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

... set of theorems, the same semantical foundations, and use the same concepts of validity and logical truth though they differ in notation, choices of primitives and axioms, diagramatic devices, modes of introduction and explication, etc. This standard logic is an enormous advance over any preceding s ...

... set of theorems, the same semantical foundations, and use the same concepts of validity and logical truth though they differ in notation, choices of primitives and axioms, diagramatic devices, modes of introduction and explication, etc. This standard logic is an enormous advance over any preceding s ...

Assumption-Based Argumentation with Preferences

... We extend ABA frameworks (L, R, A,¯¯¯) with a preference ordering 6 on the set A of assumptions to obtain ABA+ frameworks (L, R, A,¯¯¯, 6), as follows. Definition 1. An ABA+ framework is a tuple (L, R, A,¯¯¯, 6), where (L, R, A,¯¯¯) is an ABA framework and 6 is a transitive binary relation on A. We ...

... We extend ABA frameworks (L, R, A,¯¯¯) with a preference ordering 6 on the set A of assumptions to obtain ABA+ frameworks (L, R, A,¯¯¯, 6), as follows. Definition 1. An ABA+ framework is a tuple (L, R, A,¯¯¯, 6), where (L, R, A,¯¯¯) is an ABA framework and 6 is a transitive binary relation on A. We ...

Labeled Natural Deduction for Temporal Logics

... a great importance in computer science: applications include its use as a tool for the specification and verification of programs and protocols [18], in the study and development of temporal databases [39], as a framework within which to define the semantics of temporal expressions in natural langua ...

... a great importance in computer science: applications include its use as a tool for the specification and verification of programs and protocols [18], in the study and development of temporal databases [39], as a framework within which to define the semantics of temporal expressions in natural langua ...

Announcement as effort on topological spaces

... formula like Ki K̂j Ki p, for ‘agent i knows that agent j considers possible that agent i knows proposition p’. If this is true for a triple (x, Ui , Uj ), then K̂j Ki p must be true for any y ∈ Ui ; but y may not be in Uj , in which case (y, Ui , Uj ) is not well-defined: we cannot interpret K̂j Ki ...

... formula like Ki K̂j Ki p, for ‘agent i knows that agent j considers possible that agent i knows proposition p’. If this is true for a triple (x, Ui , Uj ), then K̂j Ki p must be true for any y ∈ Ui ; but y may not be in Uj , in which case (y, Ui , Uj ) is not well-defined: we cannot interpret K̂j Ki ...

Some Aspects and Examples of In nity Notions T ZF

... Proof: Let f be a noninjective surjection from x into x. Furthermore, let a be a point that witnesses the noninjectivity of f . That means: 9x; y (x = 6 y ^ f (x) = f (y) = a). Now we de ne a function g from x onto ! as follows: g(x) = minfn 2 ! : f n (x) = ag if there is such ...

... Proof: Let f be a noninjective surjection from x into x. Furthermore, let a be a point that witnesses the noninjectivity of f . That means: 9x; y (x = 6 y ^ f (x) = f (y) = a). Now we de ne a function g from x onto ! as follows: g(x) = minfn 2 ! : f n (x) = ag if there is such ...

5 model theory of modal logic

... between the (first-order) Kripke structure semantics and the (second-order) frame semantics, give rise to very distinct model theoretic flavours, each with their own tradition in the model theory of modal logic. Still, these two semantics meet through the notion of a general frame (closely related t ...

... between the (first-order) Kripke structure semantics and the (second-order) frame semantics, give rise to very distinct model theoretic flavours, each with their own tradition in the model theory of modal logic. Still, these two semantics meet through the notion of a general frame (closely related t ...

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. ...

Introduction to Computational Logic

... This time the claim involves a boolean variable x and the proof proceeds by case analysis on x. Since reflexivity performs simplification automatically, we have omitted the tactic simpl. It is important that with Coq you step back and forth in the proof script and observe what happens. This way you ...

... This time the claim involves a boolean variable x and the proof proceeds by case analysis on x. Since reflexivity performs simplification automatically, we have omitted the tactic simpl. It is important that with Coq you step back and forth in the proof script and observe what happens. This way you ...

Bridge to Abstract Mathematics: Mathematical Proof and

... countably infinite collections of sets. The main emphasis here is on standard approaches to proving set inclusion (e.g., the "choose" method) and set equality (e.g., mutual inclusion), but we manage also, through the many solved examples, to anticipate additional techniques of proof that are studied ...

... countably infinite collections of sets. The main emphasis here is on standard approaches to proving set inclusion (e.g., the "choose" method) and set equality (e.g., mutual inclusion), but we manage also, through the many solved examples, to anticipate additional techniques of proof that are studied ...

Termination of Higher-order Rewrite Systems

... Another important question is, whether the normal form of an expression is uniquely determined. Weak normalization still admits that a term reduces to two dierent normal forms. A rewrite system is con uent if for all objects r; s; t such that both r s and r t, there exists a u such that s u a ...

... Another important question is, whether the normal form of an expression is uniquely determined. Weak normalization still admits that a term reduces to two dierent normal forms. A rewrite system is con uent if for all objects r; s; t such that both r s and r t, there exists a u such that s u a ...

Sample pages 2 PDF

... Therefore, ¬A ⇒ B ∧ C ⇔ D effectively means ((¬A) ⇒ (B ∧ C)) ⇔ D. Although we can reduce brackets to a minimum, we usually use brackets to distinguish between ∧ and ∨, and between ⇒ and ⇔. Therefore, we would usually write A ∨ (B ∧ C) even if A ∨ B ∧ C would do. Similarly, we write A ⇔ (B ⇒ C) when ...

... Therefore, ¬A ⇒ B ∧ C ⇔ D effectively means ((¬A) ⇒ (B ∧ C)) ⇔ D. Although we can reduce brackets to a minimum, we usually use brackets to distinguish between ∧ and ∨, and between ⇒ and ⇔. Therefore, we would usually write A ∨ (B ∧ C) even if A ∨ B ∧ C would do. Similarly, we write A ⇔ (B ⇒ C) when ...

A joint logic of problems and propositions, a modified BHK

... Atomic formulas: problem symbols, propositional symbols (possibly depending on variables that all range over the same domain of discourse) and the constant ⊥ Formulas are of two types: problems (denoted by Greek letters) and propositions (denoted by Roman letters) Classical connectives: propositions ...

... Atomic formulas: problem symbols, propositional symbols (possibly depending on variables that all range over the same domain of discourse) and the constant ⊥ Formulas are of two types: problems (denoted by Greek letters) and propositions (denoted by Roman letters) Classical connectives: propositions ...

some results on locally finitely presentable categories

... of the 2-categories Lexop and LFP, the 2-category of l.f.p. categories, with morphisms the functors preserving limits and filtered colimits. This duality may be known to many people; most of it is already in [GU], and the remaining parts are more or less folklore. Some technical results used later a ...

... of the 2-categories Lexop and LFP, the 2-category of l.f.p. categories, with morphisms the functors preserving limits and filtered colimits. This duality may be known to many people; most of it is already in [GU], and the remaining parts are more or less folklore. Some technical results used later a ...

Interpretability formalized

... As with all sciences, mathematics aims at a better description and understanding of reality. Now, the logician asks: what is mathematical reality? Mathematics deals with numbers, functions, shapes, circles, sets, etc. But who has ever touched a number? Who has ever seen a real circle? The firm and u ...

... As with all sciences, mathematics aims at a better description and understanding of reality. Now, the logician asks: what is mathematical reality? Mathematics deals with numbers, functions, shapes, circles, sets, etc. But who has ever touched a number? Who has ever seen a real circle? The firm and u ...

Functional Dependencies in a Relational Database and

... responding dependency statement. Let t be a truth assignment, that is, a mapping that assigns to each propositional variable either the value 0 (false) or 1 (true), The propositional statement A, . . . A,,, .$ B, . . ‘ B, has truth value 0 under truth assignment t if each of A,, . . ., A,,, has the ...

... responding dependency statement. Let t be a truth assignment, that is, a mapping that assigns to each propositional variable either the value 0 (false) or 1 (true), The propositional statement A, . . . A,,, .$ B, . . ‘ B, has truth value 0 under truth assignment t if each of A,, . . ., A,,, has the ...

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" .. ...

SEQUENT SYSTEMS FOR MODAL LOGICS

... of a Gentzen sequent to modal logics have to do with the idea of defining the logical operations by means of introduction schemata (together with structural assumptions about derivability formulated in terms of structural rules). This ‘anti-realistic’ conception of the meaning of the logical operati ...

... of a Gentzen sequent to modal logics have to do with the idea of defining the logical operations by means of introduction schemata (together with structural assumptions about derivability formulated in terms of structural rules). This ‘anti-realistic’ conception of the meaning of the logical operati ...

Principia Logico-Metaphysica (Draft/Excerpt)

... Consequently, this excerpt omits the Preface, Acknowledgments, Part I (Chapters 1-6), Part II/Chapters 15–16 (which are being reworked), Part III (which is mostly unwritten), and some Appendices in Part IV. The excerpt contains references to some of this omitted content. The work is ongoing and so t ...

... Consequently, this excerpt omits the Preface, Acknowledgments, Part I (Chapters 1-6), Part II/Chapters 15–16 (which are being reworked), Part III (which is mostly unwritten), and some Appendices in Part IV. The excerpt contains references to some of this omitted content. The work is ongoing and so t ...

Introduction to Modal and Temporal Logic

... l = 0: So Γ ⊢ ψ because ψ is an axiom schema instance. By Eg 1, Ex 1, Ex 2, Eg 2, we know ∅ |= ψ for every axiom schema instance ψ, hence Γ |= ψ. Ind. Hyp. : Theorem holds for all derivations of length less than some k > 0. Ind. Step: Suppose Γ ⊢ ψ has a derivation of length k. Bottom-most rule? MP: ...

... l = 0: So Γ ⊢ ψ because ψ is an axiom schema instance. By Eg 1, Ex 1, Ex 2, Eg 2, we know ∅ |= ψ for every axiom schema instance ψ, hence Γ |= ψ. Ind. Hyp. : Theorem holds for all derivations of length less than some k > 0. Ind. Step: Suppose Γ ⊢ ψ has a derivation of length k. Bottom-most rule? MP: ...

A Judgmental Reconstruction of Modal Logic

... explanation of conjunction. We have said that a verification of A ∧ B consists of a verification of A and a verification of B. Local completeness entails that it is always possible to bring the verification of A ∧ B into this form by a local expansion. To summarize, logic is based on the notion of j ...

... explanation of conjunction. We have said that a verification of A ∧ B consists of a verification of A and a verification of B. Local completeness entails that it is always possible to bring the verification of A ∧ B into this form by a local expansion. To summarize, logic is based on the notion of j ...

Logic Part II: Intuitionistic Logic and Natural Deduction

... 2. This proof contains of a proof of a. 3. It also contains a proof of b . 4. So if we take the proof of b and put it together with the proof of a, we obtain a proof of b ...

... 2. This proof contains of a proof of a. 3. It also contains a proof of b . 4. So if we take the proof of b and put it together with the proof of a, we obtain a proof of b ...

Dialectica Interpretations A Categorical Analysis

... the papers, preprints or technical reports is preceded by a short declaration that summarizes the current status (published, accepted or submitted), contributions of the authors if more than one author, main results and relation to other work. To help the reader, each declaration also contains a sho ...

... the papers, preprints or technical reports is preceded by a short declaration that summarizes the current status (published, accepted or submitted), contributions of the authors if more than one author, main results and relation to other work. To help the reader, each declaration also contains a sho ...

code-carrying theory - Computer Science at RPI

... proofs, correctness, recursion and generic concepts became an important part of my life during the next five years. I had the privilege to be supervised by one of the world’s top researchers who taught me to be precise and patient. I will always remember him as a meticulous person about every detail ...

... proofs, correctness, recursion and generic concepts became an important part of my life during the next five years. I had the privilege to be supervised by one of the world’s top researchers who taught me to be precise and patient. I will always remember him as a meticulous person about every detail ...

a PDF file of the textbook - U of L Class Index

... Remark 1.3. You should not confuse the idea of an assertion that can be true or false with the difference between fact and opinion. Assertions will often express things that would count as facts (such as “Pierre Trudeau was born in Quebec” or “Pierre Trudeau liked almonds”), but they can also expres ...

... Remark 1.3. You should not confuse the idea of an assertion that can be true or false with the difference between fact and opinion. Assertions will often express things that would count as facts (such as “Pierre Trudeau was born in Quebec” or “Pierre Trudeau liked almonds”), but they can also expres ...