Completeness in modal logic - Lund University Publications
... We can’t have systems characterized by Kripke-frames, that do not contain K. K is as low as Kripke semantics will go (in the terminology of Hansson and Gärdenfors, to be introduced later, K determines the width of Kripke-semantics.) So what do we do if we don’t want K to be a theorem in our system? ...
... We can’t have systems characterized by Kripke-frames, that do not contain K. K is as low as Kripke semantics will go (in the terminology of Hansson and Gärdenfors, to be introduced later, K determines the width of Kripke-semantics.) So what do we do if we don’t want K to be a theorem in our system? ...
Using linear logic to reason about sequent systems
... • Since the encodings yield abstract logic programming, procedures for proof search and unification in linear logic can be used to help fashion implementations of object-logics. • Since the encoding of proof systems is natural and direct, we hope to be able to use the rich meta-theory of linear logi ...
... • Since the encodings yield abstract logic programming, procedures for proof search and unification in linear logic can be used to help fashion implementations of object-logics. • Since the encoding of proof systems is natural and direct, we hope to be able to use the rich meta-theory of linear logi ...
First-Order Intuitionistic Logic with Decidable Propositional
... shifted towards logics containing two different variants of connectives. One of them is intuitionistic, and the other is classical [Kr] Fibring logics is the most noticeable technique in this research [Ga]. This approach is less intuitive though. First, it is generally not clear where to apply which ...
... shifted towards logics containing two different variants of connectives. One of them is intuitionistic, and the other is classical [Kr] Fibring logics is the most noticeable technique in this research [Ga]. This approach is less intuitive though. First, it is generally not clear where to apply which ...
Chapter 6: The Deductive Characterization of Logic
... Note carefully that we allow zero-place rules. A well-known example in elementary logic is the reflexivity rule for identity (given “nothing”, one is entitled to write down ‘τ = τ’ for any singular term). The existence of zero-place rules is critical if we are to have a non-trivial notion of proof, ...
... Note carefully that we allow zero-place rules. A well-known example in elementary logic is the reflexivity rule for identity (given “nothing”, one is entitled to write down ‘τ = τ’ for any singular term). The existence of zero-place rules is critical if we are to have a non-trivial notion of proof, ...
Modalities in the Realm of Questions: Axiomatizing Inquisitive
... Besides our primitive connectives, we also make use of some defined ones. We write ϕ ↔ ψ for (ϕ → ψ) ∧ (ψ → ϕ) and ¬ϕ for ϕ → ⊥. Moreover, for α and β declaratives, we write α ∨ β for ¬(¬α ∧ ¬β) , and ?α for ?{α, ¬α}. Throughout the paper, we adopt the following notational convention: α, β, γ range ...
... Besides our primitive connectives, we also make use of some defined ones. We write ϕ ↔ ψ for (ϕ → ψ) ∧ (ψ → ϕ) and ¬ϕ for ϕ → ⊥. Moreover, for α and β declaratives, we write α ∨ β for ¬(¬α ∧ ¬β) , and ?α for ?{α, ¬α}. Throughout the paper, we adopt the following notational convention: α, β, γ range ...
Notes on First Order Logic
... associated with an arity: the number of arguments that are required by the function or predicate. We start by defining terms. Let V be a finite (or countable) set of variables and C a set of constants. Definition 1.1 (Terms) Let V be a set of variable, C a set of constant symbols and F a set of func ...
... associated with an arity: the number of arguments that are required by the function or predicate. We start by defining terms. Let V be a finite (or countable) set of variables and C a set of constants. Definition 1.1 (Terms) Let V be a set of variable, C a set of constant symbols and F a set of func ...
A Simple Tableau System for the Logic of Elsewhere
... the size of models of the satisfiable formulae) and we show that this problem becomes linear-time when the number of propositional variables is bounded. Although E and the well-known propositional modal S5 share numerous common features we show that E is strictly more expressive than S5 (in a sense ...
... the size of models of the satisfiable formulae) and we show that this problem becomes linear-time when the number of propositional variables is bounded. Although E and the well-known propositional modal S5 share numerous common features we show that E is strictly more expressive than S5 (in a sense ...
Henkin`s Method and the Completeness Theorem
... individual symbols Henkin means constants and infinitely many variables. It is worth pointing out that Henkin does not have the equality symbol in the alphabet. But we will see that this issue is addressed later in the paper. Task 1: Note that Henkin does not use any function symbols, which became c ...
... individual symbols Henkin means constants and infinitely many variables. It is worth pointing out that Henkin does not have the equality symbol in the alphabet. But we will see that this issue is addressed later in the paper. Task 1: Note that Henkin does not use any function symbols, which became c ...
page 139 MINIMIZING AMBIGUITY AND
... Ambiguity-adaptive logics are very close to ‘natural’ reasoning in inconsistent situations. In a lot of situations inconsistencies arise from the ambiguous meaning of non-logical terms. People who meet an inconsistency will not weaken their logic, they will suspect some words to have a double meanin ...
... Ambiguity-adaptive logics are very close to ‘natural’ reasoning in inconsistent situations. In a lot of situations inconsistencies arise from the ambiguous meaning of non-logical terms. People who meet an inconsistency will not weaken their logic, they will suspect some words to have a double meanin ...
Automata-Theoretic Model Checking Lili Anne Dworkin Advised by Professor Steven Lindell
... and the logical specification φ as a formula in a temporal logic. There are many approaches to model checking, but the technique we discuss here is the automata-theoretic approach. This strategy involves converting both the system representation and the negation of the specification into automata ov ...
... and the logical specification φ as a formula in a temporal logic. There are many approaches to model checking, but the technique we discuss here is the automata-theoretic approach. This strategy involves converting both the system representation and the negation of the specification into automata ov ...
On Herbrand`s Theorem - UCSD Mathematics
... that T S is a purely universal theory. Incidentally, the set of Sk-def axioms of the Skolem functions can be equivalently expressed as a set of universal formulas; however, they are not included in theory T S . From model-theoretic considerations, it is not difficult to see that T S contains and is ...
... that T S is a purely universal theory. Incidentally, the set of Sk-def axioms of the Skolem functions can be equivalently expressed as a set of universal formulas; however, they are not included in theory T S . From model-theoretic considerations, it is not difficult to see that T S contains and is ...
No Syllogisms for the Numerical Syllogistic
... expression of either of the forms p or p̄, where p is an atom. A literal which is an atom is said to be positive; all other literals are said to be negative. If l = p̄ is a negative literal, then we take l̄ to denote the positive literal p. An N † -formula is an expression of either of the forms (≤ ...
... expression of either of the forms p or p̄, where p is an atom. A literal which is an atom is said to be positive; all other literals are said to be negative. If l = p̄ is a negative literal, then we take l̄ to denote the positive literal p. An N † -formula is an expression of either of the forms (≤ ...
Master Thesis - Yoichi Hirai
... Middle, which states either a proposition or the negation of it is always valid. The Law of Excluded Middle asserts that either a message has reached the intended receiver or it has not reached the intended receiver. We point out that this reasoning assumes the existence of a current state of the wo ...
... Middle, which states either a proposition or the negation of it is always valid. The Law of Excluded Middle asserts that either a message has reached the intended receiver or it has not reached the intended receiver. We point out that this reasoning assumes the existence of a current state of the wo ...
Label-free Modular Systems for Classical and Intuitionistic Modal
... axioms d, t, b, 4, and 5, shown in Figure 1. In classical logic only one of the two conjuncts in each axiom shown in that Figure is needed because the other follows from De Morgan duality. However, in the intuitionistic setting both conjuncts are needed. With these five axioms one can, a priori, obt ...
... axioms d, t, b, 4, and 5, shown in Figure 1. In classical logic only one of the two conjuncts in each axiom shown in that Figure is needed because the other follows from De Morgan duality. However, in the intuitionistic setting both conjuncts are needed. With these five axioms one can, a priori, obt ...
Label-free Modular Systems for Classical and Intuitionistic Modal
... axioms d, t, b, 4, and 5, shown in Figure 1. In classical logic only one of the two conjuncts in each axiom shown in that Figure is needed because the other follows from De Morgan duality. However, in the intuitionistic setting both conjuncts are needed. With these five axioms one can, a priori, obt ...
... axioms d, t, b, 4, and 5, shown in Figure 1. In classical logic only one of the two conjuncts in each axiom shown in that Figure is needed because the other follows from De Morgan duality. However, in the intuitionistic setting both conjuncts are needed. With these five axioms one can, a priori, obt ...
Least and greatest fixed points in Ludics, CSL 2015, Berlin.
... a(~xa ) binds the variables ~xa appearing in pa . Variables which are not under the scope of a binder are free. The free variables of a design d are denoted by fv(d). We identify two designs which are α-equivalent, i.e., which are equal up to renaming of their bound variables. We denote by d[~n/~x] ...
... a(~xa ) binds the variables ~xa appearing in pa . Variables which are not under the scope of a binder are free. The free variables of a design d are denoted by fv(d). We identify two designs which are α-equivalent, i.e., which are equal up to renaming of their bound variables. We denote by d[~n/~x] ...
Classical Propositional Logic
... How to do it in the first place: suitable calculi? How to do it efficiently: search space control? How to do it optimally: reasoning support for specific theories like equality and arithmetic? ...
... How to do it in the first place: suitable calculi? How to do it efficiently: search space control? How to do it optimally: reasoning support for specific theories like equality and arithmetic? ...
PDF
... FO + IFP + W, which has a nondeterministic choice operator W, called the witness operator (Abiteboul and Vianu (1991a)). This operator chooses an arbitrary element from a set given as argument. This logic is a nondeterministic logic in that each formula de nes several di erent relations on a given s ...
... FO + IFP + W, which has a nondeterministic choice operator W, called the witness operator (Abiteboul and Vianu (1991a)). This operator chooses an arbitrary element from a set given as argument. This logic is a nondeterministic logic in that each formula de nes several di erent relations on a given s ...
Kripke completeness revisited
... accommodate also other modal logics and intuitionistic logic (Kripke 1963, 1965). The idea had several significant anticipations in the work of Arnould Bayart, Rudolf Carnap, Jaakko Hintikka, Stig Kanger, Richard Montague, Arthur Prior, and others. Questions about the originality and ultimate attrib ...
... accommodate also other modal logics and intuitionistic logic (Kripke 1963, 1965). The idea had several significant anticipations in the work of Arnould Bayart, Rudolf Carnap, Jaakko Hintikka, Stig Kanger, Richard Montague, Arthur Prior, and others. Questions about the originality and ultimate attrib ...
THE SEMANTICS OF MODAL PREDICATE LOGIC II. MODAL
... interpret constant symbols, because we need to give an interpretation of these symbols across possible worlds that respects the counterpart relations in some appropriate sense. Variables on the other hand simply denote “objects” in the domain of a given world. In the case of traditional semantics th ...
... interpret constant symbols, because we need to give an interpretation of these symbols across possible worlds that respects the counterpart relations in some appropriate sense. Variables on the other hand simply denote “objects” in the domain of a given world. In the case of traditional semantics th ...
Propositional Logic and Methods of Inference
... We can add simple causal reasoning to our rule by defining additional rules such as: (2) IF the battery is good THEN there is electricity (3) IF there is electricity and the sparkplugs are good THEN the sparkplugs will fire (4) IF the sparkplugs fire and there is gas THEN the engine will run (5) IF ...
... We can add simple causal reasoning to our rule by defining additional rules such as: (2) IF the battery is good THEN there is electricity (3) IF there is electricity and the sparkplugs are good THEN the sparkplugs will fire (4) IF the sparkplugs fire and there is gas THEN the engine will run (5) IF ...
Ans - Logic Matters
... We’ve intentionally been unspecific about the fine details of L. For example, which connectives does it have built in, and which are introduced (if at all) via definitions in terms of the basic ones? Are both the usual quantifiers built in? Does it typographically distinguish between variables occur ...
... We’ve intentionally been unspecific about the fine details of L. For example, which connectives does it have built in, and which are introduced (if at all) via definitions in terms of the basic ones? Are both the usual quantifiers built in? Does it typographically distinguish between variables occur ...
KnotandTonk 1 Preliminaries
... for Tonk fail even to define a meaningful connective, on the grounds that Tonk cannot be given semantic conditions. By exactly the same token, inferentialists might allege that the semantic conditions for Knot fail even to define a meaningful connective, on the grounds that Knot cannot be given natu ...
... for Tonk fail even to define a meaningful connective, on the grounds that Tonk cannot be given semantic conditions. By exactly the same token, inferentialists might allege that the semantic conditions for Knot fail even to define a meaningful connective, on the grounds that Knot cannot be given natu ...
Tactics for Separation Logic Abstract Andrew W. Appel INRIA Rocquencourt & Princeton University
... as Coq or Isabelle/HOL). Although the former approach appears to enjoy a nice purity and simplicity, I will advocate the two-level approach. Even when proving an imperative program, much of the reasoning is not about memory cells but concerns the abstract mathematical objects that the program’s data ...
... as Coq or Isabelle/HOL). Although the former approach appears to enjoy a nice purity and simplicity, I will advocate the two-level approach. Even when proving an imperative program, much of the reasoning is not about memory cells but concerns the abstract mathematical objects that the program’s data ...
A Well-Founded Semantics for Logic Programs with Abstract
... We assume an underlying propositional language L. A literal is either an atom (called a positive literal) or an expression of the form “not a” (called a negative literal), where a is an atom. The complement of a literal l, denoted by ¯l, is defined as the literal not a (resp. a) if l = a (resp. l = ...
... We assume an underlying propositional language L. A literal is either an atom (called a positive literal) or an expression of the form “not a” (called a negative literal), where a is an atom. The complement of a literal l, denoted by ¯l, is defined as the literal not a (resp. a) if l = a (resp. l = ...