Equivalence of the information structure with unawareness to the
... believe that agent j implicitly believes that p is false’. For any formula φ, denote the set of primitive propositions found in φ by Prim(φ). Certain formulas of the logic, called theorems, are later used to connect the propositional and set-based models. Any formula valid in the Kripke structure (t ...
... believe that agent j implicitly believes that p is false’. For any formula φ, denote the set of primitive propositions found in φ by Prim(φ). Certain formulas of the logic, called theorems, are later used to connect the propositional and set-based models. Any formula valid in the Kripke structure (t ...
Monadic Second-Order Logic with Arbitrary Monadic Predicates⋆
... For the sake of readability, we often define predicates as P = (Pn )n∈N with Pn ⊆ {0, 1}n . In such case we can see P as a language over {0, 1}, which contains exactly one word for each length. Also, we often define predicates P = (Pn )n∈N with Pn ∈ An for some finite alphabet A. This is not formall ...
... For the sake of readability, we often define predicates as P = (Pn )n∈N with Pn ⊆ {0, 1}n . In such case we can see P as a language over {0, 1}, which contains exactly one word for each length. Also, we often define predicates P = (Pn )n∈N with Pn ∈ An for some finite alphabet A. This is not formall ...
Lectures on Proof Theory - Create and Use Your home.uchicago
... of the principles of logic to be used in deriving theorems from the axioms, so that logic itself could be axiomatized. In this case, too, the timing was just right: the analysis of logic in Frege’s Begriffsschrift was just what was required. (There was a remarkable meeting here of supply and demand. ...
... of the principles of logic to be used in deriving theorems from the axioms, so that logic itself could be axiomatized. In this case, too, the timing was just right: the analysis of logic in Frege’s Begriffsschrift was just what was required. (There was a remarkable meeting here of supply and demand. ...
The Logic of Provability
... Classical first-order arithmetic with induction; also called arithmetic or PA. More formally, we take the signature of PA to have ‘0’ as a constant and ‘+’, ‘·’, and ‘<’ as binary function symbols; PA is then the theory axiomatized by the following: • ∀x(sx 6= 0) • ∀x, y(sx = sy → x = y) • For every ...
... Classical first-order arithmetic with induction; also called arithmetic or PA. More formally, we take the signature of PA to have ‘0’ as a constant and ‘+’, ‘·’, and ‘<’ as binary function symbols; PA is then the theory axiomatized by the following: • ∀x(sx 6= 0) • ∀x, y(sx = sy → x = y) • For every ...
Modal Reasoning
... 1. Atomic Harmony: x, y verify the same proposition letters 2. (a) Zig: if xRz in M, then there exists u in N with yRu and zEu. (b) Zag: if yRu in N , then there exists z in M with xRz and zEu. We notate this as M, s - N , t. ...
... 1. Atomic Harmony: x, y verify the same proposition letters 2. (a) Zig: if xRz in M, then there exists u in N with yRu and zEu. (b) Zag: if yRu in N , then there exists z in M with xRz and zEu. We notate this as M, s - N , t. ...
Boolean Connectives and Formal Proofs - FB3
... licensed by the rule. In this example there is only one step men rule, but in other examples there will be several steps. The second rule of F is Identity Elimination. It tells us th proven a sentence containing n (which we indicate by writing sentence of the form n = m, then we are justified in ass ...
... licensed by the rule. In this example there is only one step men rule, but in other examples there will be several steps. The second rule of F is Identity Elimination. It tells us th proven a sentence containing n (which we indicate by writing sentence of the form n = m, then we are justified in ass ...
Reasoning about Action and Change
... fragment of PDL is a definitional extension of the underlying stratified multimodal logic containing only atomic programs terms. (Fine & Schurz speak of atomic program terms rather than action variables). All frame completeness transfer theorems proved there apply to this fragment. This means that i ...
... fragment of PDL is a definitional extension of the underlying stratified multimodal logic containing only atomic programs terms. (Fine & Schurz speak of atomic program terms rather than action variables). All frame completeness transfer theorems proved there apply to this fragment. This means that i ...
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... he also allows for different subjective domains at each world. He goes further by using what is called neighborhood semantics, also called Montague-Scott structures (Fagin et al., 1995). As is well known, neighborhood semantics provide a more general approach for modeling knowledge than the standar ...
... he also allows for different subjective domains at each world. He goes further by using what is called neighborhood semantics, also called Montague-Scott structures (Fagin et al., 1995). As is well known, neighborhood semantics provide a more general approach for modeling knowledge than the standar ...
Verification Conditions Are Code - Electronics and Computer Science
... Also note that this property is very familiar from the study of program semantics, for example in the theory of predicate transformers, where this result would follow directly from the associativity of function composition. Working directly with program semantics is a model-theoretic approach, howev ...
... Also note that this property is very familiar from the study of program semantics, for example in the theory of predicate transformers, where this result would follow directly from the associativity of function composition. Working directly with program semantics is a model-theoretic approach, howev ...
Proof theory for modal logic
... Sometimes axioms are given using propositional variables and a rule of substitution is given that permits to substitute any formula for them. We shall instead regard axioms as schematic, so that substitution is superfluous and indeed admissible. The early study of modal logic, to the late 1950s, con ...
... Sometimes axioms are given using propositional variables and a rule of substitution is given that permits to substitute any formula for them. We shall instead regard axioms as schematic, so that substitution is superfluous and indeed admissible. The early study of modal logic, to the late 1950s, con ...
AN EXPOSITION ANS DEVELOPMENT OF KANGER`S EARLY
... sets but proper classes as extensions to the predicate symbols of L. If this were the case, then the intended interpretation of L would not be a model in the formal sense of model theory. Of course, there are interpreted first-order languages whose intended interpretations are not models in the form ...
... sets but proper classes as extensions to the predicate symbols of L. If this were the case, then the intended interpretation of L would not be a model in the formal sense of model theory. Of course, there are interpreted first-order languages whose intended interpretations are not models in the form ...
CHAPTER 7 GENERAL PROOF SYSTEMS 1 Introduction
... is called a set of axioms of S. I.e. there is an effective procedure to determine whether a given expression A ∈ E is in AX or not. Semantical link : For a given semantics M for L and its extension to E, we usually choose as AX a subset of expressions that are tautologies under the semantics M . Com ...
... is called a set of axioms of S. I.e. there is an effective procedure to determine whether a given expression A ∈ E is in AX or not. Semantical link : For a given semantics M for L and its extension to E, we usually choose as AX a subset of expressions that are tautologies under the semantics M . Com ...
possible-worlds semantics for modal notions conceived as predicates
... viewed as a predicate. This means that there is no way to assign truth and falsity to all sentences in such a way that pAq comes out as true if and only if A holds at all accessible worlds (and if some conditions are met which will be explained below). Another example is provided by McGee’s [31] Th ...
... viewed as a predicate. This means that there is no way to assign truth and falsity to all sentences in such a way that pAq comes out as true if and only if A holds at all accessible worlds (and if some conditions are met which will be explained below). Another example is provided by McGee’s [31] Th ...
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 ...
LOGIC MAY BE SIMPLE Logic, Congruence - Jean
... what is now called “lattice” (see [Ore 1936, Glivenko 1938]). Even nowadays there is a strong tendency to consider universal algebra as a general theory of structures. Some people think that algebraic structures are more fundamental than the other ones (see [Papert 1967]) and that they are the proto ...
... what is now called “lattice” (see [Ore 1936, Glivenko 1938]). Even nowadays there is a strong tendency to consider universal algebra as a general theory of structures. Some people think that algebraic structures are more fundamental than the other ones (see [Papert 1967]) and that they are the proto ...
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 ...
Bilattices and the Semantics of Logic Programming
... complete lattics, thus simplifying the technical setting somewhat. Note that the operations induced by the ≤k ordering also have a natural interpretation. Combining truth values using ⊗ amounts to the consensus approach to conflicts mentioned above, while the use of ⊕ corresponds to the accept-anyth ...
... complete lattics, thus simplifying the technical setting somewhat. Note that the operations induced by the ≤k ordering also have a natural interpretation. Combining truth values using ⊗ amounts to the consensus approach to conflicts mentioned above, while the use of ⊕ corresponds to the accept-anyth ...
Strong Completeness for Iteration
... ability in determined 2-player games. Here, a modal formula [γ]ϕ should be read as “player 1 has a strategy in the game γ to ensure an outcome where ϕ holds”. The modal language of GL is obtained by extending the program operations of PDL with the game operation dual (d ) which corresponds to a role ...
... ability in determined 2-player games. Here, a modal formula [γ]ϕ should be read as “player 1 has a strategy in the game γ to ensure an outcome where ϕ holds”. The modal language of GL is obtained by extending the program operations of PDL with the game operation dual (d ) which corresponds to a role ...
Proof Theory of Finite-valued Logics
... results can be specialized to particular logics. This idea has also found its way into the notation: Throughout this report, we use V as denoting a set of m truth values, and 2 and Q as dummies representing n-ary connectives and quantifiers, respectively. Replace V by the set of values true and fals ...
... results can be specialized to particular logics. This idea has also found its way into the notation: Throughout this report, we use V as denoting a set of m truth values, and 2 and Q as dummies representing n-ary connectives and quantifiers, respectively. Replace V by the set of values true and fals ...
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 ...
Equality in the Presence of Apartness: An Application of Structural
... the new intuitionistic concepts. We shall here use a uniform notation in which the intuitionistic notion is written with a slash over the classical one, as in a 6= b. The properties of this notion of apartness are, first, irreflexivity ¬ a 6= a, and, secondly, the “splitting” of an apartness a 6= b in ...
... the new intuitionistic concepts. We shall here use a uniform notation in which the intuitionistic notion is written with a slash over the classical one, as in a 6= b. The properties of this notion of apartness are, first, irreflexivity ¬ a 6= a, and, secondly, the “splitting” of an apartness a 6= b in ...
Robust Satisfaction - CS
... [CE81, QS81]. In order to check whether an open system satisfies a desired property, we need to check the behavior of the system with respect to an arbitrary environment [FZ88]. In the most general setting, the environment is another open system. Thus, given an open and a specification , we need to ...
... [CE81, QS81]. In order to check whether an open system satisfies a desired property, we need to check the behavior of the system with respect to an arbitrary environment [FZ88]. In the most general setting, the environment is another open system. Thus, given an open and a specification , we need to ...
Paper - Department of Computer Science and Information Systems
... SHIQO. Moreover, if a Boolean description logic has transitive roles, inverse roles and role hierarchies, then a role box can be used to define a universal role. In this case our results can be used to show the undecidability of unification relative to role boxes. This applies, for example, to the l ...
... SHIQO. Moreover, if a Boolean description logic has transitive roles, inverse roles and role hierarchies, then a role box can be used to define a universal role. In this case our results can be used to show the undecidability of unification relative to role boxes. This applies, for example, to the l ...
The logic and mathematics of occasion sentences
... ABSTRACT. The prime purpose of this paper is, first, to restore to discourse-bound occasion sentences their rightful central place in semantics and secondly, taking these as the basic propositional elements in the logical analysis of language, to contribute to the development of an adequate logic of ...
... ABSTRACT. The prime purpose of this paper is, first, to restore to discourse-bound occasion sentences their rightful central place in semantics and secondly, taking these as the basic propositional elements in the logical analysis of language, to contribute to the development of an adequate logic of ...
- Free Documents
... By the completeness of L noninterderivable and give rise to distinct and n . This is in general not so for theories. An example is the theory axiomatized by p on the one hand, and the theory T axiomatized by m p for each m, on the other. The sets p and T are the same, consisting of all nodes that to ...
... By the completeness of L noninterderivable and give rise to distinct and n . This is in general not so for theories. An example is the theory axiomatized by p on the one hand, and the theory T axiomatized by m p for each m, on the other. The sets p and T are the same, consisting of all nodes that to ...