçersiz olmaklarına neden olabilecek hatalardır. Düşünce içeriklerini bu hatalardan ayıklamak için kullanılabilecek araçlardan biri de mantıkla örülü bir eleştirel düşünme sürecidir. Eleştirel düşünmenin gerekliliği, düşünce süreçlerinde çelişkileri ayıklama yeteneği sağlarken, akıl yürütme hatalarının tespit edilmesi de geçerli argümanların oluşturulmasına katkı sağlar. Ayrıca, eleştirel düşünmenin derinlemesine anlaşılabilmesi için, mantık kurallarının öğrenilmesinin yanı sıra, kişinin düşünme süreçleri üzerinde değerlendirme yapabilme yeteneğine sahip olması gerekir. Düşünmenin kendi üzerine katlanması olarak ele alınacak yansımanın kat izleri mantık aracılığıyla belirlenir.

The U logic system is a set of four logics termed: U8, U4, U2 and U0. U8 is a unique many-valued non-classical logic system in eight variables that more closely models human reasoning. The advantage of U8 is that it can model irrelevancy, N, uncertainty, U, contradiction, 0,  Non-truth, /T, Non-falsity, /F, Non-neutrality, /N, as well as true, T, and false, F, variables.

It has been shown that U8 is sound because the modus ponens inference rule evaluates to designated values and all axioms are tautologies. Furthermore, a Canonical Model construction and proof by contradiction have been employed to show that U8 is complete. As U8 is sound and complete it qualifies as a robust formal logic.

The word "suppose" occupies a unique position in mathematical reasoning. Far from being casual language, it functions as a precise logical tool for introducing assumptions, developing hypothetical reasoning, and guiding proofs. This article explores the epistemological and logical reality of "suppose" in mathematics. Drawing from formal logic, proof theory, and classical examples, the study analyzes how "suppose" differs from related terms such as "let" and "assume". The article further examines its philosophical implications in Platonism, Constructivism, and Proof by Contradiction. Ultimately, "suppose" is revealed as an indispensable linguistic operator that bridges natural language and rigorous mathematical deduction.

In the last several years the computational complexity of classical planning and HTN planning have been studied. But in both cases it is assumed that the planner has complete knowledge about the initial state. Recently, there has been proposal to use 'sensing' actions to plan in presence of incompleteness. In this paper we study the complexity of planning in such cases. In our study we use the action description language proposed in 1993 by M. Gelfond and V. Lifschitz and its extensions. The language allows planning in the situations with complete information. It is known that, if we consider only plans of feasible (polynomial) length, the planning problem for such situations is NP-complete: even checking whether a given objective is attainable from a given initial state is NP-complete. In this paper, we show that the planning problem in presence of incompleteness is indeed harder: it belongs to the next level of complexity hierarchy (in precise terms, it is complete). To overcome the complexity of this problem, C. Baral and T. Son have proposed several approximations. We show that under certain conditions, one of these approximations -O-approximation -makes the problem NP-complete (thus indeed reducing its complexity).

We introduce more basic axioms with which we are able to prove some "axioms" of Propositional Logic. We use the symbols from my other article: "Introduction to Logical Structures". Logical Structures (SrL) are graphs with doubly labelled vertices with edges carrying symbols. The proofs are very mechanical and does not require ingenuity to construct. It is easy to see that in order to transform information, it has to be chopped up. Just look at a kid playing with blocks with letters on them: he has to break up the word into letters to assemble another word. Within SrL we take as our "atoms" propositions with chopped up relations attached to them. We call the results: (incomplete) "structures". We play it safe by only allowing relations among propositions to be choppable. We will see whether this is the correct way of chopping up sentences (it seems to be). This is where our Attractors (Repulsors) and Stoppers come in. Attractors that face away from each other repels and so break a relation between the two propositions. Then a Stopper attaches to the chopped up relation to indicate it can't reconnect. So it is possible to infer sentences from sentences. The rules I stumbled upon, to implement this, seems to be consistent. Modus Ponens (MP) is found in most logics. I can prove MP without resorting to Metalogical 1 notions like the definition of truth, Models, Frames and Possible Worlds. The definition of truth seems to presuppose MP: i.e. it presupposes MP just on the "is valid" specifier, rather than the "is proveable" specifier. Therefore the usual proof of MP is circular.

The theory of abstract algebraic logic aims at drawing a strong bridge between logic and universal algebra, namely by generalizing the well known connection between classical propositional logic and Boolean algebras. Despite of its successfulness, the current scope of application of the theory is rather limited. Namely, logics with a many-sorted language simply fall out from its scope. Herein, we propose a way to extend the existing theory in order to deal also with many-sorted logics, by capitalizing on the theory of many-sorted equational logic. Besides showing that a number of relevant concepts and results extend to this generalized setting, we also analyze in detail the examples of first-order logic and the paraconsistent logic C1 of da Costa.

Questions concerning the proof-theoretic strength of classical versus nonclassical theories of truth have received some attention recently. A particularly convenient case study concerns classical and nonclassical axiomatizations of fixed-point semantics. It is known that nonclassical axiomatizations in four-or three-valued logics are substantially weaker than their classical counterparts. In this paper we consider the addition of a suitable conditional to First-Degree Entailment -a logic recently studied by Hannes Leitgeb under the label HYPE. We show in particular that, by formulating the theory PKF over HYPE, one obtains a theory that is sound with respect to fixed-point models, while being proof-theoretically on a par with its classical counterpart KF. Moreover, we establish that also its schematic extension -in the sense of Feferman -is as strong as the schematic extension of KF, thus matching the strength of predicative analysis.

In comparing classical and non-classical solutions to the semantic paradoxes arguments relying on strength have been influential. In this paper I argue that non-classical solutions should preserve the proof-theoretic strength of classical solutions. Leitgeb's logic of HYPE is then presented as an interesting possibility to strengthen FDE with a suitable conditional. It is shown that HYPE allows for a non-classical Kripkean theory of truth, called KFL, that is strong enough for the relevant purposes and has additional attractive properties.

Interactive theorem proving systems for mathematics require user interfaces which allow for user interaction that is as natural as possible. However, this interaction is often limited by the traditional calculi underlying most theorem proving systems. This is particularly problematic with respect to the application of assertions and intuitive presentation of proof states. In this paper we show how a more flexible user interaction can be realized when traditional calculi for classical logic are replaced by a less restrictive reasoning engine, the recently developed CORE [2] system. We describe the task level which is built on top of the CORE system and combines the Proof by Pointing approach [5] with a flexible mechanism for the application of assertions that avoids decomposition and abstracts from the syntactical form of an assertion. We demonstrate how proof steps that are difficult to implement in other systems, like forward application of assertions, are quite naturally supported by the underlying CORE system and are therefore straightforward to realize at the task level.

Non-classical logics are used in a wide spectrum of disciplines, including artificial intelligence, computer science, mathematics, and philosophy. The de-facto standard infrastructure for automated theorem proving, the TPTP World, currently supports only classical logics. Similar standards for non-classical logic reasoning do not exist (yet). This hampers practical development of reasoning systems, and limits their interoperability and application. This paper describes the latest extension of the TPTP World, which provides languages and infrastructure for reasoning in non-classical logics. The extensions integrate seamlessly with the existing TPTP World.

We propose a method which allows us to develop tableaux modulo theories using the principles of superdeduction, among which the theory is used to enrich the deduction system with new deduction rules. This method is presented in the framework of the Zenon automated theorem prover, and is applied to the set theory of the B method. This allows us to provide another prover to Atelier B, which can be used to verify B proof rules in particular. We also propose some benchmarks, in which this prover is able to automatically verify a part of the rules coming from the database maintained by Siemens IC-MOL.

We propose a formal and mechanized framework which consists in verifying proof rules of the B method, which cannot be automatically proved by the elementary prover of Atelier B and using an external automated theorem prover called Zenon. This framework contains in particular a set of tools, named BCARe and developed by Siemens IC-MOL, which relies on a deep embedding of the B theory within the logic of the Coq proof assistant. This toolkit allows us to automatically generate the required properties to be checked for a given proof rule. Currently, this tool chain is able to automatically verify a part of the derived rules of the B-Book, as well as some added rules coming from Atelier B and the rule database maintained by Siemens IC-MOL.

Classical logical inference engines assume the consistency of the ontologies they reason with. Conclusions drawn from an inconsistent ontology by classical inference may be completely meaningless. An inconsistency reasoner is one which is able to return meaningful answers to queries, given an inconsistent ontology. In this chapter, we propose a general framework for reasoning with inconsistent ontologies. We present the formal definitions of soundness, meaningfulness, local completeness, and maximality of an inconsistency reasoner. We propose and investigate a pre-processing algorithm, discuss the strategies of inconsistency reasoning based on pre-defined selection functions dealing with concept relevance. We have implemented a system called PION (Processing Inconsistent ONtologies) for reasoning with inconsistent ontologies. We discuss how the syntactic relevance can be used for PION. In this chapter, we also report the preliminary experiments with PION.

EU-IST Integrated Project (IP) IST-2003-506826 SEKT Deliverable D3.4.1.1 (WP3.4) This document is an informal deliverable provided to SEKT WP3 partners. In this document, a general framework for reasoning with inconsistent ontologies is proposed. An inconsistency reasoner is one which is able to return meaningful answers to queries, given an inconsistent ontology. The formal definitions of soundness, meaningfulness, local completeness, and maximal completeness of an inconsistency reasoner are introduced. A pre-processing algorithm and a strategy for inconsistency reasoning based on linear extensions and selection functions are investigated. This document also contains a chapter that discusses the design of reasoners with inconsistent ontologies (RIO). A RIO architecture, which is based on the DIG description logic interface, is proposed.

Slater has argued that there cannot be any truly paraconsistent logics, because it's always more plausible to suppose whatever "negation" symbol is used in the language is not a real negation, than to accept the paraconsistent reading. In this paper I neither endorse nor dispute Slater's argument concerning negation; instead, my aim is to show that as an argument against paraconsistency, it misses (some of) the target. A important class of paraconsistent logics -the preservationist logics -are not subject to this objection. In addition I show that if we identify logics by means of consequence relations, at least one dialetheic logic can be reinterpreted in preservationist (non-dialetheic) terms. Thus the interest of paraconsistent consequence relations -even those that emerge from dialetheic approaches -does not depend on the tenability of dialetheism. Of course, if dialetheism is defensible, then paraconsistent logic will be required to cope with it. But the existence (and interest) of paraconsistent logics does not depend on a defense of dialetheism.

John Locke y Benito Jerónimo Feijoo enseñaron lógica en alguna etapa de sus vidas. Esto ha influido para que la lógica esté presente en sus obras. Algunas diferencias en temas de lógica están condicionadas por el nominalismo de Locke y el realismo de Feijoo. El monje español tiene mayor conciencia del valor intrínseco de la lógica formal que el médico inglés. Pero ambos coinciden en recomendar la brevedad en la enseñanza de la misma y en resaltar la importancia de la lógica natural.

John Locke and Benito Jerónimo Feijoo taught logic during a period of their lives. This fact contributed to logic being present in their respective works. Locke’s nominalism and Feijoo’s realism explain some of their differences on the subject. The Spanish monk shows more awareness than the English physician of the instrinsic value of formal logic. Both thinkers recommend teaching logic only briefly, and they also emphasize the importance of natural logic.

George Boole (1815-1864) is rightly known as a logician, the author of an algebra of logic, even if he did not conceive it quite in the same way as we know it. The calculus on classes and the calculus of propositions, that he set out to be equivalent, were at the core of his algebra of logic. After more than twenty centuries where logic was associated to the analysis of language, it was handled for the first time with mathematical symbols. However, the known Boole's algebra is not the one Boole produced. In this paper, I would like to move apart from the classical recurring approach to history -often referred to as Whiggist -, and to contextualize Boole's work so as to brighten the main trends which guided this renewal of how to deal with logic. Boole was also a mathematician, and since his Mathematical Analysis of Logic (1847), his work was very close to the symbolical way of thinking algebra, as it was developed in Cambridge by "The Analytics" in order to found usual algebraic practices. An Investigation on the Laws of Thought (1854) enunciated a more systematic view of this project for logic, devised as a whole system, facing what was at stake for the theory of knowledge with the strong rise of experimental sciences. Clearly, Boole wanted to stand out from empiricism, and from metaphysics as well, and to maintain an absolute meaning to formal sciences. And I will scrutinize his philosophical references so as to precise how his enterprise answer these issues, relating to necessary reasoning, and to probable reasoning as well.

This paper investigates the applications of Labelled Deductive Systems (LDS) as a framework for the development of a computational formalism for the description of active historical databases. This formalism is then applied to the study of retroactive updates. The propositional temporal logic USV/'Z for temporal validity is presented and we prove its soundness and completeness. The historical data model is derived from an LDS framework for this logic. The mechanism for the execution of active temporal rules is described, showing how retroactive updates can cause the appearance of nonsupported actions. An extension of LISWZ into a non-monotonic LDS is applied for the detection of such actions.

The aim of this work is to show how to compute an extra hypothesis H to an unproved sequent Γ ? G, such that: • Γ, H G is provable; • H is not trivial. • If Γ G (which is not known a priory) then Γ, H G has a much simpler proof. Due to the last item, this is not exactly the usual abductive reasoning found in literature, for the latter requires the input sequent not to be derivable. The idea is that Γ is a contextual database, containing background knowledge, G is a goal formula representing some fact or evidence that one wants to explain or prove, and H is an hypothesis that explains the evidence in the presence of the background knowledge or that facilitates the proof of G from Γ. We show how this task is related to the problem of computing non-analytic cuts. Several algorithms are provided that compute efficient proofs with non-analytic cuts via abductive reasoning. This is a joint work with Marcello D'Agostino and Dov Gabbay.

It is well understood how to compute the average or centroid of a set of numeric values, as well as their variance. In this way we handle inconsistent measurements of the same property. We wish to solve the analogous problem on qualitative data: How to compute the "average" or consensus of a set of affirmations on a non-numeric fact, as reported for instance by different Web sites? What is the most likely truth among a set of inconsistent assertions about the same attribute? Given a set (a bag, in fact) of statements about a qualitative feature, this paper provides a method, based in the theory of confusion, to assess the most plausible value or "consensus" value. It is the most likely value to be true, given the information available. We also compute the inconsistency of the bag, which measures how far apart the testimonies in the bag are. All observers are equally credible, so differences arise from perception errors, due to the limited accuracy of the individual findings (the limited information extracted by the examination method from the observed reality). Our approach differs from classical logic, which considers a set of assertions to be either consistent (True, or 1) or inconsistent (False, or 0), and it does not use Fuzzy Logic.

It is well understood how to compute the average or centroid of a set of numeric values, as well as their variance. In this way we handle inconsistent measurements of the same property. We wish to solve the analogous problem on qualitative data: How to compute the "average" or consensus of a set of affirmations on a non-numeric fact, as reported for instance by different Web sites? What is the most likely truth among a set of inconsistent assertions about the same attribute? Given a set (a bag, in fact) of statements about a qualitative feature, this paper provides a method, based in the theory of confusion, to assess the most plausible value or "consensus" value. It is the most likely value to be true, given the information available. We also compute the inconsistency of the bag, which measures how far apart the testimonies in the bag are. All observers are equally credible, so differences arise from perception errors, due to the limited accuracy of the individual findings (the limited information extracted by the examination method from the observed reality). Our approach differs from classical logic, which considers a set of assertions to be either consistent (True, or 1) or inconsistent (False, or 0), and it does not use Fuzzy Logic.

Constructive Zermelo-Fraenkel Set Theory, CZF, has emerged as a standard reference theory that relates to constructive predicative mathematics as ZFC relates to classical Cantorian mathematics. A hallmark of this theory is that it possesses a type-theoretic model. Aczel showed that it has a formulae-as-types interpretation in Martin-Löf's intuitionist theory of types . This paper, though, is concerned with a rather different interpretation. It is shown that Kleene realizability provides a self-validating semantics for CZF, viz. this notion of realizability can be formalized in CZF and demonstrably in CZF it can be verified that every theorem of CZF is realized. This semantics, then, is put to use in establishing several equiconsistency results. Specifically, augmenting CZF by well-known principles germane to Russian constructivism and Brouwer's intuitionism turns out to engender theories of equal proof-theoretic strength with the same stock of provably recursive functions.

We show how to use Logical Structures (of ref. [1]) in a variety of

settings. It is of use in mathematics to show explicit formal reasoning, and

especially important in detective work and arguing in a court of law. It is also

useful to express knowledge so that someone else can check the correctness of the

Structures before and after executing any operators. It is also useful in solving

logical puzzles (for pleasure or as a test of mental competence). What we need to

do is to fix our thinking to a commonly agreed on symbolisation (words and letters

are not clear, precise, varied and focused enough). Logical Structures are graphs with

singly or doubly labelled vertices and labelled (by symbols) edges. The levels of

reasoning are of special importance and are made explicit. They are of great help in

reasoning correctly. We prove AND introduction, and other "axioms" of propositional

logic using properties of Attractors and Stoppers. Other axioms of propositional logic

can also be proved. We note that all the axioms of Zermelo-Fraenkel set theory can be

expressed entirely in symbols of Structural Logic.

Repenser l'identité : au-delà du "A est A"
de la logique classique
Depuis Parménide, la philosophie occidentale privilégie l'unité absolue et l'identité fermée sur
elle-même. Le fameux Principe d'Identité - "A est A et seulement A" - structure notre pensée
logique mais appauvrit notre compréhension de l'être concret.
Et si cette vision était fondamentalement incomplète ?
Dans ce texte, je propose une alternative révolutionnaire : le Principe d'Alter-Identité (PAI),
qui reconnaît que tout être réel est constitué d'altérités. Contrairement au Principe d'Identité
classique qui exclut l'autre par principe, le PAI l'intègre structurellement : A est A / X.NON-
A.
Cette formule simple révèle une vérité troublante : de l'atome d'hydrogène (composé d'un
noyau et d'un électron) aux organismes vivants, rien dans la nature n'existe comme unité pure.
Tout est composition, relation, multiplicité organisée.
Deux logiques qui façonnent le monde
Cette opposition conceptuelle se manifeste concrètement dans nos vies :
• La logique identitaire du PI anime nos instincts de conservation, notre système
immunitaire, les mouvements politiques conservateurs
• La logique altéritaire du PAI pousse vers l'expansion, l'exploration, l'innovation - de
l'attraction sexuelle aux conquêtes spatiales d'Elon Musk
Une grille de lecture pour notre époque
Ce cadre théorique éclaire d'un jour nouveau les tensions contemporaines : pourquoi certains
se replient-ils sur l'identité nationale quand d'autres embrassent la mondialisation ? Pourquoi
les GAFAM expansent-ils sans limite pendant que d'autres prônent le retour aux "vraies
valeurs" ?
La réponse tient peut-être dans cette dualité fondamentale entre deux principes qui, loin de
s'exclure, se complètent comme les deux pieds d'une marche vers l'avenir.
Une philosophie pour demain
En repensant les bases logiques de notre ontologie, nous ouvrons la voie à une "métaphysique
du Pluriel" plus fidèle à l'expérience vécue et aux découvertes scientifiques contemporaines.
Un texte dense mais accessible, nourri de références classiques et d'exemples contemporains,
pour quiconque s'interroge sur les fondements de notre rapport à l'identité et à l'altérité

The structure-preference (SP) order is a way of defining argument preference relations in structured argumentation theory that takes into account how arguments are constructed. The SP order was first introduced in the context of endowing Brewka’s prioritised default logic (PDL) with sound and complete argumentation semantics. In this paper, we further articulate the underlying intuitions of the SP order in terms of how an agent should construct arguments. We also compare the SP order to other argument preference relations and illustrate the different results one would obtain. Finally, we prove that the SP order allows for the original version of PDL to satisfy Brewka’s and Eiter’s postulates.

The purpose of this paper is to apply Crispin Wright's criteria and various axes of objectivity to mathematics. I test the criteria and the objectivity of mathematics against each other. Along the way, various issues concerning general logic and epistemology are encountered.

The use of the universal and existential quantifiers with the capability to express the concept of at least k or all but k, for a nonnegative integer k, has been thoroughly studied in various kinds of logics. In classical logic there are counting quantifiers, in modal logics graded modalities, in description logics number restrictions. Recently, the complexity issues related to the decidability of the μcalculus, when the universal and existential quantifiers are augmented with graded modalities, have been investigated by Kupfermann, Sattler and Vardi. They have shown that this problem is ExpTime-complete. In this paper we consider another extension of modal logic, the Computational Tree Logic CTL, augmented with graded modalities generalizing standard quantifiers and investigate the complexity issues, with respect to the model-checking problem. We consider a system model represented by a pointed Kripke structure K and give an algorithm to solve the model-checking problem running in time O(|K| • |ϕ|) which is hence tight for the problem (where |ϕ| is the number of temporal and boolean operators and does not include the values occurring in the graded modalities). In this framework, the graded modalities express the ability to generate a user-defined number of counterexamples (or evidences) to a specification ϕ given in CTL. However these multiple counterexamples can partially overlap, that is they may share some behavior. We have hence investigated the case when all of them are completely disjoint. In this case we prove that the model-checking problem is both NP-hard and coNP-hard and give an algorithm for solving it running in polynomial space. We have thus studied a fragment of this graded-CTL logic, and have proved that the model-checking problem is solvable in polynomial time.

Em primeiro lugar, devo agradecimento ao meu orientador, Professor Marcelo E. Coniglio, por ter sugerido o tema, colaborado na elaboração do projeto e me introduzido ao tema das combinações entre lógicas, de onde nasceu este trabalho, além de não ter cessado de me ensinar e orientar desde que ingressei no Mestrado. Aos demais professores do CLE, especialmente os professores Ítala M. L. D'Ottaviano e Walter A Carnielli, por aulas, comentários, conselhos e constante solicitude. Aos colegas do CLE por críticas e sugestões. À Professora Maria da Paz, que me iniciou no estudo da Lógica e me orientou no começo da minha vida acadêmica, e cuja intervenção acabou me trazendo à Unicamp. Aos demais professores do departamento de Filosofia da UFRN, em especial os do grupo de Lógica, pelos conhecimentos e inspiração que me passaram. Agradeço à minha família pelo incondicional apoio e encorajamento. Em especial à minha mãe, Sheyla, minha avó, Ivete, minha tia Alberta e meu tio Sivam, sem cujo apoio eu jamais teria conseguido chegar a defender esta tese. A todos aqueles que, desconhecendo o fato da minha existência, possibilitaram esta dissertação, pelos seus próprios estudos e idéias que povoam a lista de referências do trabalho. Sobretudo agradeço à minha esposa, Luciana, a quem palavras jamais farão justiça. E a Deus, por tê-la posto em meu caminho

We present a system for textual inference (the task of inferring whether a sentence follows from another text) that uses learning and a logical-formula semantic representation of the text. More precisely, our system begins by parsing and then transforming sentences into a logical formula-like representation similar to the one used by . An abductive theorem prover then tries to find the minimum "cost" set of assumptions necessary to show that one statement follows from the other. These costs reflect how likely different assumptions are, and are learned automatically using information from syntactic/semantic features and from linguistic resources such as WordNet. If one sentence follows from the other given only highly plausible, low cost assumptions, then we conclude that it can be inferred. Our approach can be viewed as combining statistical machine learning and classical logical reasoning, in the hope of marrying the robustness and scalability of learning with the preciseness and elegance of logical theorem proving. We give experimental results from the recent PASCAL RTE 2005 challenge competition on recognizing textual inferences, where a system using this inference algorithm achieved the highest confidence weighted score.

The paper aims at contributing to the problem of translating natural (ethnic) language into the framework of formal logic in a structure-preserving way. There are several problems with encoding knowledge in logic-derived formalisms. Among the most difficult are those connected with natural language quantifiers. Even for relatively simple natural language sentences there are readings hard to encode in the classical formal logic. The solution we propose is to expand the language of the classical first-order logic with a number of new quantifier symbols (corresponding to the natural language quantifiers possibly co-occurring in a sentence) with a nonclassical semantics/interpretation similar in a way to the Mostowski's generalized quantifier, i.e. as a family of subsets of the Cartesian product of variable ranges. We claim that the logic-based formalism involving such nonclassical quantifiers may be considered a good knowledge representation tool closer to the natural language then the classical logic.

This paper treats decision problexs for the intuitionistic logic without weakening rule FL,,. First, the cut elimination theorem for FL,, will be s h o ~~n . Using this fact and Mripke's method, it will be proved that the propositional FL,, is decidable. On the other hand, the predicate FL,, t ~i l l be shown to be undecidable by reducing the decision problem to that of the intuitionistic predicate logic. In recent pears. various studies have been done concerning logics lacking some or ail of structural rules. (For more irnformation. see ,) This paper will be devoted to stucly of the decision problem of the logic FL,,, which is obtained from the intuitionistic Iogic by deleting the -weakening rule. The decidability of the propositional FL,, will be shown bj-using a method originally developed for relevant logics. Next, the nndecidability of the prerlicate FL,, millbe proved by reducing its decision problem to that of the intuixionistic

In our joint paper (KO) with H. Kihara, we discuss comprehensively inter- polation properties and Beth definability properties of substructural logics, and their algebraic characterizations in comparison with various forms of amalgamation properties of varieties of residuated lattices. This is a brief exposition of some of main results of the paper (KO), We start from an observation on two results on classical logic. The first one says that classical logic has both Craig's interpolation property (CIP) and Beth definability property (BDP). The second says that the CIP of classical logic is derived from the amalgamation property (AP) of the class of Boolean algebras, and vice versa. Connections of these three properties can be shown in a more general setting. For instance, the CIP and the BDP are shown to be interderivable for each classical regular logic (see e.g. (GM)). Also, as shown by Maksi- mova (see also (GM)), for each superintuitionistic logic the CIP implies the AP (in ...

We define the semantics of the modal predicate logic introduced in Part I and prove its soundness and strong completeness with respect to appropriate structures. These semantical tools allow us to give a simple proof that the main conservation requirement articulated in Part I, Section 1, is met as it follows directly from Theorem 5.1 below. Section numbering is consecutive to that of Part I. The bibliography at the end applies only to Part II. We will freely use notation and results from Part I. Moreover, in what follows ∀A will denote the canonical universal closure of A, that is, (∀y 1 ) • • • (∀y n )A where y 1 , . . . , y n are all the free variables of A in alphabetical order. Thus ∀A is the same expression as A if the latter is closed. We may also abbreviate (∀y 1 ) • • • (∀y n ) by (∀ y). In general, a denotes a 1 , . . . , a n , where n is either unimportant or is clear from the context. Theorem 5.1. If A is a wff and T is a classical theory, then T 2A implies that T A, classically. The converse also holds by the derived rule "WN" (cf. Part I, Metatheorem 4.2).

We consider the problem of specifying and computing consistent answers to queries against databases that do not satisfy given integrity constraints. This is done by simultaneously embedding the database and the integrity constraints, which are mutually inconsistent in classical logic, into a theory in annotated predicate calculus -a logic that allows non trivial reasoning in the presence of inconsistency. In this way, several goals are achieved: (a) A logical specification of the class of all minimal "repairs" of the original database, and the ability to reason about them; (b) The ability to distinguish between consistent and inconsistent information in the database; and (c) The development of computational mechanisms for retrieving consistent query answers, i.e., answers that are not affected by the violation of the integrity constraints.

We review the relationship between abstract machines for (call-byname or call-by-value) λ-calculi (extended with Felleisen's C) and sequent calculus, reintroducing on the way Curien-Herbelin's syntactic kit of the duality of computation. We provide a term language for a presentation of LK (with conjunction, disjunction, and negation), and we transcribe cut elimination as (non confluent) rewriting. A key slogan, which may appear here in print for the first time, is that commutative cut elimination rules are explicit substitution propagation rules. We then describe the focalised proof search discipline (in the classical setting), and narrow down the language and the rewriting rules to a confluent calculus (a variant of the second author's focalising system L). We then define a game of patterns and counterpatterns, leading us to a fully focalised finitary syntax for a synthetic presentation of classical logic, that provides a quotient on (focalised) proofs, abstracting out the order of decomposition of negative connectives.

Klasik Mantığa Giriş (Notlar) - Mahmut Boyuneğmez
Bu notlar, klasik mantığın temel kavramlarını ve ilkelerini tanıtmayı amaçlar. Mantık, doğru akıl yürütmenin kurallarını inceleyen bir disiplindir ve bilim, matematik, felsefe gibi alanlarda temel bir araçtır. Bu metin önermeler, bağlaçlar, niceleyiciler ve çıkarsama yöntemlerini kapsar. Her bölüm, tanımlar, örnekler ve doğruluk tablolarıyla desteklenerek mantıksal düşünme becerilerini geliştirmeyi hedefler.