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... among the symbols of x. An atom is a Boolean expression representing an atom (minimal nonzero element) of the free Boolean algebra on generators B. Formally, an atom of B = fb1 : : : bk g is a string of literals c1 c2 ck , where each ci 2 fbi big. This assumes an arbitrary but xed order ...
... among the symbols of x. An atom is a Boolean expression representing an atom (minimal nonzero element) of the free Boolean algebra on generators B. Formally, an atom of B = fb1 : : : bk g is a string of literals c1 c2 ck , where each ci 2 fbi big. This assumes an arbitrary but xed order ...
Using linear logic to reason about sequent systems ?
... [Mil93]) is complete. Thus, Forum makes it possible to claim that all of linear logic can be seen as an abstract logic programming language [MNPS91]. Forum has been used to specify a number of computation systems, ranging from objectoriented languages [DM95], imperative programming features [Mil96,C ...
... [Mil93]) is complete. Thus, Forum makes it possible to claim that all of linear logic can be seen as an abstract logic programming language [MNPS91]. Forum has been used to specify a number of computation systems, ranging from objectoriented languages [DM95], imperative programming features [Mil96,C ...
Argument construction and reinstatement in logics for
... stand in explicit conflict. Motivated by examples like this, Prakken and Sartor define a notion of conflict among arguments in a way that takes their strict extensions into account; in the current, simplified setting, the idea behind their definition can be presented as follows. First, where α is an ...
... stand in explicit conflict. Motivated by examples like this, Prakken and Sartor define a notion of conflict among arguments in a way that takes their strict extensions into account; in the current, simplified setting, the idea behind their definition can be presented as follows. First, where α is an ...
Using linear logic to reason about sequent systems
... In [Mil96], Miller proposed moving from intuitionistic logic to the more expressive setting of linear logic to capture the more general setting of sequent calculus proof system. This use of linear logic has been future explored in [Ric98,Pim01,MP]. In this paper we consider the structure of proofs i ...
... In [Mil96], Miller proposed moving from intuitionistic logic to the more expressive setting of linear logic to capture the more general setting of sequent calculus proof system. This use of linear logic has been future explored in [Ric98,Pim01,MP]. In this paper we consider the structure of proofs i ...
Introduction to Modal and Temporal Logic
... Classical (Two-Valued) Nature of Kripke Semantics Lemma 1 For any Kripke model hW, R, ϑi, any w ∈ W and any formula ϕ, either ϑ(w, ϕ) = t or else ϑ(w, ϕ) = f . Proof: Pick any Kripke model hW, R, ϑi, any w ∈ W , and any formula ϕ. Proceed by induction on the length l of ϕ. Base Case l = 1: If ϕ is ...
... Classical (Two-Valued) Nature of Kripke Semantics Lemma 1 For any Kripke model hW, R, ϑi, any w ∈ W and any formula ϕ, either ϑ(w, ϕ) = t or else ϑ(w, ϕ) = f . Proof: Pick any Kripke model hW, R, ϑi, any w ∈ W , and any formula ϕ. Proceed by induction on the length l of ϕ. Base Case l = 1: If ϕ is ...
Relevant and Substructural Logics
... and with hypotheses from among the set X. A proof from hypotheses is simply a list of formulas, each of which is either an hypothesis, an axiom, or one which follows from earlier formulas in the list by means of a rule. In Orlov’s system, the only rule is modus ponens. We will see later that this is ...
... and with hypotheses from among the set X. A proof from hypotheses is simply a list of formulas, each of which is either an hypothesis, an axiom, or one which follows from earlier formulas in the list by means of a rule. In Orlov’s system, the only rule is modus ponens. We will see later that this is ...
AGM Postulates in Arbitrary Logics: Initial Results and - FORTH-ICS
... The effects of the underlying KR-scheme in the considered belief change operation are obvious. One cannot start considering the properties of a belief change operation unless at least some of the properties of the underlying KR system are fixed. Under any given KR formalism, several types of operati ...
... The effects of the underlying KR-scheme in the considered belief change operation are obvious. One cannot start considering the properties of a belief change operation unless at least some of the properties of the underlying KR system are fixed. Under any given KR formalism, several types of operati ...
Introduction to first order logic for knowledge representation
... In describing a phenomena or a portion of the world, we adopt a language. The phrases of this language are used to describe objects of the real worlds, their properties, and facts that holds. This language can be informal (natural language, graphical language, icons, etc...) or a formal (logical lan ...
... In describing a phenomena or a portion of the world, we adopt a language. The phrases of this language are used to describe objects of the real worlds, their properties, and facts that holds. This language can be informal (natural language, graphical language, icons, etc...) or a formal (logical lan ...
Nominal Monoids
... nominal sets were used to prove independence of the axiom of choice, and other axioms. In Computer Science, they have been rediscovered by Gabbay and Pitts in [7], as an elegant formalism for modeling name binding. Since then, nominal sets have become a lively topic in semantics. They were also inde ...
... nominal sets were used to prove independence of the axiom of choice, and other axioms. In Computer Science, they have been rediscovered by Gabbay and Pitts in [7], as an elegant formalism for modeling name binding. Since then, nominal sets have become a lively topic in semantics. They were also inde ...
On the Construction of Analytic Sequent Calculi for Sub
... are logics that are strictly contained (as consequence relations) in classical logic. Thus, by choosing a subset of axioms and derivation rules that are derivable in (some proof system for) classical logic, one easily obtains a (proof system for a) sub-classical logic. Various important and useful n ...
... are logics that are strictly contained (as consequence relations) in classical logic. Thus, by choosing a subset of axioms and derivation rules that are derivable in (some proof system for) classical logic, one easily obtains a (proof system for a) sub-classical logic. Various important and useful n ...
Gödel`s Theorems
... 1. What Gödel’s Theorems say and specifies how the operations of addition and multiplication work. He has then done all he needs to do to make it the case that Goldbach’s conjecture is true (or false, as the case may be). Of course, that last remark is far too fanciful for comfort. We may find it ...
... 1. What Gödel’s Theorems say and specifies how the operations of addition and multiplication work. He has then done all he needs to do to make it the case that Goldbach’s conjecture is true (or false, as the case may be). Of course, that last remark is far too fanciful for comfort. We may find it ...
Carnap and Quine on the analytic-synthetic - Philsci
... Carnap is that Carnap’s notion of analyticity may be too narrow. I will conclude, pace Quine and Carnap, that a broad notion of analyticity may be philosophically useful. In the next section I start by clarifying the analytic-synthetic distinction to be employed in the historical analysis, and to el ...
... Carnap is that Carnap’s notion of analyticity may be too narrow. I will conclude, pace Quine and Carnap, that a broad notion of analyticity may be philosophically useful. In the next section I start by clarifying the analytic-synthetic distinction to be employed in the historical analysis, and to el ...
Notes on the Science of Logic
... to be used in proving something. It should also be referred to by name or number. The difference between axioms and definitions is this: We do not call something a definition unless we have reason to believe that, relative to axioms, the definition satisfies the usual criteria of eliminability and n ...
... to be used in proving something. It should also be referred to by name or number. The difference between axioms and definitions is this: We do not call something a definition unless we have reason to believe that, relative to axioms, the definition satisfies the usual criteria of eliminability and n ...
Modal fixpoint logic: some model theoretic questions
... Now, from what we have seen, the µ-calculus seems a well-suited specification language, as it combines a great expressive power and manageable decision procedures. But there is a drawback: the µ-calculus is probably not the most understandable way to specify behaviors. Most people would have a diffi ...
... Now, from what we have seen, the µ-calculus seems a well-suited specification language, as it combines a great expressive power and manageable decision procedures. But there is a drawback: the µ-calculus is probably not the most understandable way to specify behaviors. Most people would have a diffi ...
S. P. Odintsov “REDUCTIO AD ABSURDUM” AND LUKASIEWICZ`S
... Le playing a key role in studying the structure of the class Jhn (see [21]). On the other hand, the isomorphs were induced by mappings which can be identified in a natural way with modalities of L, which easily implies the fact that Le is definitially equivalent to the positive fragment of L and that ...
... Le playing a key role in studying the structure of the class Jhn (see [21]). On the other hand, the isomorphs were induced by mappings which can be identified in a natural way with modalities of L, which easily implies the fact that Le is definitially equivalent to the positive fragment of L and that ...
Characterizations of stable model semantics for logic programs with
... We note that disjunctive programs with aggregates have been studied previously by Faber et al. (2004) and Pelov and Truszczynski (2004), where aggregates do not appear in the heads of program rules. In the second application, we show that our abstract representation of constraint atoms provides a me ...
... We note that disjunctive programs with aggregates have been studied previously by Faber et al. (2004) and Pelov and Truszczynski (2004), where aggregates do not appear in the heads of program rules. In the second application, we show that our abstract representation of constraint atoms provides a me ...
relevant reasoning as the logical basis of
... extensional notion of material implication (denoted by → in this paper) which is defined as A→B =df ¬(A∧¬B) or A→B =df ¬A∨B. However, the material implication is just a truth-function of its antecedent and consequent but not requires that there must exist a necessarily relevant and/or conditional re ...
... extensional notion of material implication (denoted by → in this paper) which is defined as A→B =df ¬(A∧¬B) or A→B =df ¬A∨B. However, the material implication is just a truth-function of its antecedent and consequent but not requires that there must exist a necessarily relevant and/or conditional re ...
A Cut-Free Calculus for Second
... by 0V and 1V the maximal and minimal elements (respectively) of V with respect to ≤. The operations minV , maxV , inf V and supV are defined as usual (where minV ∅ = 1V and maxV ∅ = 0V ). For every pair of elements u1 , u2 ∈ V , u1 →V u2 is defined to be 1V if u1 ≤ u2 , and u2 otherwise. The relatio ...
... by 0V and 1V the maximal and minimal elements (respectively) of V with respect to ≤. The operations minV , maxV , inf V and supV are defined as usual (where minV ∅ = 1V and maxV ∅ = 0V ). For every pair of elements u1 , u2 ∈ V , u1 →V u2 is defined to be 1V if u1 ≤ u2 , and u2 otherwise. The relatio ...
Forking in simple theories and CM-triviality Daniel Palacín Cruz
... In section 1.2, the class of supersimple theories is introduced, as well as the two fundamental ranks for such theories: The Lascar rank and the D-rank. We characterize supersimplicity in terms of such ranks. Of course, we also mention the breathtaking theorem of Buechler, Pillay, and Wagner: elimin ...
... In section 1.2, the class of supersimple theories is introduced, as well as the two fundamental ranks for such theories: The Lascar rank and the D-rank. We characterize supersimplicity in terms of such ranks. Of course, we also mention the breathtaking theorem of Buechler, Pillay, and Wagner: elimin ...
On perturbations of continuous structures - HAL
... There is another hurdle though, lying in the way we treat unbounded structures such as Banach spaces in continuous logic. The method of splitting such a structure into a many-sorted structure, each sort consisting of a bounded ball, as suggested in [BU], is unsatisfactory when it is the norm (and th ...
... There is another hurdle though, lying in the way we treat unbounded structures such as Banach spaces in continuous logic. The method of splitting such a structure into a many-sorted structure, each sort consisting of a bounded ball, as suggested in [BU], is unsatisfactory when it is the norm (and th ...
Modular Construction of Complete Coalgebraic Logics
... structor. The associated language is similar to the standard modal language over the empty set of atomic propositions. However, this language will be interpreted over Id-coalgebras, which provide a trivial model of deterministic systems. (3) If M is a (possibly infinite) set of modal operators with ...
... structor. The associated language is similar to the standard modal language over the empty set of atomic propositions. However, this language will be interpreted over Id-coalgebras, which provide a trivial model of deterministic systems. (3) If M is a (possibly infinite) set of modal operators with ...
Handling Exceptions in nonmonotonic reasoning
... Note that γ1 is complete because g2 is rejected and g3 is excluded by γ1 . γ2 is complete because g1 is rejected and g4 is excluded by γ2 . The Exceptions-First Principle8 captures a very important feature of reasoning with propositions subject to exceptions. Exceptions stipulate meta conditions to ...
... Note that γ1 is complete because g2 is rejected and g3 is excluded by γ1 . γ2 is complete because g1 is rejected and g4 is excluded by γ2 . The Exceptions-First Principle8 captures a very important feature of reasoning with propositions subject to exceptions. Exceptions stipulate meta conditions to ...
Informal Proceedings of the 30th International Workshop on
... A rule A/B is passive L if its premise A is not uni able in L. Passive rules are admissible in every logic. A logic L is Almost Structurally Complete, ASC, if every admissible rule in L which is not passive is derivable (e.g. all extensions S4.3 are ASC). Projective uni cation implies ASC (or SC). L ...
... A rule A/B is passive L if its premise A is not uni able in L. Passive rules are admissible in every logic. A logic L is Almost Structurally Complete, ASC, if every admissible rule in L which is not passive is derivable (e.g. all extensions S4.3 are ASC). Projective uni cation implies ASC (or SC). L ...
Deep Sequent Systems for Modal Logic
... Labelled systems are formulated in a hybrid language which not only contains modalities but also variables and an accessibility relation. There are some concerns about incorporating the semantics into the syntax of a proof system in this way. Avron discusses them in [1], for example. However, even w ...
... Labelled systems are formulated in a hybrid language which not only contains modalities but also variables and an accessibility relation. There are some concerns about incorporating the semantics into the syntax of a proof system in this way. Avron discusses them in [1], for example. However, even w ...
Linear Contextual Modal Type Theory
... proof theoretical account of availability, truth, and contextual validity in form of a sequent calculus. Next we prove the admissibility of cut, which guarantees the existence of canonical proofs in linear contextual modal logic. The cutelimination result of this section is formalized and machineche ...
... proof theoretical account of availability, truth, and contextual validity in form of a sequent calculus. Next we prove the admissibility of cut, which guarantees the existence of canonical proofs in linear contextual modal logic. The cutelimination result of this section is formalized and machineche ...