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 ...

Graphical Representation of Canonical Proof: Two case studies

... and cut elimination corresponds to computation. If cut reduction is confluent, then the computation it embodies is deterministic, which in many cases means the proof system may be employed, more or less directly, as a language of computation. One of Girard’s original motivations for proof nets was t ...

... and cut elimination corresponds to computation. If cut reduction is confluent, then the computation it embodies is deterministic, which in many cases means the proof system may be employed, more or less directly, as a language of computation. One of Girard’s original motivations for proof nets was t ...

Limitations

... Structural Induction • Useful for proving properties about languages. • Useful for proving properties about algorithms defined on languages. • A generalization of the induction you know and love. ...

... Structural Induction • Useful for proving properties about languages. • Useful for proving properties about algorithms defined on languages. • A generalization of the induction you know and love. ...

a semantic perspective - Institute for Logic, Language and

... combination of basic modal formulas, or (most interesting of all) a formula prefixed by a diamond or a box. There is redundancy in the way we have defined basic modal languages: we don’t need all these boolean connectives as primitives, and it will follow from the satisfaction definition given below ...

... combination of basic modal formulas, or (most interesting of all) a formula prefixed by a diamond or a box. There is redundancy in the way we have defined basic modal languages: we don’t need all these boolean connectives as primitives, and it will follow from the satisfaction definition given below ...

Principia Logico-Metaphysica (Draft/Excerpt)

... In this and subsequent chapters, our metalanguage makes use of informal notions and principles about numbers and sets so as to more precisely articulate and render certain definitions. These are not primitive notions or principles of our metaphysical system; ultimately, they are to be understood by ...

... In this and subsequent chapters, our metalanguage makes use of informal notions and principles about numbers and sets so as to more precisely articulate and render certain definitions. These are not primitive notions or principles of our metaphysical system; ultimately, they are to be understood by ...

Characterstics of Ternary Semirings

... ternary rings. D. Madhusudhana Rao and G. Srinvasa Rao [9, 10] introduced some special element in ternary semiring and studied about ternary semirings. Our main purpose in this paper is to characterize the ternary semirings. ...

... ternary rings. D. Madhusudhana Rao and G. Srinvasa Rao [9, 10] introduced some special element in ternary semiring and studied about ternary semirings. Our main purpose in this paper is to characterize the ternary semirings. ...

Ribbon Proofs - A Proof System for the Logic of Bunched Implications

... O’Hearn, Reynolds, Yang and others ([24, 29, 40]) in which the resources concerned are computational, such as memory cells; this provides a powerful practical motivation for studying the logic. However the two inference systems in [32] do not directly reflect these semantic intuitions about BI, and ...

... O’Hearn, Reynolds, Yang and others ([24, 29, 40]) in which the resources concerned are computational, such as memory cells; this provides a powerful practical motivation for studying the logic. However the two inference systems in [32] do not directly reflect these semantic intuitions about BI, and ...

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. ...

SEQUENT SYSTEMS FOR MODAL LOGICS

... [I]ntroductions represent, as it were, the ‘definitions’ of the symbols concerned, and the eliminations are no more, in the final analysis, than the consequences of these definitions. To qualify as a definition of a logical operation, an introduction schema must satisfy certain adequacy criteria. Su ...

... [I]ntroductions represent, as it were, the ‘definitions’ of the symbols concerned, and the eliminations are no more, in the final analysis, than the consequences of these definitions. To qualify as a definition of a logical operation, an introduction schema must satisfy certain adequacy criteria. Su ...

Labeled Natural Deduction for Temporal Logics

... that temporal logics have been studied for many years, their theoretical analysis is far from being concluded. In particular, a satisfactory proof-theoretical analysis for temporal logics is still lacking. This is especially true in the case of branchingtime logics, as shown by the fact that for one ...

... that temporal logics have been studied for many years, their theoretical analysis is far from being concluded. In particular, a satisfactory proof-theoretical analysis for temporal logics is still lacking. This is especially true in the case of branchingtime logics, as shown by the fact that for one ...

Relevant and Substructural Logics

... and with hypotheses from among the set X. A proof from hypotheses is simply a list of formulas, each of which is either an hypothesis, an axiom, or one which follows from earlier formulas in the list by means of a rule. In Orlov’s system, the only rule is modus ponens. We will see later that this is ...

... and with hypotheses from among the set X. A proof from hypotheses is simply a list of formulas, each of which is either an hypothesis, an axiom, or one which follows from earlier formulas in the list by means of a rule. In Orlov’s system, the only rule is modus ponens. We will see later that this is ...

Logic Part II: Intuitionistic Logic and Natural Deduction

... in many elds of mathematics, there are contradictory propositions from which anything is derivable ...

... in many elds of mathematics, there are contradictory propositions from which anything is derivable ...

Functional Dependencies in a Relational Database and

... In our example, there are two atoms X (namely, A and B) that satisfy a, and two atoms X (namely, A and D ) that satisfy b. The only atom X that satisfies both a and b is A. (If there were several atoms X that satisfied both a and b, the algorithm would now arbitrarily select one of them.) In the nex ...

... In our example, there are two atoms X (namely, A and B) that satisfy a, and two atoms X (namely, A and D ) that satisfy b. The only atom X that satisfies both a and b is A. (If there were several atoms X that satisfied both a and b, the algorithm would now arbitrarily select one of them.) In the nex ...

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 ...

proof terms for classical derivations

... of p. The second assumed p twice (tagging these assumptions with x and y), conjoined their result (in that order) and disharged them in turn (also in that order). Seeing this, you may realise that there are two other proofs of the same formula. One where the conjuncts are formed in the other order ( ...

... of p. The second assumed p twice (tagging these assumptions with x and y), conjoined their result (in that order) and disharged them in turn (also in that order). Seeing this, you may realise that there are two other proofs of the same formula. One where the conjuncts are formed in the other order ( ...

Classical Propositional Logic

... A Henkin-style Completeness Proof for Natural Deduction Computability ...

... A Henkin-style Completeness Proof for Natural Deduction Computability ...

code-carrying theory - Computer Science at RPI

... reachable, his door was always open. We held hundreds of meetings, sometimes discussing for hours either in his office or through email. There were times that he even worked harder on this project than I did. He is a good friend, a thoughtful person, and a real problem solver. I will always miss his ...

... reachable, his door was always open. We held hundreds of meetings, sometimes discussing for hours either in his office or through email. There were times that he even worked harder on this project than I did. He is a good friend, a thoughtful person, and a real problem solver. I will always miss his ...

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 ...

relevance logic - Consequently.org

... usage to an ordinary English word (even requiring in this technical usage that ‘implication’ be a metalinguistic relation between sentences), the point is that relevance logicians by and large believe we are using ‘implies’ in the ordinary non-technical sense, in which a sentence like (3) might be t ...

... usage to an ordinary English word (even requiring in this technical usage that ‘implication’ be a metalinguistic relation between sentences), the point is that relevance logicians by and large believe we are using ‘implies’ in the ordinary non-technical sense, in which a sentence like (3) might be t ...

Termination of Higher-order Rewrite Systems

... resulting expression may be rewritten again and again, giving rise to a reduction sequence. Such a sequence can be seen as a computation. Certain expressions are considered as results, or normal forms. A computation terminates successfully when such a normal form has been reached. This situation can ...

... resulting expression may be rewritten again and again, giving rise to a reduction sequence. Such a sequence can be seen as a computation. Certain expressions are considered as results, or normal forms. A computation terminates successfully when such a normal form has been reached. This situation can ...

Inductive Types in Constructive Languages

... This dissertation deals with constructive languages: languages for the formal expression of mathematical constructions. The concept of construction does not only encompass computations, as expressed in programming languages, but also propositions and proofs, as expressed in a mathematical logic, and ...

... This dissertation deals with constructive languages: languages for the formal expression of mathematical constructions. The concept of construction does not only encompass computations, as expressed in programming languages, but also propositions and proofs, as expressed in a mathematical logic, and ...

Everything Else Being Equal: A Modal Logic for Ceteris Paribus

... the first half of the 20th century, especially in economics and social choice theory [28]. In logic, Halldén [17] initiated a field of research that was subsequently championed in von Wright [36], which is usually taken to be the seminal work in preference logic. The present paper presents a modal l ...

... the first half of the 20th century, especially in economics and social choice theory [28]. In logic, Halldén [17] initiated a field of research that was subsequently championed in von Wright [36], which is usually taken to be the seminal work in preference logic. The present paper presents a modal l ...

Discrete Mathematics for Computer Science Some Notes

... such a course and I will try justifying this assertion below. The main reason is that, based on my experience of more than twenty five years of teaching, I have found that the majority of the students find it very difficult to present an argument in a rigorous fashion. The notion of a proof is somet ...

... such a course and I will try justifying this assertion below. The main reason is that, based on my experience of more than twenty five years of teaching, I have found that the majority of the students find it very difficult to present an argument in a rigorous fashion. The notion of a proof is somet ...

A Course in Modal Logic - Sun Yat

... concepts, whereas these differences are actually supported by different intuitive semantics (even philosophical background). The reader who is interested in this can refer to the related literature. (Ⅱ) The cardinality of At is finite or countable infinite, but, in fact, most of results given in thi ...

... concepts, whereas these differences are actually supported by different intuitive semantics (even philosophical background). The reader who is interested in this can refer to the related literature. (Ⅱ) The cardinality of At is finite or countable infinite, but, in fact, most of results given in thi ...

In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are a set of axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano. These axioms have been used nearly unchanged in a number of metamathematical investigations, including research into fundamental questions of whether number theory is consistent and complete.The need to formalize arithmetic was not well appreciated until the work of Hermann Grassmann, who showed in the 1860s that many facts in arithmetic could be derived from more basic facts about the successor operation and induction. In 1881, Charles Sanders Peirce provided an axiomatization of natural-number arithmetic. In 1888, Richard Dedekind proposed another axiomatization of natural-number arithmetic, and in 1889 Peano published a more precisely formulated version of them as a collection of axioms in his book, The principles of arithmetic presented by a new method (Latin: Arithmetices principia, nova methodo exposita).The Peano axioms contain three types of statements. The first axiom asserts the existence of at least one member of the set ""number"". The next four are general statements about equality; in modern treatments these are often not taken as part of the Peano axioms, but rather as axioms of the ""underlying logic"". The next three axioms are first-order statements about natural numbers expressing the fundamental properties of the successor operation. The ninth, final axiom is a second order statement of the principle of mathematical induction over the natural numbers. A weaker first-order system called Peano arithmetic is obtained by explicitly adding the addition and multiplication operation symbols and replacing the second-order induction axiom with a first-order axiom schema.