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Ground Nonmonotonic Modal Logics - Dipartimento di Informatica e
... Recently, there have been a number of attempts to reconcile fix-point and semantic characterizations of modal nonmonotonic logics. In particular, Schwarz [30] proposed a semantics for McDermott and Doyle’s logics. However, the notion of minimal knowledge underlying the above cited works is stronger ...
... Recently, there have been a number of attempts to reconcile fix-point and semantic characterizations of modal nonmonotonic logics. In particular, Schwarz [30] proposed a semantics for McDermott and Doyle’s logics. However, the notion of minimal knowledge underlying the above cited works is stronger ...
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... encode sentences as numbers, so that Peano Arithmetic, being a theory of numbers, may indirectly talk about its own sentences. These issues are sidestepped in a theory of sets such as ZFC, because virtually any mathematical object, including systems of sentences, can already be thought of as sets, s ...
... encode sentences as numbers, so that Peano Arithmetic, being a theory of numbers, may indirectly talk about its own sentences. These issues are sidestepped in a theory of sets such as ZFC, because virtually any mathematical object, including systems of sentences, can already be thought of as sets, s ...
Sets, Logic, Computation
... facts, and a store of methods and techniques, and this text covers both. Some students won’t need to know some of the results we discuss outside of this course, but they will need and use the methods we use to establish them. The Löwenheim-Skolem theorem, say, does not often make an appearance in co ...
... facts, and a store of methods and techniques, and this text covers both. Some students won’t need to know some of the results we discuss outside of this course, but they will need and use the methods we use to establish them. The Löwenheim-Skolem theorem, say, does not often make an appearance in co ...
Henkin`s Method and the Completeness Theorem
... We start by getting acquainted with the notation of Henkin’s original source. Henkin mainly uses the notation from the classic monograph by Church [1], which was considered as standard by the time Henkin’s paper appeared in print. At the time, the first-order logic was called the first-order functio ...
... We start by getting acquainted with the notation of Henkin’s original source. Henkin mainly uses the notation from the classic monograph by Church [1], which was considered as standard by the time Henkin’s paper appeared in print. At the time, the first-order logic was called the first-order functio ...
A non-standard Semantics for Inexact Knowledge with Introspection
... commonly used to represent situations of social knowledge, for instance in game theory, due to their well-known correspondence with partition models of information (Osborne & Rubinstein 1994). An important feature of these models is the fact that they represent a notion of precise or exact knowledge ...
... commonly used to represent situations of social knowledge, for instance in game theory, due to their well-known correspondence with partition models of information (Osborne & Rubinstein 1994). An important feature of these models is the fact that they represent a notion of precise or exact knowledge ...
What is "formal logic"?
... von Neumann 1927, see Church, 1956, p.158). Once we have this concept, we can present a proof system where axioms and rules are schemes, then the substitution theorem appears rather as a axiom, expressing the formal character of logic. In fact in the 1950s, the substitution theorem was explicitly st ...
... von Neumann 1927, see Church, 1956, p.158). Once we have this concept, we can present a proof system where axioms and rules are schemes, then the substitution theorem appears rather as a axiom, expressing the formal character of logic. In fact in the 1950s, the substitution theorem was explicitly st ...
Classical Propositional Logic
... DPLL and the refined CDCL algorithm are the practically best methods for PL The resolution calculus (Robinson 1969) has been introduced as a basis for automated theorem proving in first-order logic. We will see it in detail in the first-order logic part of this lecture Refined versions are still the ...
... DPLL and the refined CDCL algorithm are the practically best methods for PL The resolution calculus (Robinson 1969) has been introduced as a basis for automated theorem proving in first-order logic. We will see it in detail in the first-order logic part of this lecture Refined versions are still the ...
Advanced Logic —
... P ROOF. Suppose IND is true, but for a reductio, suppose that LNP is not. Then there is some subset B ⊆ N such that: (1) B is not empty; and (2) B has no least element. Let A = N\B. Clearly 0 ∈ / B (for then 0 would be the least element), so 0 ∈ A. Moreover, since B has no least element, this just m ...
... P ROOF. Suppose IND is true, but for a reductio, suppose that LNP is not. Then there is some subset B ⊆ N such that: (1) B is not empty; and (2) B has no least element. Let A = N\B. Clearly 0 ∈ / B (for then 0 would be the least element), so 0 ∈ A. Moreover, since B has no least element, this just m ...
Continuous Markovian Logic – From Complete ∗ Luca Cardelli
... relies on the fact that for a fixed integer q there exists a finite number of integers p such that p/q ∈ [0, 1] (see [23]), a series of nontrivial additional problems rise in the stochastic case. The construction of a small model for a consistent CML-formula is the cornerstone of this paper supporti ...
... relies on the fact that for a fixed integer q there exists a finite number of integers p such that p/q ∈ [0, 1] (see [23]), a series of nontrivial additional problems rise in the stochastic case. The construction of a small model for a consistent CML-formula is the cornerstone of this paper supporti ...
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... intuitionistic framework. We use a squash operator for this purpose. This operator can creates a proposition stating that a certain type is non-empty without providing an inhabitant, i.e. squash “forgets” proofs. It was first introduced in [6] and also used in [15] and MetaPRL system [9,10]. Using sq ...
... intuitionistic framework. We use a squash operator for this purpose. This operator can creates a proposition stating that a certain type is non-empty without providing an inhabitant, i.e. squash “forgets” proofs. It was first introduced in [6] and also used in [15] and MetaPRL system [9,10]. Using sq ...
CS1231 - Lecture 09
... cardinals describe ‘how many elements’ in INFINITE sets as well. As such it describes how infinite the set is. – So a ‘cardinal number’ is a more generalized definition of a ‘number’ as you have been taught in your previous math education. ...
... cardinals describe ‘how many elements’ in INFINITE sets as well. As such it describes how infinite the set is. – So a ‘cardinal number’ is a more generalized definition of a ‘number’ as you have been taught in your previous math education. ...
Logic for Computer Science. Lecture Notes
... Finally we have to state clearly what kind of opinions (sentences) can be formulated in the language we deal with and, moreover, which of those opinions are true (valid), and which are false (invalid). Now we can investigate the subject of reasoning via the validity of expressed opinions. Such an ab ...
... Finally we have to state clearly what kind of opinions (sentences) can be formulated in the language we deal with and, moreover, which of those opinions are true (valid), and which are false (invalid). Now we can investigate the subject of reasoning via the validity of expressed opinions. Such an ab ...
Introduction to Artificial Intelligence
... Example: If A stands for “It is raining today” and B for “It is cold today” and these are both true, then A ∧ B is true. If B represents “It is hot today” and this is false, then A ∧ B is false. Definition of Interpretation A function I : Σ → {t, f }, which assigns a truth value to every proposition ...
... Example: If A stands for “It is raining today” and B for “It is cold today” and these are both true, then A ∧ B is true. If B represents “It is hot today” and this is false, then A ∧ B is false. Definition of Interpretation A function I : Σ → {t, f }, which assigns a truth value to every proposition ...
Sets, Logic, Computation
... facts, and a store of methods and techniques, and this text covers both. Some students won’t need to know some of the results we discuss outside of this course, but they will need and use the methods we use to establish them. The Löwenheim-Skolem theorem, say, does not often make an appearance in co ...
... facts, and a store of methods and techniques, and this text covers both. Some students won’t need to know some of the results we discuss outside of this course, but they will need and use the methods we use to establish them. The Löwenheim-Skolem theorem, say, does not often make an appearance in co ...
Modalities in the Realm of Questions: Axiomatizing Inquisitive
... Importantly, conjunction, implication, and the modalities are allowed to apply to interrogatives as well as declaratives. Also, notice how the two syntactic categories are intertwined: from a sequence of declaratives, clause (iii) allows us to form a basic interrogative, from which more complex inte ...
... Importantly, conjunction, implication, and the modalities are allowed to apply to interrogatives as well as declaratives. Also, notice how the two syntactic categories are intertwined: from a sequence of declaratives, clause (iii) allows us to form a basic interrogative, from which more complex inte ...
The substitutional theory of logical consequence
... of these models. Models have set-sized domains, while the intended interpretation, if it could be conceived as a model, cannot be limited by any cardinality. Similarly, logical truth defined as truth in all models does not imply truth simpliciter. If logical truth is understood as truth under all in ...
... of these models. Models have set-sized domains, while the intended interpretation, if it could be conceived as a model, cannot be limited by any cardinality. Similarly, logical truth defined as truth in all models does not imply truth simpliciter. If logical truth is understood as truth under all in ...
Notes on Classical Propositional Logic
... Proof It is easy to see that if the conditions are met, every line of a proof must be a tautology, hence the last line, which is what the proof proves. In fact a stronger result can be proved, which we leave to you as Exercise 5.1. So, we will make sure to choose axiom schemes whose instances are ta ...
... Proof It is easy to see that if the conditions are met, every line of a proof must be a tautology, hence the last line, which is what the proof proves. In fact a stronger result can be proved, which we leave to you as Exercise 5.1. So, we will make sure to choose axiom schemes whose instances are ta ...
Insights into Modal Slash Logic and Modal Decidability
... such that the domain M 0 = {(x0 , x1 , . . . , xn ) ∈ M n+1 : x0 = w and n < ω and Rxi xi+1 for all 0 ≤ i < n}, the accessibility relation R0 satisfies h(w, x1 , . . . , xn ), (w, y1 , . . . , ym )i ∈ R0 iff (m = n + 1 and xi = yi for all 1 ≤ i ≤ n and Rxn ym ), and the valuation V 0 satisfies (w, x ...
... such that the domain M 0 = {(x0 , x1 , . . . , xn ) ∈ M n+1 : x0 = w and n < ω and Rxi xi+1 for all 0 ≤ i < n}, the accessibility relation R0 satisfies h(w, x1 , . . . , xn ), (w, y1 , . . . , ym )i ∈ R0 iff (m = n + 1 and xi = yi for all 1 ≤ i ≤ n and Rxn ym ), and the valuation V 0 satisfies (w, x ...
Computability theoretic classifications for classes of structures
... Abelian p-groups. That Abelian p-groups are effectively Σ-small follows from work of Khisamiev [Khi04]. Equivalence structures. We refer the reader to [Mon10b, Section 4.2] for an analysis of the ∃-types on equivalence structures. Trees (as partial orderings). By ’trees’ we mean downward closed subs ...
... Abelian p-groups. That Abelian p-groups are effectively Σ-small follows from work of Khisamiev [Khi04]. Equivalence structures. We refer the reader to [Mon10b, Section 4.2] for an analysis of the ∃-types on equivalence structures. Trees (as partial orderings). By ’trees’ we mean downward closed subs ...
Kripke completeness revisited
... relation, as those needed in temporal logic, the canonical accessibility relation need not be irreflexive; Some extra devices, such as the one called bulldozing have to be used to obtain an irreflexive frame from the canonical one (cf. Bull and Segerberg 1984, 2001). The criticism of insufficient fo ...
... relation, as those needed in temporal logic, the canonical accessibility relation need not be irreflexive; Some extra devices, such as the one called bulldozing have to be used to obtain an irreflexive frame from the canonical one (cf. Bull and Segerberg 1984, 2001). The criticism of insufficient fo ...
An Institution-Independent Generalization of Tarski`s Elementary
... each signature, a set of sentences, a category of models, and a satisfaction relation. Sentences have translations, and models have reducts, along signature morphisms; the translations and reducts express the sentence- and model- modifications under change of notation from one language to another. S ...
... each signature, a set of sentences, a category of models, and a satisfaction relation. Sentences have translations, and models have reducts, along signature morphisms; the translations and reducts express the sentence- and model- modifications under change of notation from one language to another. S ...
Propositions as Types - Informatics Homepages Server
... cursive function is λ-definable, and vice-versa. The proof was outlined by Church [8] and published in detail by Kleene [36]. Rather than mollifying Gödel, this result caused him to doubt that his own definition was correct! Things stood at an impasse. Meanwhile, at Cambridge, Alan Turing, a studen ...
... cursive function is λ-definable, and vice-versa. The proof was outlined by Church [8] and published in detail by Kleene [36]. Rather than mollifying Gödel, this result caused him to doubt that his own definition was correct! Things stood at an impasse. Meanwhile, at Cambridge, Alan Turing, a studen ...
Modal Languages and Bounded Fragments of Predicate Logic
... What precisely are fragments of classical first-order logic showing “modal” behaviour? Perhaps the most influential answer is that of Gabbay 1981, which identifies them with so-called “finite-variable fragments”, using only some fixed finite number of variables (free or bound). This view-point has b ...
... What precisely are fragments of classical first-order logic showing “modal” behaviour? Perhaps the most influential answer is that of Gabbay 1981, which identifies them with so-called “finite-variable fragments”, using only some fixed finite number of variables (free or bound). This view-point has b ...
A Deduction Method Complete for Refutation and Finite Satis ability
... their speci cations. Model generators can be applied to (a logic representation of) the programs to generate \samples", or \cases" in which a requirement is violated. These samples can then be used for correcting the programs under development. Clearly Occam's razor applies: The simplest samples are ...
... their speci cations. Model generators can be applied to (a logic representation of) the programs to generate \samples", or \cases" in which a requirement is violated. These samples can then be used for correcting the programs under development. Clearly Occam's razor applies: The simplest samples are ...