Normal modal logics (Syntactic characterisations)
... In common with most modern approaches, we will define systems of modal logic (‘modal logics’ or just ‘logics’ for short) in rather abstract terms — a system of modal logic is just a set of formulas satisfying certain closure conditions. A formula A is a theorem of the system Σ simply when A ∈ Σ. Whi ...
... In common with most modern approaches, we will define systems of modal logic (‘modal logics’ or just ‘logics’ for short) in rather abstract terms — a system of modal logic is just a set of formulas satisfying certain closure conditions. A formula A is a theorem of the system Σ simply when A ∈ Σ. Whi ...
Equality in the Presence of Apartness: An Application of Structural
... In classical logic, each universal axiom can be converted, through reduction to conjunctive normal form, to a conjunction of regular formulas, and therefore all universal axioms can be converted into regular rules. By using intuitionistic logic, we have instead a limitation to those axioms that are ...
... In classical logic, each universal axiom can be converted, through reduction to conjunctive normal form, to a conjunction of regular formulas, and therefore all universal axioms can be converted into regular rules. By using intuitionistic logic, we have instead a limitation to those axioms that are ...
Using linear logic to reason about sequent systems
... here is to encode object-level sequents into meta-level sequents as directly as possible, and since contexts in Forum are either multisets or sets, we will not be able to represent sequents that make use of lists. It is unlikely, for example, that non-commutative object-logics can be encoded into ou ...
... here is to encode object-level sequents into meta-level sequents as directly as possible, and since contexts in Forum are either multisets or sets, we will not be able to represent sequents that make use of lists. It is unlikely, for example, that non-commutative object-logics can be encoded into ou ...
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 ...
A Proof of Nominalism. An Exercise in Successful
... for a first-order language in the same language, as is shown in Hintikka and Sandu (1999). It might also be at the bottom of Zermelo’s unfortunate construal of the axiom of choice as a non-logical, mathematical assumption. Systematically speaking, and even more importantly, the version of the axiom ...
... for a first-order language in the same language, as is shown in Hintikka and Sandu (1999). It might also be at the bottom of Zermelo’s unfortunate construal of the axiom of choice as a non-logical, mathematical assumption. Systematically speaking, and even more importantly, the version of the axiom ...
Using linear logic to reason about sequent systems ?
... here is to encode object-level sequents into meta-level sequents as directly as possible, and since contexts in Forum are either multisets or sets, we will not be able to represent sequents that make use of lists. It is unlikely, for example, that non-commutative object-logics can be encoded into ou ...
... here is to encode object-level sequents into meta-level sequents as directly as possible, and since contexts in Forum are either multisets or sets, we will not be able to represent sequents that make use of lists. It is unlikely, for example, that non-commutative object-logics can be encoded into ou ...
Deep Sequent Systems for Modal Logic
... approaches. It allows to capture a wide class of modal logics and does so systematically. In many important cases it yields systems which are natural and easy to use, which have good structural properties like contractionadmissibility and invertibility of all rules, and which give rise to decision p ...
... approaches. It allows to capture a wide class of modal logics and does so systematically. In many important cases it yields systems which are natural and easy to use, which have good structural properties like contractionadmissibility and invertibility of all rules, and which give rise to decision p ...
pdf
... elements. Moreover, they take awareness with respect to domain elements, not formulas; that is, agents are (un)aware of objects (i.e., domain elements), not formulas. They also allow different domains at different worlds; more precisely, they allow an agent to have a subjective view of what the set ...
... elements. Moreover, they take awareness with respect to domain elements, not formulas; that is, agents are (un)aware of objects (i.e., domain elements), not formulas. They also allow different domains at different worlds; more precisely, they allow an agent to have a subjective view of what the set ...
Conjunctive normal form - Computer Science and Engineering
... boolean formula expressed in Conjunctive Normal Form, such that the formula is true. The k-SAT problem is the problem of finding a satisfying assignment to a boolean formula expressed in CNF in which each disjunction contains at most k variables. 3-SAT is NP-complete (like any other k-SAT problem wi ...
... boolean formula expressed in Conjunctive Normal Form, such that the formula is true. The k-SAT problem is the problem of finding a satisfying assignment to a boolean formula expressed in CNF in which each disjunction contains at most k variables. 3-SAT is NP-complete (like any other k-SAT problem wi ...
Sequent Calculus in Natural Deduction Style
... we present avoids such useless steps altogether. The calculus we present is characterized by the following two properties: First, two-premiss rules have independent contexts, corresponding to the independent treatment of assumptions in natural deduction. The structure of a calculus with independent ...
... we present avoids such useless steps altogether. The calculus we present is characterized by the following two properties: First, two-premiss rules have independent contexts, corresponding to the independent treatment of assumptions in natural deduction. The structure of a calculus with independent ...
CS389L: Automated Logical Reasoning Lecture 1
... Formulas F1 and F2 are equivalent (written F1 ⇔ F2 ) iff for all interpretations I , I |= F1 ↔ F2 F1 ⇔ F2 iff F1 ↔ F2 is valid ...
... Formulas F1 and F2 are equivalent (written F1 ⇔ F2 ) iff for all interpretations I , I |= F1 ↔ F2 F1 ⇔ F2 iff F1 ↔ F2 is valid ...
Insights into Modal Slash Logic and Modal Decidability
... If R is a binary relation, write R+ for the transitive closure of R and R∗ for the reflexive transitive closure of R. A modal structure is treelike if its accessibility relation R satisfies: (i) there is a unique element r ∈ M , the root of the model, such that for all x ∈ M , R∗ rx; (ii) every elem ...
... If R is a binary relation, write R+ for the transitive closure of R and R∗ for the reflexive transitive closure of R. A modal structure is treelike if its accessibility relation R satisfies: (i) there is a unique element r ∈ M , the root of the model, such that for all x ∈ M , R∗ rx; (ii) every elem ...
Lecture 09
... – BASIS STEP: P(2) is true since 2 itself is prime. – INDUCTIVE STEP: The inductive hypothesis is P(j) is true for all integers j with 2 ≤ j ≤ k. To show that P(k + 1) must be true under this assumption, two cases need to be considered: • If k + 1 is prime, then P(k + 1) is true. • Otherwise, k + 1 ...
... – BASIS STEP: P(2) is true since 2 itself is prime. – INDUCTIVE STEP: The inductive hypothesis is P(j) is true for all integers j with 2 ≤ j ≤ k. To show that P(k + 1) must be true under this assumption, two cases need to be considered: • If k + 1 is prime, then P(k + 1) is true. • Otherwise, k + 1 ...
pdf file
... E1 is the extension vindicated by common sense while E2 is an anomalous extension. The principle that states that the derivation of an exception has priority over the derivation of the default to which it is an exception, the situation on pattern of example 2, has been called in [9] the exceptions-f ...
... E1 is the extension vindicated by common sense while E2 is an anomalous extension. The principle that states that the derivation of an exception has priority over the derivation of the default to which it is an exception, the situation on pattern of example 2, has been called in [9] the exceptions-f ...
The Expressive Power of Modal Dependence Logic
... logic extends standard modal logic with team semantics by modal dependence atoms, =(p1 , . . . , pn , q). The intuitive meaning of the formula =(p1 , . . . , pn , q) is that within a team the truth value of the proposition q is functionally determined by the truth values of the propositions p1 , . . ...
... logic extends standard modal logic with team semantics by modal dependence atoms, =(p1 , . . . , pn , q). The intuitive meaning of the formula =(p1 , . . . , pn , q) is that within a team the truth value of the proposition q is functionally determined by the truth values of the propositions p1 , . . ...
Chapter 6: The Deductive Characterization of Logic
... straightforward; in intro logic, the semantic characterization of SL is given in terms of truth tables, whereas the deductive characterization of SL is given in terms of derivations in a natural deduction system. Whereas truth tables are easy to describe in a logically and mathematically rigorous ma ...
... straightforward; in intro logic, the semantic characterization of SL is given in terms of truth tables, whereas the deductive characterization of SL is given in terms of derivations in a natural deduction system. Whereas truth tables are easy to describe in a logically and mathematically rigorous ma ...
Building explicit induction schemas for cyclic induction reasoning
... Abstract. In the setting of classical first-order logic with inductive predicates, two kinds of sequent-based induction reasoning are distinguished: cyclic and structural. Proving their equivalence is of great theoretical and practical interest for the automated reasoning community. In [3,4], it has ...
... Abstract. In the setting of classical first-order logic with inductive predicates, two kinds of sequent-based induction reasoning are distinguished: cyclic and structural. Proving their equivalence is of great theoretical and practical interest for the automated reasoning community. In [3,4], it has ...
Decidability for some justification logics with negative introspection
... explicit reasons for an agent’s knowledge. Formally, justification logic includes statements of the form t : F meaning F is known for reason t instead of the simple 2F meaning F is known. The evidence term t in the statement t : F can represent an informal justification why F is known or a formal ma ...
... explicit reasons for an agent’s knowledge. Formally, justification logic includes statements of the form t : F meaning F is known for reason t instead of the simple 2F meaning F is known. The evidence term t in the statement t : F can represent an informal justification why F is known or a formal ma ...
Inference and Proofs - Dartmouth Math Home
... We concluded our last section with a proof that the sum of two even numbers is even. That proof contained several crucial ingredients. First, we introduced symbols for members of the universe of integers. In other words, rather than saying “suppose we have two integers,” we introduced symbols for th ...
... We concluded our last section with a proof that the sum of two even numbers is even. That proof contained several crucial ingredients. First, we introduced symbols for members of the universe of integers. In other words, rather than saying “suppose we have two integers,” we introduced symbols for th ...
Bounded Proofs and Step Frames - Università degli Studi di Milano
... principle and elements from Γ as well as modus ponens, necessitation and inferences from Ax (again notice that uniform substitution cannot be applied to members of Γ ). We need some care when replacing a logic L with an inference system Ax, because we want global consequence relation to be preserve ...
... principle and elements from Γ as well as modus ponens, necessitation and inferences from Ax (again notice that uniform substitution cannot be applied to members of Γ ). We need some care when replacing a logic L with an inference system Ax, because we want global consequence relation to be preserve ...
Justifying Underlying Desires for Argument
... the importance of the mechanisms for generating some abstract goals for reconciliation. Although they show guidelines for reconciliatory solutions to their thoughtful story, however, they do not give a formal and general method for finding underlying desires in accordance with well known scheme of p ...
... the importance of the mechanisms for generating some abstract goals for reconciliation. Although they show guidelines for reconciliatory solutions to their thoughtful story, however, they do not give a formal and general method for finding underlying desires in accordance with well known scheme of p ...
On presenting monotonicity and on EA=>AE (pdf file)
... a convention for indicating its use. On page 93 of [1], a hint explicitly states the use of monotonicity of ∧ and ∃ in a weakening step, but on pages 73 and 77, monotonicity of ∀ -body is used without mention. We think the authors just didn’t anticipate the frequent use of monotonicity and the probl ...
... a convention for indicating its use. On page 93 of [1], a hint explicitly states the use of monotonicity of ∧ and ∃ in a weakening step, but on pages 73 and 77, monotonicity of ∀ -body is used without mention. We think the authors just didn’t anticipate the frequent use of monotonicity and the probl ...
Tableau-based decision procedure for the full
... knowledge, no efficient decision procedures for logics studied ...
... knowledge, no efficient decision procedures for logics studied ...
full text (.pdf)
... logic, foreshadowing Kripke’s [1963; 1965] formulation of similar state-based semantics for these logics (see [Artemov 2001]). Kripke models also form the basis of the standard semantics of DL (see [Harel et al. 2000]), although as mentioned, DL does not realize the intuitionistic nature of partial ...
... logic, foreshadowing Kripke’s [1963; 1965] formulation of similar state-based semantics for these logics (see [Artemov 2001]). Kripke models also form the basis of the standard semantics of DL (see [Harel et al. 2000]), although as mentioned, DL does not realize the intuitionistic nature of partial ...
Godel`s Proof
... conclusions, and here I would like to point out one key difference. In their “Concluding Reflections,” Nagel and Newman argue that from Gödel’s discoveries it follows that computers—“calculating machines,” as they call them—are in principle incapable of reasoning as flexibly as we humans reason, a re ...
... conclusions, and here I would like to point out one key difference. In their “Concluding Reflections,” Nagel and Newman argue that from Gödel’s discoveries it follows that computers—“calculating machines,” as they call them—are in principle incapable of reasoning as flexibly as we humans reason, a re ...