Dependence Logic
... Some immediate observations can be made using Definition 5. We first note that the empty team ∅ is of the type of any formula, as (φ, ∅, 1) ∈ T holds for all φ. In fact: Lemma 7 For all φ and M we have (φ, ∅, 1) ∈ T and (φ, ∅, 0) ∈ T . Proof. Inspection of definition 5 reveals that all the necessary ...
... Some immediate observations can be made using Definition 5. We first note that the empty team ∅ is of the type of any formula, as (φ, ∅, 1) ∈ T holds for all φ. In fact: Lemma 7 For all φ and M we have (φ, ∅, 1) ∈ T and (φ, ∅, 0) ∈ T . Proof. Inspection of definition 5 reveals that all the necessary ...
Loop Formulas for Circumscription - Joohyung Lee
... Fages’ theorem and extended it to programs with nested expressions (in the sense of [Lifschitz et al., 1999]) in the bodies of rules. Instead of looking for conditions that will guarantee the equivalence between the completion semantics and the answer set semantics, Lin and Zhao [2002] considered ho ...
... Fages’ theorem and extended it to programs with nested expressions (in the sense of [Lifschitz et al., 1999]) in the bodies of rules. Instead of looking for conditions that will guarantee the equivalence between the completion semantics and the answer set semantics, Lin and Zhao [2002] considered ho ...
The Relative Efficiency of Propositional Proof
... for propositional proof systems which will be used in the rest of this paper. The letter n will always stand for an adequate set of propositional connectives which are binary, unary, or nullary (have two, one, or zero arguments). Adequate here means that every truth function can be expressed by form ...
... for propositional proof systems which will be used in the rest of this paper. The letter n will always stand for an adequate set of propositional connectives which are binary, unary, or nullary (have two, one, or zero arguments). Adequate here means that every truth function can be expressed by form ...
Handling Exceptions in nonmonotonic reasoning
... of this very simple defeasible axiomatic basis. Some argue that only g2 should be derived and others argue for splitting the expansions: one applying g1 and other applying g2 . Translating the example in Reiter’s default logic (according with Def. 2.4), it applies only the second rule, and this has ...
... of this very simple defeasible axiomatic basis. Some argue that only g2 should be derived and others argue for splitting the expansions: one applying g1 and other applying g2 . Translating the example in Reiter’s default logic (according with Def. 2.4), it applies only the second rule, and this has ...
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 ...
Finite Presentations of Infinite Structures: Automata and
... Automatic structures are structures whose functions and relations are represented by finite automata. Informally, a relational structure A = (A, R1 , . . . , Rm ) is automatic if we can find a regular language Lδ ⊆ Σ ∗ (which provides names for the elements of A) and a function ν : Lδ → A mapping e ...
... Automatic structures are structures whose functions and relations are represented by finite automata. Informally, a relational structure A = (A, R1 , . . . , Rm ) is automatic if we can find a regular language Lδ ⊆ Σ ∗ (which provides names for the elements of A) and a function ν : Lδ → A mapping e ...
Beyond Quantifier-Free Interpolation in Extensions of Presburger
... these interpolants expensive. This is not by accident. In fact, in this paper we first show that interpolation of QPA+UP in general requires the use of quantifiers, even if the input formulae are themselves free of quantifiers. In order to solve this problem, we study fragments of QPA+UP that are clos ...
... these interpolants expensive. This is not by accident. In fact, in this paper we first show that interpolation of QPA+UP in general requires the use of quantifiers, even if the input formulae are themselves free of quantifiers. In order to solve this problem, we study fragments of QPA+UP that are clos ...
The Development of Mathematical Logic from Russell to Tarski
... strive for. Let us consider first Pieri’s description of his work on the axiomatization of geometry, which had been carried out independently of Hilbert’s famous Foundations of Geometry ( 1899). In his presentation to the International Congress of Philosophy in 1900, Pieri emphasized that the study ...
... strive for. Let us consider first Pieri’s description of his work on the axiomatization of geometry, which had been carried out independently of Hilbert’s famous Foundations of Geometry ( 1899). In his presentation to the International Congress of Philosophy in 1900, Pieri emphasized that the study ...
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... computing. Most of that work has used standard Kripke structures to model knowledge, where an agent knows a fact ϕ if ϕ is true in all the worlds that the agent considers possible. While this approach has proved useful for many applications, it suffers from a serious shortcoming, known as the logica ...
... computing. Most of that work has used standard Kripke structures to model knowledge, where an agent knows a fact ϕ if ϕ is true in all the worlds that the agent considers possible. While this approach has proved useful for many applications, it suffers from a serious shortcoming, known as the logica ...
Understanding SPKI/SDSI Using First-Order Logic
... A formal semantics for SPKI/SDSI defines a class of queries that can be asked against a set of SPKI/SDSI statements, together with an entailment relation that determines whether a query follows from a set of SPKI/SDSI statements. A good formal semantics should achieve the following four goals. First ...
... A formal semantics for SPKI/SDSI defines a class of queries that can be asked against a set of SPKI/SDSI statements, together with an entailment relation that determines whether a query follows from a set of SPKI/SDSI statements. A good formal semantics should achieve the following four goals. First ...
LPF and MPLω — A Logical Comparison of VDM SL and COLD-K
... This allows a large class of recursive and inductive definitions of functions and predicates to be expressed as formulae of MPLω . This was first sketched in [KR89, Section 4] and later worked out in detail by Renardel de Lavalette in [Ren89]. If A is a formula, then the term ιx : S (A) can be forme ...
... This allows a large class of recursive and inductive definitions of functions and predicates to be expressed as formulae of MPLω . This was first sketched in [KR89, Section 4] and later worked out in detail by Renardel de Lavalette in [Ren89]. If A is a formula, then the term ιx : S (A) can be forme ...
Decidability for some justification logics with negative introspection
... Artemov et al. [4] introduced the first justification logic with negative introspection. The current formulation, however, of justified negative introspection has been independently developed by Pacuit [21] and Rubtsova [23]. They prove several initial results for justification logics with negative ...
... Artemov et al. [4] introduced the first justification logic with negative introspection. The current formulation, however, of justified negative introspection has been independently developed by Pacuit [21] and Rubtsova [23]. They prove several initial results for justification logics with negative ...
Introduction to Modal Logic - CMU Math
... Fix M = hW , R, V i. We will define now what it means for M to model a modal formula ϕ at some world w . M |=w P if and only if w ∈ V (P). M |=w ¬P if and only if M 6|=w P. We decide if M |=w ϕ where ϕ = ψ ∧ θ, ϕ = ψ ∨ θ, or ψ → θ by looking it up in the truth table. M |=w ϕ if and only if for ever ...
... Fix M = hW , R, V i. We will define now what it means for M to model a modal formula ϕ at some world w . M |=w P if and only if w ∈ V (P). M |=w ¬P if and only if M 6|=w P. We decide if M |=w ϕ where ϕ = ψ ∧ θ, ϕ = ψ ∨ θ, or ψ → θ by looking it up in the truth table. M |=w ϕ if and only if for ever ...
An Interpolating Theorem Prover
... We will use the notation φ Γ to indicate that all variables and uninterpreted function symbols occurring in φ also occur in Γ. A term x is local with respect to a pair (A, B) if it contains a variable or uninterpreted function symbol not occurring in B (in other words x 6 B) and global otherwise ...
... We will use the notation φ Γ to indicate that all variables and uninterpreted function symbols occurring in φ also occur in Γ. A term x is local with respect to a pair (A, B) if it contains a variable or uninterpreted function symbol not occurring in B (in other words x 6 B) and global otherwise ...
A THEOREM-PROVER FOR A DECIDABLE SUBSET OF DEFAULT
... E’=Th(“fly(Max),bird(Max)=>fly(Max)) “bird(Max). ...
... E’=Th(“fly(Max),bird(Max)=>fly(Max)) “bird(Max). ...
Modal logic and the approximation induction principle
... A context C[] denotes a formula containing one occurrence of []. The formula C[φ ] is obtained by replacing this occurrence of [] by the formula φ . It is well-known, and easy to see, that φ ⇒ ψ yields C[φ ] ⇒ C[ψ] for all contexts C[] over HML+ (here ϕ ⇒ ψ denotes that for any state s, s |= ϕ ⇒ s | ...
... A context C[] denotes a formula containing one occurrence of []. The formula C[φ ] is obtained by replacing this occurrence of [] by the formula φ . It is well-known, and easy to see, that φ ⇒ ψ yields C[φ ] ⇒ C[ψ] for all contexts C[] over HML+ (here ϕ ⇒ ψ denotes that for any state s, s |= ϕ ⇒ s | ...
Proof analysis beyond geometric theories: from rule systems to
... The applicability of the method of proof analysis to logics characterized by a relational semantics has brought a wealth of applications to the proof theory of non-classican logics, including provability logic (Negri 2005), substructural logic (Negri 2008), intermediate logics (Dyckhoff and Negri 20 ...
... The applicability of the method of proof analysis to logics characterized by a relational semantics has brought a wealth of applications to the proof theory of non-classican logics, including provability logic (Negri 2005), substructural logic (Negri 2008), intermediate logics (Dyckhoff and Negri 20 ...
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... of Principia), that it presents in this respect a considerable step backwards as compared with Frege. What is missing, above all, is a precise statement of the syntax of the formalism. Syntactical considerations are omitted even in cases where they are necessary for the cogency of the proofs, in par ...
... of Principia), that it presents in this respect a considerable step backwards as compared with Frege. What is missing, above all, is a precise statement of the syntax of the formalism. Syntactical considerations are omitted even in cases where they are necessary for the cogency of the proofs, in par ...
Relevant deduction
... An imporant idea in philosophy of science, deriving from Popper, is that of the verisimilitude of theories. Although the ultimate aim of science is to give true theories, most of them are true only in an ‘approximative’ sense, but strictly speaking false. For example, Newton’s theory, although succe ...
... An imporant idea in philosophy of science, deriving from Popper, is that of the verisimilitude of theories. Although the ultimate aim of science is to give true theories, most of them are true only in an ‘approximative’ sense, but strictly speaking false. For example, Newton’s theory, although succe ...
Geometric Modal Logic
... proposition is simply necessary. Speaking of something as ‘possibly possible’, we implicitly let the variation system itself vary, we shift from a given system of possibility into a frame inside which this system is only one among others, and we say that respectively to some other system, such or su ...
... proposition is simply necessary. Speaking of something as ‘possibly possible’, we implicitly let the variation system itself vary, we shift from a given system of possibility into a frame inside which this system is only one among others, and we say that respectively to some other system, such or su ...
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... sentences containing them are to be translated into sentences not containing them. In order to be sure, however, that (or for what expressions) this translation is possible and uniquely determined and that (or to what extent) the rules of inference apply also to the new kind of expressions, it is ne ...
... sentences containing them are to be translated into sentences not containing them. In order to be sure, however, that (or for what expressions) this translation is possible and uniquely determined and that (or to what extent) the rules of inference apply also to the new kind of expressions, it is ne ...
HKT Chapters 1 3
... R on A is called nullary, unary (or monadic), binary (or dyadic), ternary, or n-ary if its arity is 0, 1, 2, 3, or n, respectively. A unary relation on A is just a subset of A. The empty relation ∅ is the relation containing no tuples. It can be considered as a relation of any desired arity; all oth ...
... R on A is called nullary, unary (or monadic), binary (or dyadic), ternary, or n-ary if its arity is 0, 1, 2, 3, or n, respectively. A unary relation on A is just a subset of A. The empty relation ∅ is the relation containing no tuples. It can be considered as a relation of any desired arity; all oth ...
1992-Ideal Introspective Belief
... This is a simple translation of DL into a minimal AE logic. It is the same as the translation in [5] (except for the use of a A Q instead of a), but there it was necessary to limit the extensions of the AE logic to strongly grounded ones, a syntactic method based on the form of the premises. No such ...
... This is a simple translation of DL into a minimal AE logic. It is the same as the translation in [5] (except for the use of a A Q instead of a), but there it was necessary to limit the extensions of the AE logic to strongly grounded ones, a syntactic method based on the form of the premises. No such ...
On presenting monotonicity and on EA=>AE (pdf file)
... These manipulations do not change the monotonicity of the position of z . Now, comes a straightforward proof by induction on the structure of the more restricted expressions E , which will rely on monotonic/antimonotonic properties (1), (2), and (3). Incidently, when teaching calculational logic and ...
... These manipulations do not change the monotonicity of the position of z . Now, comes a straightforward proof by induction on the structure of the more restricted expressions E , which will rely on monotonic/antimonotonic properties (1), (2), and (3). Incidently, when teaching calculational logic and ...