First-Order Theorem Proving and Vampire
... Which of the following statements are true? 1. First-order logic is an extension of propositional logic; 2. First-order logic is NP-complete. 3. First-order logic is PSPACE-complete. 4. First-order logic is decidable. 5. In first-order logic you can use quantifiers over sets. 6. One can axiomatise i ...
... Which of the following statements are true? 1. First-order logic is an extension of propositional logic; 2. First-order logic is NP-complete. 3. First-order logic is PSPACE-complete. 4. First-order logic is decidable. 5. In first-order logic you can use quantifiers over sets. 6. One can axiomatise i ...
a-logic - Digital Commons@Wayne State University
... To solve or eliminate the problems confronting standard logic, fundamental changes in basic concepts are required. To be sure, most traditional forms of valid arguments (vs. theorems), are logically valid according to both standard and A-logic. Further, A-logic includes the exact set of “theorems” w ...
... To solve or eliminate the problems confronting standard logic, fundamental changes in basic concepts are required. To be sure, most traditional forms of valid arguments (vs. theorems), are logically valid according to both standard and A-logic. Further, A-logic includes the exact set of “theorems” w ...
Announcement as effort on topological spaces
... set of states and O is a collection of subsets of X (not necessarily a topology, however topological spaces constitute a particular case of subset spaces).1 The elements of O are considered as possible observations or possible observation sets, and the formulas are interpreted not only with respect ...
... set of states and O is a collection of subsets of X (not necessarily a topology, however topological spaces constitute a particular case of subset spaces).1 The elements of O are considered as possible observations or possible observation sets, and the formulas are interpreted not only with respect ...
Graphical Representation of Canonical Proof: Two case studies
... forms of proofs in this formalism are free of bureaucracy, and also canonical from a semantic perspective: for a suitable notion of normal form, they correspond one–to–one with morphisms in free Cartesian closed categories (see e.g. [69]). Another example are Girard’s proof nets for multiplicative l ...
... forms of proofs in this formalism are free of bureaucracy, and also canonical from a semantic perspective: for a suitable notion of normal form, they correspond one–to–one with morphisms in free Cartesian closed categories (see e.g. [69]). Another example are Girard’s proof nets for multiplicative l ...
Structural Proof Theory
... of proof. Sequent calculus, instead, has been developed in various directions. One line leads from Gentzen through Ketonen, Kleene, Dragalin, and Troelstra to what are known as contraction-free systems of sequent calculus. Each of these logicians added some essential discovery, until a gem emerged. ...
... of proof. Sequent calculus, instead, has been developed in various directions. One line leads from Gentzen through Ketonen, Kleene, Dragalin, and Troelstra to what are known as contraction-free systems of sequent calculus. Each of these logicians added some essential discovery, until a gem emerged. ...
Color - Alex Kocurek
... on all L-formulas (i.e., they are L-equivalent), and (b) they disagree in LTS on α. To make showing that such models are L-equivalent easier, we can appeal to the notion of a bisimulation.16 The notion of a bisimulation for first-order modal logic has not been discussed much until recently.17 Below, ...
... on all L-formulas (i.e., they are L-equivalent), and (b) they disagree in LTS on α. To make showing that such models are L-equivalent easier, we can appeal to the notion of a bisimulation.16 The notion of a bisimulation for first-order modal logic has not been discussed much until recently.17 Below, ...
Ribbon Proofs - A Proof System for the Logic of Bunched Implications
... The structure of these proofs extends the notion of box proofs (or ‘Fitch proofs’[10]), but to deal with BI’s substructural nature an additional notion of ‘ribbon’ is used. This technique could be applied in other substructural logics, although we will not attempt that in the present work. Ribbon pr ...
... The structure of these proofs extends the notion of box proofs (or ‘Fitch proofs’[10]), but to deal with BI’s substructural nature an additional notion of ‘ribbon’ is used. This technique could be applied in other substructural logics, although we will not attempt that in the present work. Ribbon pr ...
Interpretability formalized
... and for different purposes. A famous and well known example is an interpretation of hyperbolic geometry in Euclidean geometry (e.g., the Beltrami-Klein model, see, for example, [Gre96]) to show the relative consistency of non-Euclidean geometry. Another example, no less famous, is Gödel’s interpret ...
... and for different purposes. A famous and well known example is an interpretation of hyperbolic geometry in Euclidean geometry (e.g., the Beltrami-Klein model, see, for example, [Gre96]) to show the relative consistency of non-Euclidean geometry. Another example, no less famous, is Gödel’s interpret ...
Predicate Logic
... Binding variables and scope When a quantifier is used on the variable x we say that this occurrence of x is bound. When the occurrence of a variable is not bound by a quantifier or set to a particular value, the variable is said to be free. The part of a logical expression to which a quantifier is a ...
... Binding variables and scope When a quantifier is used on the variable x we say that this occurrence of x is bound. When the occurrence of a variable is not bound by a quantifier or set to a particular value, the variable is said to be free. The part of a logical expression to which a quantifier is a ...
A Unified View of Induction Reasoning for First-Order Logic
... and implicit induction principles, [16, 22, 27, 30, 40, 56] being among the most notable. Other studies have been conducted to reduce the gap between them. Protzen [42] proposed a proof strategy to perform lazy induction on particular explicit induction proofs. Kapur and Subramaniam [29] devised a m ...
... and implicit induction principles, [16, 22, 27, 30, 40, 56] being among the most notable. Other studies have been conducted to reduce the gap between them. Protzen [42] proposed a proof strategy to perform lazy induction on particular explicit induction proofs. Kapur and Subramaniam [29] devised a m ...
Labeled Natural Deduction for Temporal Logics
... A.1 Proofs of Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 ...
... A.1 Proofs of Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 ...
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 ...
Discrete Mathematics
... Introduction Rules And If Li proves P and Lj proves Q, then write from Li Lj have Lk : "P ∧ Q" .. Or (1) If Li proves P , then write from Li have Lk : "P ∨ Q" .. Or (2) If Li proves Q, then write from Li have Lk : "P ∨ Q" .. ...
... Introduction Rules And If Li proves P and Lj proves Q, then write from Li Lj have Lk : "P ∧ Q" .. Or (1) If Li proves P , then write from Li have Lk : "P ∨ Q" .. Or (2) If Li proves Q, then write from Li have Lk : "P ∨ Q" .. ...
JUXTAPOSITION - Brown University
... §1. Introduction. Methods of combining logics are of great interest.1 Formal systems that result from the combination of multiple logical systems into a single system have applications in mathematics, linguistics, and computer science. For example, there are many applications for logics with multipl ...
... §1. Introduction. Methods of combining logics are of great interest.1 Formal systems that result from the combination of multiple logical systems into a single system have applications in mathematics, linguistics, and computer science. For example, there are many applications for logics with multipl ...
A survey on Interactive Theorem Proving
... provide communication with other users and systems; the essential point, however, is that this new medium is active, whereas paper, for instance, is not. A.Asperti ...
... provide communication with other users and systems; the essential point, however, is that this new medium is active, whereas paper, for instance, is not. A.Asperti ...
Hilbert`s Program Then and Now
... Mathematical Society entitled “Axiomatic Thought” [1918a]. It is his first published contribution to mathematical foundations since 1905. In it, he again emphasized the requirement of consistency proofs for axiomatic systems: “The chief requirement of the theory of axioms must go farther [than merel ...
... Mathematical Society entitled “Axiomatic Thought” [1918a]. It is his first published contribution to mathematical foundations since 1905. In it, he again emphasized the requirement of consistency proofs for axiomatic systems: “The chief requirement of the theory of axioms must go farther [than merel ...
Foundations of Databases - Free University of Bozen
... • The extensional schema of P , edb(P ), consists of all extensional relations of P • The intensional schema of P , idb(P ), consists of all intensional relations of P • The schema of P , sch(P ), is the union of edb(P ) and idb(P ). Remarks. ...
... • The extensional schema of P , edb(P ), consists of all extensional relations of P • The intensional schema of P , idb(P ), consists of all intensional relations of P • The schema of P , sch(P ), is the union of edb(P ) and idb(P ). Remarks. ...
A Logical Foundation for Session
... Over the years, computation systems have evolved from monolithic single-threaded machines to concurrent and distributed environments with multiple communicating threads of execution, for which writing correct programs becomes substantially harder than in the more traditional sequential setting. Thes ...
... Over the years, computation systems have evolved from monolithic single-threaded machines to concurrent and distributed environments with multiple communicating threads of execution, for which writing correct programs becomes substantially harder than in the more traditional sequential setting. Thes ...
Hilbert`s Program Then and Now - Philsci
... the philosophy and foundations of mathematics. It was formulated in the early 1920s by German mathematician David Hilbert (1862–1943), and was pursued by him and his collaborators at the University of Göttingen and elsewhere in the 1920s and 1930s. Briefly, Hilbert’s proposal called for a new found ...
... the philosophy and foundations of mathematics. It was formulated in the early 1920s by German mathematician David Hilbert (1862–1943), and was pursued by him and his collaborators at the University of Göttingen and elsewhere in the 1920s and 1930s. Briefly, Hilbert’s proposal called for a new found ...
Simply Logical: Intelligent Reasoning by Example
... question ‘How is this done in Prolog?’, but rather on the question ‘How should I solve this problem, were I to start from scratch?’ In other words, the philosophy of this book is ‘teaching by showing, learning by doing’. This should not be taken to imply that the book is devoid of all theoretical un ...
... question ‘How is this done in Prolog?’, but rather on the question ‘How should I solve this problem, were I to start from scratch?’ In other words, the philosophy of this book is ‘teaching by showing, learning by doing’. This should not be taken to imply that the book is devoid of all theoretical un ...
.pdf
... This paper explores the design of programming logics in which assumptions about the environment can be given explicitly. Such logics allow us to prove that all feasible behaviors of a program satisfy a property, where the characterization of what is feasible is now explicit and subject to change. We ...
... This paper explores the design of programming logics in which assumptions about the environment can be given explicitly. Such logics allow us to prove that all feasible behaviors of a program satisfy a property, where the characterization of what is feasible is now explicit and subject to change. We ...
Full Text - Institute for Logic, Language and Computation
... a formal apparatus to this conception. Just after the Second World War, the field became autonomous with GoodmanÕs seminal article, ÒThe problem of counterfactual conditionalsÓ, which highlights specific problems. Furthermore, the central task is to give an account of the use of conditionals in natu ...
... a formal apparatus to this conception. Just after the Second World War, the field became autonomous with GoodmanÕs seminal article, ÒThe problem of counterfactual conditionalsÓ, which highlights specific problems. Furthermore, the central task is to give an account of the use of conditionals in natu ...
How to Go Nonmonotonic Contents David Makinson
... acting on sets A of formulae to give larger sets Cn(A). In effect, the operation gathers together all the formulae that are consequences of given premises. The two representations of classical consequence are trivially interchangeable. Given a relation |-, we may define the operation Cn by setting C ...
... acting on sets A of formulae to give larger sets Cn(A). In effect, the operation gathers together all the formulae that are consequences of given premises. The two representations of classical consequence are trivially interchangeable. Given a relation |-, we may define the operation Cn by setting C ...
The Premiss-Based Approach to Logical Aggregation Franz Dietrich & Philippe Mongin
... Our results are general enough to cover Mongin’s theorem as a particular case. They also cover several existing theorems, which correspond to another particular case, in which every formula is a premiss; see, e.g., Dietrich and List (2007, forthcoming), Dokow and Holzman (2005, 2006), Gärdenfors (2 ...
... Our results are general enough to cover Mongin’s theorem as a particular case. They also cover several existing theorems, which correspond to another particular case, in which every formula is a premiss; see, e.g., Dietrich and List (2007, forthcoming), Dokow and Holzman (2005, 2006), Gärdenfors (2 ...
Abella: A System for Reasoning about Relational Specifications
... Types in Abella are the simple types; such types are either primitive types or built from two types using the arrow type constructor →. The type constructor → associates to the right, so every type in Abella can be written in the form τ1 → · · · → τn → b (for n ≥ 0) where b is an atomic type that is ...
... Types in Abella are the simple types; such types are either primitive types or built from two types using the arrow type constructor →. The type constructor → associates to the right, so every type in Abella can be written in the form τ1 → · · · → τn → b (for n ≥ 0) where b is an atomic type that is ...