Modal Logics of Submaximal and Nodec Spaces 1 Introduction
... examples see Lemma 3.1 below. We also recall that a space X is called an I-space if ddX = ∅. It is pointed out in [3] that for a space X the following three conditions are equivalent: (i) X is an I-space; (ii) X is nodec and (weakly) scattered; (iii) X is submaximal and (weakly) scattered. Examples ...
... examples see Lemma 3.1 below. We also recall that a space X is called an I-space if ddX = ∅. It is pointed out in [3] that for a space X the following three conditions are equivalent: (i) X is an I-space; (ii) X is nodec and (weakly) scattered; (iii) X is submaximal and (weakly) scattered. Examples ...
Everything Else Being Equal: A Modal Logic for Ceteris Paribus
... of the notions we develop later, but also as a foundational standard against which we can evaluate our own results. In Section 3, we present a basic modal logic of weak and strict preference interpreted in ordered models of possible worlds, we discuss its expressive power and we provide a complete a ...
... of the notions we develop later, but also as a foundational standard against which we can evaluate our own results. In Section 3, we present a basic modal logic of weak and strict preference interpreted in ordered models of possible worlds, we discuss its expressive power and we provide a complete a ...
MATH20302 Propositional Logic
... is why it makes sense to apply these propositional connectives to propositional variables as well as to propositions. So now the formal definition. We start with a collection, p, q, r, p0 , p1 , ... of symbols which we call propositional variables. Then we define, by induction, the propositional ter ...
... is why it makes sense to apply these propositional connectives to propositional variables as well as to propositions. So now the formal definition. We start with a collection, p, q, r, p0 , p1 , ... of symbols which we call propositional variables. Then we define, by induction, the propositional ter ...
The Fundamental Theorem of World Theory
... Note that, given Coherence and some basic modal and propositional logic, the Equivalence Principle is equivalent to: The Leibniz Principle It is necessary that p if and only if p is true at every possible world. More formally, in terms of the language at hand: LP p ↔ ∀w(w |= p) Given this equivalen ...
... Note that, given Coherence and some basic modal and propositional logic, the Equivalence Principle is equivalent to: The Leibniz Principle It is necessary that p if and only if p is true at every possible world. More formally, in terms of the language at hand: LP p ↔ ∀w(w |= p) Given this equivalen ...
Towards NP−P via Proof Complexity and Search
... short propositional proofs is an NP -complete problem, and hence a feasible method to find optimal propositional proofs will also give a feasible algorithm to find (short) proofs in any proof system. There are a variety of propositional proof systems that are commonly studied. Three of the most basi ...
... short propositional proofs is an NP -complete problem, and hence a feasible method to find optimal propositional proofs will also give a feasible algorithm to find (short) proofs in any proof system. There are a variety of propositional proof systems that are commonly studied. Three of the most basi ...
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... We can imagine a simpler principle if we work in a sub-logic in which we do not keep track of all the evidence. This kind of sub-logic is called classical logic, and we will examine it more later in the course. {Least Number Principle:} ∃x : N.A(x) ⇒ ∃y : N.(A(y)&∀z : N.z < y ⇒∼ A(z)). There are man ...
... We can imagine a simpler principle if we work in a sub-logic in which we do not keep track of all the evidence. This kind of sub-logic is called classical logic, and we will examine it more later in the course. {Least Number Principle:} ∃x : N.A(x) ⇒ ∃y : N.(A(y)&∀z : N.z < y ⇒∼ A(z)). There are man ...
Formale Methoden der Softwaretechnik Formal methods of software
... The problem with this proof is step 8. In this step we have used step 3, a step that occurs within an earlier subproof. But it turns out that this sort of justification—one that reaches back inside a subproof that has already ended—is not legitimate. To understand why it’s not legitimate, we need to ...
... The problem with this proof is step 8. In this step we have used step 3, a step that occurs within an earlier subproof. But it turns out that this sort of justification—one that reaches back inside a subproof that has already ended—is not legitimate. To understand why it’s not legitimate, we need to ...
Frege`s Other Program
... induction can then be proved. But more work needs to be done if Peano Arithmetic is to be recovered. In particular, after laying down the language and its semantics, we can formulate the three characteristic axioms of the system. These axioms are essentially nonlogical in nature (although, arguably, ...
... induction can then be proved. But more work needs to be done if Peano Arithmetic is to be recovered. In particular, after laying down the language and its semantics, we can formulate the three characteristic axioms of the system. These axioms are essentially nonlogical in nature (although, arguably, ...
Problems on Discrete Mathematics1
... The equivalence symbol ≡ above means: a → b∧¬c is to be interpreted as a → (b ∧ (¬c)), and a → (b ∧ (¬c)) can be abbreviated as a → b ∧ ¬c. We can alternatively use one of them without introducing ambiguity. Associativity : ∧ and ∨ are left associative; → and ↔ are right associative. For example, a ...
... The equivalence symbol ≡ above means: a → b∧¬c is to be interpreted as a → (b ∧ (¬c)), and a → (b ∧ (¬c)) can be abbreviated as a → b ∧ ¬c. We can alternatively use one of them without introducing ambiguity. Associativity : ∧ and ∨ are left associative; → and ↔ are right associative. For example, a ...
logic for computer science - Institute for Computing and Information
... reasoning and of methods of attaching meaning to them. So there are strong parallels between formal computer science and logic. Both involve the study of formal systems and ways of giving them meaning (semantics). However in Logic you study a wider variety of formal systems than you do in Computer S ...
... reasoning and of methods of attaching meaning to them. So there are strong parallels between formal computer science and logic. Both involve the study of formal systems and ways of giving them meaning (semantics). However in Logic you study a wider variety of formal systems than you do in Computer S ...
Document
... of the same type in the given implicit typing. This last is not a serious restriction as we can always replace other constants by terms h(c) using a fresh function symbol h. ...
... of the same type in the given implicit typing. This last is not a serious restriction as we can always replace other constants by terms h(c) using a fresh function symbol h. ...
The Development of Categorical Logic
... in a topos, the axiom of choice implies that the topos is Boolean. This means that, in IZF, the axiom of choice implies the law of excluded middle. This latter formulation of Diaconescu’s result was refined by Goodman and Myhill (1978) to show that, in IZF, the law of excluded middle follows from th ...
... in a topos, the axiom of choice implies that the topos is Boolean. This means that, in IZF, the axiom of choice implies the law of excluded middle. This latter formulation of Diaconescu’s result was refined by Goodman and Myhill (1978) to show that, in IZF, the law of excluded middle follows from th ...
Logic and Proof Jeremy Avigad Robert Y. Lewis Floris van Doorn
... From this perspective, logic is not so much a language for asserting truth, but a language for describing possible states of affairs. In other words, logic provides a specification language, with expressions that can be true or false depending on how we interpret the symbols that are allowed to vary. ...
... From this perspective, logic is not so much a language for asserting truth, but a language for describing possible states of affairs. In other words, logic provides a specification language, with expressions that can be true or false depending on how we interpret the symbols that are allowed to vary. ...
22c:145 Artificial Intelligence
... statement: “every interpretation that makes A ∧ B true, makes C also true.” A ∧ B /I C is a mathematical abbreviation standing for the statement: “I derives C from A ∧ B ”. ...
... statement: “every interpretation that makes A ∧ B true, makes C also true.” A ∧ B /I C is a mathematical abbreviation standing for the statement: “I derives C from A ∧ B ”. ...
Introduction to Logic
... The term “logic” may be, very roughly and vaguely, associated with something like “correct thinking”. Aristotle defined a syllogism as “discourse in which, certain things being stated something other than what is stated follows of necessity from their being so.” And, in fact, this intuition not only ...
... The term “logic” may be, very roughly and vaguely, associated with something like “correct thinking”. Aristotle defined a syllogism as “discourse in which, certain things being stated something other than what is stated follows of necessity from their being so.” And, in fact, this intuition not only ...
Logic in the Finite - CIS @ UPenn
... now subject a sentence ' 2 L to the following e ective procedure: successively test whether A1 satis es '; A2 satis es '; : : : ; at the rst stage where the outcome is negative, halt the procedure and return the answer \no." Clearly, this procedure yields the correct answer to the query \is ' valid ...
... now subject a sentence ' 2 L to the following e ective procedure: successively test whether A1 satis es '; A2 satis es '; : : : ; at the rst stage where the outcome is negative, halt the procedure and return the answer \no." Clearly, this procedure yields the correct answer to the query \is ' valid ...
thèse - IRIT
... In this respect, we first introduce a monotonic modal logic called MEM that is powerful enough to characterise the existence of an equilibrium model as well as the consequence relation in equilibrium models. The logic MEM thus captures the minimisation attitude that is central in the definition of e ...
... In this respect, we first introduce a monotonic modal logic called MEM that is powerful enough to characterise the existence of an equilibrium model as well as the consequence relation in equilibrium models. The logic MEM thus captures the minimisation attitude that is central in the definition of e ...
Default Reasoning in a Terminological Logic
... the sense that they only allow for monadic predicate symbols, a limited use of negation and no disjunction at all. For our purposes, it is also essential to observe that their monotonic fragment is far less expressive than TLs as, having no term constructors in their syntactic apparatus, they only a ...
... the sense that they only allow for monadic predicate symbols, a limited use of negation and no disjunction at all. For our purposes, it is also essential to observe that their monotonic fragment is far less expressive than TLs as, having no term constructors in their syntactic apparatus, they only a ...
... to a protocol in the class), then A! inference is the appropriate type of inference, where A! consists of all Kripke structures corresponding to protocols. Many of the inference rules that arise in practice are actually A’ rules for some natural choice of A. In fact, the way one typically checks tha ...