Elementary Logic
... A (propositional) sequent is an expression of the form Γ ` ∆, where Γ = A1 , A2 , · · · , Am and ∆ = B1 , B2 , · · · , Bn are finite (possibly empty) sequences of (propositional) formulae. In a sequent Γ ` ∆, Γ is called the antecedent (also context) and ∆ the consequent. Note: many authors prefer t ...
... A (propositional) sequent is an expression of the form Γ ` ∆, where Γ = A1 , A2 , · · · , Am and ∆ = B1 , B2 , · · · , Bn are finite (possibly empty) sequences of (propositional) formulae. In a sequent Γ ` ∆, Γ is called the antecedent (also context) and ∆ the consequent. Note: many authors prefer t ...
Modal Logic - Web Services Overview
... Syntax of Modal Logic (□ and ◊) Formulae in (propositional) Modal Logic ML: • The Language of ML contains the Language of Propositional Calculus, i.e. if P is a formula in Propositional Calculus, then P is a formula in ML. • If and are formulae in ML, then ...
... Syntax of Modal Logic (□ and ◊) Formulae in (propositional) Modal Logic ML: • The Language of ML contains the Language of Propositional Calculus, i.e. if P is a formula in Propositional Calculus, then P is a formula in ML. • If and are formulae in ML, then ...
Taming method in modal logic and mosaic method in temporal logic
... We want to apply the mosaic method for proving decidability and Hilbertstyle completeness of temporal logics over linear flows of time. The mosaic approach serves as a general method to prove decidability of certain frames of logic. The main key is to show that the existence of a model is equivalent ...
... We want to apply the mosaic method for proving decidability and Hilbertstyle completeness of temporal logics over linear flows of time. The mosaic approach serves as a general method to prove decidability of certain frames of logic. The main key is to show that the existence of a model is equivalent ...
On the Notion of Coherence in Fuzzy Answer Set Semantics
... as least fixpoint of a logic program, it has been due to an excess of information in the program (possibly erroneous information). As a result, rejecting noncoherent interpretations seems convenient as well. An important remark is that coherence can be interpreted with an empirical sense and that th ...
... as least fixpoint of a logic program, it has been due to an excess of information in the program (possibly erroneous information). As a result, rejecting noncoherent interpretations seems convenient as well. An important remark is that coherence can be interpreted with an empirical sense and that th ...
Suszko`s Thesis, Inferential Many-Valuedness, and the
... universe of interpretation into two subsets of elements: distinguished ...
... universe of interpretation into two subsets of elements: distinguished ...
The Expressive Power of Modal Dependence Logic
... is that within a team the truth value of the proposition q is functionally determined by the truth values of the propositions p1 , . . . , pn . Modal dependence logic is a first step toward combining functional dependences and modal logic. The logic however lacks the ability to express temporal depe ...
... is that within a team the truth value of the proposition q is functionally determined by the truth values of the propositions p1 , . . . , pn . Modal dependence logic is a first step toward combining functional dependences and modal logic. The logic however lacks the ability to express temporal depe ...
Document
... : Show that for all A M(P), every interpretation I: I |= P implies I |= A. Let us consider Herbrand interpretation IH = {A | A ground atom and I |= A}. Then, I |= P I |= A ← B1, ... , Bn for all A ← B1, ... , Bn ground(P) if I |= B1, ... , Bn then I |= A for all A ← B1, ... , Bn ground(P) ...
... : Show that for all A M(P), every interpretation I: I |= P implies I |= A. Let us consider Herbrand interpretation IH = {A | A ground atom and I |= A}. Then, I |= P I |= A ← B1, ... , Bn for all A ← B1, ... , Bn ground(P) if I |= B1, ... , Bn then I |= A for all A ← B1, ... , Bn ground(P) ...
pdf file
... the semantics of the theory) principle. It applies to all contexts independently on the subject matter, might it be temporal projection, causal relations, frame problem, or any other sort of epistemic situation which requires practical or common sense reasoning. To implement the exceptions-first pri ...
... the semantics of the theory) principle. It applies to all contexts independently on the subject matter, might it be temporal projection, causal relations, frame problem, or any other sort of epistemic situation which requires practical or common sense reasoning. To implement the exceptions-first pri ...
An Overview of Intuitionistic and Linear Logic
... Let N = p1 · p2 · . . . · pk + 1. Note that none of pi divide N. Now, either N is prime or it can be factored into primes. So there is at least one prime q that divides N, but q can’t be any of the pi ’s, so q is a prime number not in the list, which contradicts our assumption. ...
... Let N = p1 · p2 · . . . · pk + 1. Note that none of pi divide N. Now, either N is prime or it can be factored into primes. So there is at least one prime q that divides N, but q can’t be any of the pi ’s, so q is a prime number not in the list, which contradicts our assumption. ...
From Syllogism to Common Sense Normal Modal Logic
... ‣ A U B: A is true until B becomes true ‣ G = ‘always’ , F = ‘eventually’, ‣ liveness properties state that something good keeps happening: ...
... ‣ A U B: A is true until B becomes true ‣ G = ‘always’ , F = ‘eventually’, ‣ liveness properties state that something good keeps happening: ...
Handling Exceptions in nonmonotonic reasoning
... 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 motivated some authors to “correct” default logic proposing variations which split ...
... 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 motivated some authors to “correct” default logic proposing variations which split ...
article in press - School of Computer Science
... monadic two-variable guarded fragment GF 2mon of classical first-order logic, where guard relations satisfy conditions that can be expressed as monadic second-order definable closure constraints, is decidable. Our contribution is a slight generalisation of this result to account for conditions which ...
... monadic two-variable guarded fragment GF 2mon of classical first-order logic, where guard relations satisfy conditions that can be expressed as monadic second-order definable closure constraints, is decidable. Our contribution is a slight generalisation of this result to account for conditions which ...
An Introduction to Prolog Programming
... The search tree built up by Prolog when trying to answer a query corresponds to a logic proof using resolution, which is a very efficient deduction system for Horn formulas. A short introduction can be found in the notes; for more details refer to theoretically oriented books on logic programming. ...
... The search tree built up by Prolog when trying to answer a query corresponds to a logic proof using resolution, which is a very efficient deduction system for Horn formulas. A short introduction can be found in the notes; for more details refer to theoretically oriented books on logic programming. ...
SORT LOGIC AND FOUNDATIONS OF MATHEMATICS 1
... a model. But in smaller models (1.4) is simply false, even though ϕ may have models. So (1.4) does not really express the existence of a model for ϕ. The situation would be different if we allowed “∃P ∃R1 . . . ∃Rn ” to refer to outside the model. In sort logic, which we will introduce in detail bel ...
... a model. But in smaller models (1.4) is simply false, even though ϕ may have models. So (1.4) does not really express the existence of a model for ϕ. The situation would be different if we allowed “∃P ∃R1 . . . ∃Rn ” to refer to outside the model. In sort logic, which we will introduce in detail bel ...
full text (.pdf)
... For this reason, models for one language can be viewed as models for the other. We base S on + instead of ∗ because the resulting deductive system is cleaner—it contains no contraction rule1 . This is perhaps due to the fact that + can be viewed as a more primitive operation than ∗ . A test is eithe ...
... For this reason, models for one language can be viewed as models for the other. We base S on + instead of ∗ because the resulting deductive system is cleaner—it contains no contraction rule1 . This is perhaps due to the fact that + can be viewed as a more primitive operation than ∗ . A test is eithe ...
Complexity of Recursive Normal Default Logic 1. Introduction
... are several such conditions in the published literature. Some of these will be used below. These include the notion of stratification [ABW88] and its generalization, local stratification [Prz88]. These conditions guarantee (when described in the language of nonmonotonic rule systems) the existence o ...
... are several such conditions in the published literature. Some of these will be used below. These include the notion of stratification [ABW88] and its generalization, local stratification [Prz88]. These conditions guarantee (when described in the language of nonmonotonic rule systems) the existence o ...
General Dynamic Dynamic Logic
... Theorem 4.11 in [12], first noted in [20], which states that any dynamic operator whose effect on a model can be described in PDL (without Kleene’s iteration operator ∗) can be reduced to the underlying modal logic using essentially only the standard axioms of PDL. We show how this idea can be used ...
... Theorem 4.11 in [12], first noted in [20], which states that any dynamic operator whose effect on a model can be described in PDL (without Kleene’s iteration operator ∗) can be reduced to the underlying modal logic using essentially only the standard axioms of PDL. We show how this idea can be used ...
Can Modalities Save Naive Set Theory?
... see Forster (1995) for motivations for admitting a universal set, and for an overview of existing approaches to such set theories. In this paper, we approach set theories based on (Comp2) from an abstract perspective, considering different principles governing 2 in the form of different quantified m ...
... see Forster (1995) for motivations for admitting a universal set, and for an overview of existing approaches to such set theories. In this paper, we approach set theories based on (Comp2) from an abstract perspective, considering different principles governing 2 in the form of different quantified m ...
Nonmonotonic Reasoning - Computer Science Department
... reasoning is true in all intended interpretations (or models) in which the premises are true. A ”completeness and correctness theorem” for a system says that the ”safe” rules of deduction in the textbooks generate exactly all those conclusions from premises which are true in every interpretation in ...
... reasoning is true in all intended interpretations (or models) in which the premises are true. A ”completeness and correctness theorem” for a system says that the ”safe” rules of deduction in the textbooks generate exactly all those conclusions from premises which are true in every interpretation in ...
A Well-Founded Semantics for Logic Programs with Abstract
... While ASP assumes that solutions are given by answer sets, well-founded models (Van Gelder, Ross, and Schlipf 1991) have been found to be very useful as well. First, computing the well-founded model of a normal logic program is tractable. This compares to the NP-completeness of computing an answer s ...
... While ASP assumes that solutions are given by answer sets, well-founded models (Van Gelder, Ross, and Schlipf 1991) have been found to be very useful as well. First, computing the well-founded model of a normal logic program is tractable. This compares to the NP-completeness of computing an answer s ...
Turner`s Logic of Universal Causation, Propositional Logic, and
... Turner’s logic of universal causation [17], called UCL, is a nonmonotonic modal logic that generalizes McCain and Turner’s causal action theories [15]. The idea is to use the modal operator C to specify the statement that a proposition is “caused”. For instance, ψ ⊃ Cφ says that φ is caused whenever ...
... Turner’s logic of universal causation [17], called UCL, is a nonmonotonic modal logic that generalizes McCain and Turner’s causal action theories [15]. The idea is to use the modal operator C to specify the statement that a proposition is “caused”. For instance, ψ ⊃ Cφ says that φ is caused whenever ...
Tactics for Separation Logic Abstract Andrew W. Appel INRIA Rocquencourt & Princeton University
... portions are disjoint, and their union forms the entire heap reasoned about by P ∗ Q. This is therefore a linear logic, in that in general P ∗ Q does not entail P ∗ P ∗ Q or vice versa. When doing machine-checked proofs of imperative programs, one faces a choice: one could implement Hoare logic (or ...
... portions are disjoint, and their union forms the entire heap reasoned about by P ∗ Q. This is therefore a linear logic, in that in general P ∗ Q does not entail P ∗ P ∗ Q or vice versa. When doing machine-checked proofs of imperative programs, one faces a choice: one could implement Hoare logic (or ...
Proof theory of witnessed G¨odel logic: a
... introducing a general criterion for a first-order logic expressed Hilbert-style not to admit any ∃-analytic calculus. Our criterion applies to a large class of logics including Gw , the fragment2 of Łukasiewicz logic axiomatized in [18], and intuitionistic logic extended with the quantifiers of clas ...
... introducing a general criterion for a first-order logic expressed Hilbert-style not to admit any ∃-analytic calculus. Our criterion applies to a large class of logics including Gw , the fragment2 of Łukasiewicz logic axiomatized in [18], and intuitionistic logic extended with the quantifiers of clas ...
Classical First-Order Logic Introduction
... propositional logic, but also the symbols ∃ and ∀ for “there exists” and “for all”, along with various symbols to represent variables, constants, functions, and relations. ...
... propositional logic, but also the symbols ∃ and ∀ for “there exists” and “for all”, along with various symbols to represent variables, constants, functions, and relations. ...
First-Order Logic, Second-Order Logic, and Completeness
... basically a two-sorted first-order logic. This would also explain the apparent tension between the above mentioned results and Lindström’s theorem: No logic that goes beyond the expressive power of FOL satisfies both the compactness and the Löwenheim-Skolem theorem.12 It is also worth noting that a ...
... basically a two-sorted first-order logic. This would also explain the apparent tension between the above mentioned results and Lindström’s theorem: No logic that goes beyond the expressive power of FOL satisfies both the compactness and the Löwenheim-Skolem theorem.12 It is also worth noting that a ...