Specifying and Verifying Fault-Tolerant Systems
... ord For each general g, the value of ord [g] denotes the order chosen by general g, or the string “?” if g has not yet chosen an order. now The variable now will be a real number that denotes the current time. In our informal discussion, we describe the values that variables will have in a behavior ...
... ord For each general g, the value of ord [g] denotes the order chosen by general g, or the string “?” if g has not yet chosen an order. now The variable now will be a real number that denotes the current time. In our informal discussion, we describe the values that variables will have in a behavior ...
Preferences and Unrestricted Rebut
... Theorem 2. Let E be the grounded extension of (Ar , def ). E is closed under subarguments. That is, if A ∈ E, then ∀A0 ∈ Sub(A), A0 ∈ E. Proof. Let A ∈ E and A0 ∈ Sub(A). Since E is admissible and A ∈ E, then every defeater B of A is defeated by an argument in E. Now, let B be an arbitrary defeater ...
... Theorem 2. Let E be the grounded extension of (Ar , def ). E is closed under subarguments. That is, if A ∈ E, then ∀A0 ∈ Sub(A), A0 ∈ E. Proof. Let A ∈ E and A0 ∈ Sub(A). Since E is admissible and A ∈ E, then every defeater B of A is defeated by an argument in E. Now, let B be an arbitrary defeater ...
beliefrevision , epistemicconditionals andtheramseytest
... certainty, new evidence may undermine one’s original certainties without being logically incompatible with them. For instance, if you are certain that most F’s are G’s, your certainty may well be undermined by repeated observations of F’s that are non-G’s. Clearly, such a nonBayesian behavior would ...
... certainty, new evidence may undermine one’s original certainties without being logically incompatible with them. For instance, if you are certain that most F’s are G’s, your certainty may well be undermined by repeated observations of F’s that are non-G’s. Clearly, such a nonBayesian behavior would ...
Hilbert`s Program Then and Now - Philsci
... logic-free intuitive capacity which guarantees certainty of knowledge about formulas and proofs arrived at by such syntactic operations. Mathematics itself, however, operates with abstract concepts, e.g., quantifiers, sets, functions, and uses logical inference based on principles such as mathematic ...
... logic-free intuitive capacity which guarantees certainty of knowledge about formulas and proofs arrived at by such syntactic operations. Mathematics itself, however, operates with abstract concepts, e.g., quantifiers, sets, functions, and uses logical inference based on principles such as mathematic ...
Reading
... is, strictly speaking, a distinct principle from HP, since HP = AX≈Y and HP2 = AY≈X (further, neither HP → HP2 nor HP2 → HP is a logical truth, since HP and HP2 involve distinct abstraction operators @X≈Y and @Y≈X). We shall address subtle issues regarding the equivalence of abstraction principles b ...
... is, strictly speaking, a distinct principle from HP, since HP = AX≈Y and HP2 = AY≈X (further, neither HP → HP2 nor HP2 → HP is a logical truth, since HP and HP2 involve distinct abstraction operators @X≈Y and @Y≈X). We shall address subtle issues regarding the equivalence of abstraction principles b ...
The Computer Modelling of Mathematical Reasoning Alan Bundy
... This book started as notes for a postgraduate course in Mathematical Reasoning given in the Department of Artificial Intelligence at Edinburgh from 1979 onwards. Students on the course are drawn from a wide range of backgrounds: Psychology, Computer Science, Mathematics, Education, etc. The first dr ...
... This book started as notes for a postgraduate course in Mathematical Reasoning given in the Department of Artificial Intelligence at Edinburgh from 1979 onwards. Students on the course are drawn from a wide range of backgrounds: Psychology, Computer Science, Mathematics, Education, etc. The first dr ...
Duplication of directed graphs and exponential blow up of
... patterns might vary from theorem to theorem. If we consider now formal systems of deduction, we might see that they forbid the creation of some of these patterns, and if so we will never be able in a feasible time to show inside them some of their theorems. At the moment we know a relatively small n ...
... patterns might vary from theorem to theorem. If we consider now formal systems of deduction, we might see that they forbid the creation of some of these patterns, and if so we will never be able in a feasible time to show inside them some of their theorems. At the moment we know a relatively small n ...
Completeness theorems and lambda
... The problem of impredicativity was formulated through discussions between Poincaré and Russell Poincaré argued that impredicative definitions are “circular” and should be avoided More than 100 years later, this is still one of the most important problem in proof theory ...
... The problem of impredicativity was formulated through discussions between Poincaré and Russell Poincaré argued that impredicative definitions are “circular” and should be avoided More than 100 years later, this is still one of the most important problem in proof theory ...
logic for the mathematical
... Actually, in that argument, the word “should” is probably better left out. Mostly, we want to deal with statements which simply state some kind of claimed fact, statements which are clearly either true or false, though which of the two might not be easy to determine. Such statements are often called ...
... Actually, in that argument, the word “should” is probably better left out. Mostly, we want to deal with statements which simply state some kind of claimed fact, statements which are clearly either true or false, though which of the two might not be easy to determine. Such statements are often called ...
Mathematical Logic
... It is an easy exercise to show that the usual equality axioms can be derived. All these axioms can be seen as special cases of a general scheme, that of an inductively defined predicate, which is defined by some introduction rules and one elimination rule. We will study this kind of definition in fu ...
... It is an easy exercise to show that the usual equality axioms can be derived. All these axioms can be seen as special cases of a general scheme, that of an inductively defined predicate, which is defined by some introduction rules and one elimination rule. We will study this kind of definition in fu ...
Lecture Notes on the Lambda Calculus
... What is a function? In modern mathematics, the prevalent notion is that of “functions as graphs”: each function f has a fixed domain X and codomain Y , and a function f : X → Y is a set of pairs f ⊆ X × Y such that for each x ∈ X, there exists exactly one y ∈ Y such that (x, y) ∈ f . Two functions f ...
... What is a function? In modern mathematics, the prevalent notion is that of “functions as graphs”: each function f has a fixed domain X and codomain Y , and a function f : X → Y is a set of pairs f ⊆ X × Y such that for each x ∈ X, there exists exactly one y ∈ Y such that (x, y) ∈ f . Two functions f ...
proof terms for classical derivations
... Variables annotate assumptions, and the term constructors of pairing and λ-abstraction correspond to the introduction of conjunctions and conditionals respectively. Now the terms corresponding to the proofs bear the marks of the different proof behaviour. The first proof took an assumption p to p ∧ ...
... Variables annotate assumptions, and the term constructors of pairing and λ-abstraction correspond to the introduction of conjunctions and conditionals respectively. Now the terms corresponding to the proofs bear the marks of the different proof behaviour. The first proof took an assumption p to p ∧ ...
Discrete Mathematics for Computer Science Some Notes
... P then Q). Some authors use the symbol → and write an implication as P → Q. We do not like to use this notation because the symbol → is already used in the notation for functions (f : A → B). We will mostly use the symbol ⇒. We also have the atomic statements ⊥ (falsity), which corresponds to false ...
... P then Q). Some authors use the symbol → and write an implication as P → Q. We do not like to use this notation because the symbol → is already used in the notation for functions (f : A → B). We will mostly use the symbol ⇒. We also have the atomic statements ⊥ (falsity), which corresponds to false ...
The History of Categorical Logic
... continuous functions, the category T opGrp of topological groups with continuous homomorphisms and the category Ban of Banach spaces with linear transformations with norm at most 1. This is a surprisingly short list of examples. They give more examples by defining the notion of a subcategory in the ...
... continuous functions, the category T opGrp of topological groups with continuous homomorphisms and the category Ban of Banach spaces with linear transformations with norm at most 1. This is a surprisingly short list of examples. They give more examples by defining the notion of a subcategory in the ...
The Deduction Rule and Linear and Near
... formulas, each of which is one of the Ai ’s, is an axiom, or is inferred by modus ponens from earlier formulas, such that B is the final formula of the proof. Although we have not specified the axiom schemata to be used in a Frege proof system, it is easy to see that different choices of axiom schem ...
... formulas, each of which is one of the Ai ’s, is an axiom, or is inferred by modus ponens from earlier formulas, such that B is the final formula of the proof. Although we have not specified the axiom schemata to be used in a Frege proof system, it is easy to see that different choices of axiom schem ...
Horn Belief Contraction: Remainders, Envelopes and Complexity
... update a set of beliefs when new information is obtained that may be inconsistent with the current beliefs (Fermé and Hansson 2011; Hansson 1999; Peppas 2008). The standard approach, called the AGM approach after (Alchourrón, Gärdenfors, and Makinson 1985), is to formulate postulates that need to ...
... update a set of beliefs when new information is obtained that may be inconsistent with the current beliefs (Fermé and Hansson 2011; Hansson 1999; Peppas 2008). The standard approach, called the AGM approach after (Alchourrón, Gärdenfors, and Makinson 1985), is to formulate postulates that need to ...
Goal-directed Proof Theory
... The concept of goal directed computation we adopt can also be seen as a generalization of the notion of uniform proof as introduced in [Miller et al. 91]. As far as we know, a goal-directed presentation have been given of (fragments of) intuitionistic logic [Gabbay and Reyle 84],[Miller 89], [McCart ...
... The concept of goal directed computation we adopt can also be seen as a generalization of the notion of uniform proof as introduced in [Miller et al. 91]. As far as we know, a goal-directed presentation have been given of (fragments of) intuitionistic logic [Gabbay and Reyle 84],[Miller 89], [McCart ...
Horn Belief Contraction: Remainders, Envelopes and Complexity
... this holds for full meet contraction and, in an asymptotic sense, for most maxichoice and most partial meet contractions, both based on remainders and on weak remainders.2 Our result is based on a new blow-up result for computing Horn envelopes (also called Horn LUB’s), which was obtained in connect ...
... this holds for full meet contraction and, in an asymptotic sense, for most maxichoice and most partial meet contractions, both based on remainders and on weak remainders.2 Our result is based on a new blow-up result for computing Horn envelopes (also called Horn LUB’s), which was obtained in connect ...
1 Introduction to Categories and Categorical Logic
... developed. Thus for example, Top is the context for general topology, Grp is the context for group theory, etc. On the other hand, the last two examples illustrate that many important mathematical structures themselves appear as categories of particular kinds. The fact that two such different kinds ...
... developed. Thus for example, Top is the context for general topology, Grp is the context for group theory, etc. On the other hand, the last two examples illustrate that many important mathematical structures themselves appear as categories of particular kinds. The fact that two such different kinds ...