Normal modal logics (Syntactic characterisations)
... Systems of modal logic 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 ...
... Systems of modal logic 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 ...
Remarks on Second-Order Consequence
... only that each informally proven theorem be provable by means of the calculus (in other words, when formalizing, we do not mean to be true to proofs, but to theorems). As soon as we state this demand we see the difficulty it involves, for if the notion of an informal theorem turned out to be open-en ...
... only that each informally proven theorem be provable by means of the calculus (in other words, when formalizing, we do not mean to be true to proofs, but to theorems). As soon as we state this demand we see the difficulty it involves, for if the notion of an informal theorem turned out to be open-en ...
Admissible rules in the implication-- negation fragment of intuitionistic logic
... Γ , namely ⊢L σ ϕ for all ϕ ∈ Γ , and (ii) Γ ⊢L σ ψ → ψ and Γ ⊢L ψ → σ ψ for all ψ ∈ FmL . (We also say that ϕ ∈ FmL is L-projective if {ϕ} is L-projective to conform with Ghilardi’s definition.) Moreover, such a σ is also a most general L-unifier for Γ in the sense that for any other L-unifier σ1 f ...
... Γ , namely ⊢L σ ϕ for all ϕ ∈ Γ , and (ii) Γ ⊢L σ ψ → ψ and Γ ⊢L ψ → σ ψ for all ψ ∈ FmL . (We also say that ϕ ∈ FmL is L-projective if {ϕ} is L-projective to conform with Ghilardi’s definition.) Moreover, such a σ is also a most general L-unifier for Γ in the sense that for any other L-unifier σ1 f ...
First-Order Loop Formulas for Normal Logic Programs
... graph of P , written GP , is the infinite graph (V, E), where V is the set of atoms that do not mention any constants other than those in P , and for any A, A0 ∈ V , (A, A0 ) ∈ E if there is a rule (1) in P and a substitution θ such that hθ = A and bθ = A0 for some b ∈ Body. A finite non-empty subse ...
... graph of P , written GP , is the infinite graph (V, E), where V is the set of atoms that do not mention any constants other than those in P , and for any A, A0 ∈ V , (A, A0 ) ∈ E if there is a rule (1) in P and a substitution θ such that hθ = A and bθ = A0 for some b ∈ Body. A finite non-empty subse ...
cl-ch9
... Closed formulas, which are also called sentences, have truth values, true or false, when supplied with an interpretation. But they may have different truth values under different interpretations. For our original example (9)–(12), on the genealogical interpretation we have since named G (and equally ...
... Closed formulas, which are also called sentences, have truth values, true or false, when supplied with an interpretation. But they may have different truth values under different interpretations. For our original example (9)–(12), on the genealogical interpretation we have since named G (and equally ...
Proofs as Efficient Programs - Dipartimento di Informatica
... distinct proofs (in particular those corresponding to unnatural algorithms, in computer science terminology), proving completeness does not say much about the usefulness of the considered system as a programming language. On the other hand, soundness is a key property: any representable function is ...
... distinct proofs (in particular those corresponding to unnatural algorithms, in computer science terminology), proving completeness does not say much about the usefulness of the considered system as a programming language. On the other hand, soundness is a key property: any representable function is ...
How Does Resolution Works in Propositional Calculus and
... x, A(x) implies B(x). In this formula universal quantifier“” applies over the entire formula (A(x)x)). Hence (A(x)x)) is the scope of the quantifier. Note-1x P(x) means “for all x, P of x is true” Example: x Happy(x) If the universe of discourse is people, then this means that everyone is ...
... x, A(x) implies B(x). In this formula universal quantifier“” applies over the entire formula (A(x)x)). Hence (A(x)x)) is the scope of the quantifier. Note-1x P(x) means “for all x, P of x is true” Example: x Happy(x) If the universe of discourse is people, then this means that everyone is ...
PDF
... the tableau cannot be extended any further, because all formulas have been decomposed. Since the propositional tableau method terminates after finitely many steps, this was an easy thing to define. In the first-order case, however, we have to be a bit more careful. We know that because of γ-formulas ...
... the tableau cannot be extended any further, because all formulas have been decomposed. Since the propositional tableau method terminates after finitely many steps, this was an easy thing to define. In the first-order case, however, we have to be a bit more careful. We know that because of γ-formulas ...
Intuitionistic modal logic made explicit
... the -modality in families of so-called justification terms. Instead of formulas A, meaning that A is known, justification logics include formulas t : A, meaning that A is known for reason t. Artemov’s original semantics for the first justification logic, the Logic of Proofs LP, was a provability s ...
... the -modality in families of so-called justification terms. Instead of formulas A, meaning that A is known, justification logics include formulas t : A, meaning that A is known for reason t. Artemov’s original semantics for the first justification logic, the Logic of Proofs LP, was a provability s ...
Deciding Global Partial-Order Properties
... be exploited to reduce the state-space explosion problem: the cost of generating at least one representative per equivalence class is typically significantly less than the cost of generating all interleavings [5,9, 10, 151. If the specification could distinguish between two sequences of the same equ ...
... be exploited to reduce the state-space explosion problem: the cost of generating at least one representative per equivalence class is typically significantly less than the cost of generating all interleavings [5,9, 10, 151. If the specification could distinguish between two sequences of the same equ ...
On Gabbay`s temporal fixed point operator
... A = ¬Y q → Y Y q are the first Y q and the Y Y q; neither the two occurrences of q nor the subformula Y q of the Y Y q are basic subformulas of A. The normal subformulas of A are just the first Y q, ¬Y q, Y Y q and A. Lemma 3.6 Let A be any normal formula. Then YA is equivalent to a normal formula B ...
... A = ¬Y q → Y Y q are the first Y q and the Y Y q; neither the two occurrences of q nor the subformula Y q of the Y Y q are basic subformulas of A. The normal subformulas of A are just the first Y q, ¬Y q, Y Y q and A. Lemma 3.6 Let A be any normal formula. Then YA is equivalent to a normal formula B ...
EVERYONE KNOWS THAT SOMEONE KNOWS
... on a plane at a certain point (x, y) with agents facing a certain direction θ [3]. Variables in this case could be interpreted as ranging over all possible triples (x, y, θ). Although it might be natural to assume that the number of agents is finite, the number of viewpoints that an agent can have i ...
... on a plane at a certain point (x, y) with agents facing a certain direction θ [3]. Variables in this case could be interpreted as ranging over all possible triples (x, y, θ). Although it might be natural to assume that the number of agents is finite, the number of viewpoints that an agent can have i ...
S2 - CALCULEMUS.ORG
... finite models cannot be axiomatizable except in the case of very pure vocabularies. As a result, until the questions and problems concerning researches on the foundations of computer science arose, most of the mathematicians did not pay too much attention to semantics restricted to finite models. Th ...
... finite models cannot be axiomatizable except in the case of very pure vocabularies. As a result, until the questions and problems concerning researches on the foundations of computer science arose, most of the mathematicians did not pay too much attention to semantics restricted to finite models. Th ...
Algebraizing Hybrid Logic - Institute for Logic, Language and
... on at least one branch of the tableau will be satisfiable by label too. Remark 2.4.2. The systematic tableau construction is defined in [5]. Roughly speaking, this construction is needed in order to prove strong completeness. Theorem 2.4.1. ([5]) Any consistent set of formulas in countable language ...
... on at least one branch of the tableau will be satisfiable by label too. Remark 2.4.2. The systematic tableau construction is defined in [5]. Roughly speaking, this construction is needed in order to prove strong completeness. Theorem 2.4.1. ([5]) Any consistent set of formulas in countable language ...
The unintended interpretations of intuitionistic logic
... his mathematical and philosophical talent was understood and appreciated by his thesis adviser D. J. Korteweg. In 1908 Korteweg advised Brouwer, after Brouwer completed his thesis, to devote some time to “proper” mathematics, as opposed to foundations, so as to earn recognition and become eligible f ...
... his mathematical and philosophical talent was understood and appreciated by his thesis adviser D. J. Korteweg. In 1908 Korteweg advised Brouwer, after Brouwer completed his thesis, to devote some time to “proper” mathematics, as opposed to foundations, so as to earn recognition and become eligible f ...
GLukG logic and its application for non-monotonic reasoning
... We considered the above points only for pragmatical reasons. We did not considered at first to be able to handled contradictory programs. The first partial answer was that we could use modal logic S5 but representing as long as we use ¬a for the definition of such negation operator, see [14,15]. Th ...
... We considered the above points only for pragmatical reasons. We did not considered at first to be able to handled contradictory programs. The first partial answer was that we could use modal logic S5 but representing as long as we use ¬a for the definition of such negation operator, see [14,15]. Th ...
Logic - UNM Computer Science
... Definition 8 The implication of two propositions p and q is “if p then q”, and is denoted by p → q. In an implication, the proposition p is called the premise and the proposition q is called the conclusion. The truth table for implication is: p T T F F ...
... Definition 8 The implication of two propositions p and q is “if p then q”, and is denoted by p → q. In an implication, the proposition p is called the premise and the proposition q is called the conclusion. The truth table for implication is: p T T F F ...
Cylindric Modal Logic - Homepages of UvA/FNWI staff
... To start with the first point, let us consider (multi-)modal logic; here correspondence theory (cf. van Benthem [7]) studies the relation between modal and classical formalisms as languages for the same class of Kripke structures. The usual direction in correspondence theory is to start with a varia ...
... To start with the first point, let us consider (multi-)modal logic; here correspondence theory (cf. van Benthem [7]) studies the relation between modal and classical formalisms as languages for the same class of Kripke structures. The usual direction in correspondence theory is to start with a varia ...
Classical Logic and the Curry–Howard Correspondence
... proceed line by line, with each line derived from those preceding it by means of some inference rule. Nowadays such logics are known as ‘Hilbert systems’. This format can be somewhat cumbersome and inelegant, both because it does not follow the reasoning-patterns of ordinary mathematics and because ...
... proceed line by line, with each line derived from those preceding it by means of some inference rule. Nowadays such logics are known as ‘Hilbert systems’. This format can be somewhat cumbersome and inelegant, both because it does not follow the reasoning-patterns of ordinary mathematics and because ...
A systematic proof theory for several modal logics
... where the two rules not labelled with ↑ or ↓ are self-dual. 2. The entire up-fragment , ie. the rules labelled with ↑, is admissible. This is shown, in op. cit. and discussed in more detail below, by means of a translation from cut-free proofs of the sequent calculus into proofs of system KS, that i ...
... where the two rules not labelled with ↑ or ↓ are self-dual. 2. The entire up-fragment , ie. the rules labelled with ↑, is admissible. This is shown, in op. cit. and discussed in more detail below, by means of a translation from cut-free proofs of the sequent calculus into proofs of system KS, that i ...
Properties of Independently Axiomatizable Bimodal Logics
... Let EL denote the lattice of extensions of a modal logic. We have defined an operation − ⊗ − : (EK)2 → EK2 . ⊗ is a -homomorphism in both arguments. There are certain easy properties of this map which are noteworthy. Fixing the second argument we can study the map − ⊗ M : EK → EK2 . This is a -h ...
... Let EL denote the lattice of extensions of a modal logic. We have defined an operation − ⊗ − : (EK)2 → EK2 . ⊗ is a -homomorphism in both arguments. There are certain easy properties of this map which are noteworthy. Fixing the second argument we can study the map − ⊗ M : EK → EK2 . This is a -h ...
Turner`s Logic of Universal Causation, Propositional Logic, and
... Propositional Logic, and Logic Programming Jianmin Ji1 and Fangzhen Lin2 ...
... Propositional Logic, and Logic Programming Jianmin Ji1 and Fangzhen Lin2 ...
On Probability of First Order Formulas in a Given Model
... of (individual) variables and SP, 2F, C are sets of predicate, function and constant symbols, respectively. We construct formulas of the language i f of atomic formulas in the common way, by means of logical connectives ~~|, A , V , ->,<->, quantifiers V, 3 and the symbol = of equality. We use also ...
... of (individual) variables and SP, 2F, C are sets of predicate, function and constant symbols, respectively. We construct formulas of the language i f of atomic formulas in the common way, by means of logical connectives ~~|, A , V , ->,<->, quantifiers V, 3 and the symbol = of equality. We use also ...
Many-Valued Models
... interesting general problem and has received attention from several different areas. In this tutorial we present an elementary but general approach on small finite models, showing their relevance and reviewing some elementary methods and techniques on their uses. There are many significant names in ...
... interesting general problem and has received attention from several different areas. In this tutorial we present an elementary but general approach on small finite models, showing their relevance and reviewing some elementary methods and techniques on their uses. There are many significant names in ...
Interpreting Lattice-Valued Set Theory in Fuzzy Set Theory
... This paper presents a comparison of two axiomatic set theories over two non-classical logics. In particular, it suggests an interpretation of lattice-valued set theory as defined in [16] by S. Titani in fuzzy set theory as defined in [11] by authors of this paper. There are many different conception ...
... This paper presents a comparison of two axiomatic set theories over two non-classical logics. In particular, it suggests an interpretation of lattice-valued set theory as defined in [16] by S. Titani in fuzzy set theory as defined in [11] by authors of this paper. There are many different conception ...