a-logic - Digital Commons@Wayne State University
... he standard logic today is the logic of Frege’s Begriffschrift (1879) and subsequently of Russell and Whitehead’s great book, Principia Mathematica (1913) of Quine’s Mathematical Logic (1940) and Methods of Logic (4th ed.,1982) and of hundreds of other textbooks and treatises which have the same set ...
... he standard logic today is the logic of Frege’s Begriffschrift (1879) and subsequently of Russell and Whitehead’s great book, Principia Mathematica (1913) of Quine’s Mathematical Logic (1940) and Methods of Logic (4th ed.,1982) and of hundreds of other textbooks and treatises which have the same set ...
Independence logic and tuple existence atoms
... Definition R relation, ~x , ~y , ~z tuples of attributes. Then R |= ~x ~y | ~z if and only if, for all r , r 0 ∈ R such that r (~x ) = r 0 (~x ) there exists a r 00 ∈ R such that r 00 (~x ~y ) = r (~x ~y ) and r 00 (~x ~z ) = r (~x ~z ). Huge literature on the topic; If ~x ~y ~z contains all attri ...
... Definition R relation, ~x , ~y , ~z tuples of attributes. Then R |= ~x ~y | ~z if and only if, for all r , r 0 ∈ R such that r (~x ) = r 0 (~x ) there exists a r 00 ∈ R such that r 00 (~x ~y ) = r (~x ~y ) and r 00 (~x ~z ) = r (~x ~z ). Huge literature on the topic; If ~x ~y ~z contains all attri ...
Predicate Logic
... This means that ∀xP (x) ∧ ∀xQ(x) is true. If ∀xP (x) ∧ ∀xQ(x) is true, then ∀x(P (x) ∧ Q(x)) is true. Proof: Suppose that ∀xP (x) ∧ ∀xQ(x) is true. It follows that ∀xP (x) is true and ∀xQ(x) is true. So, if a is in the domain, then P (a) is true and Q(a) is true. It follows that if a is in the domai ...
... This means that ∀xP (x) ∧ ∀xQ(x) is true. If ∀xP (x) ∧ ∀xQ(x) is true, then ∀x(P (x) ∧ Q(x)) is true. Proof: Suppose that ∀xP (x) ∧ ∀xQ(x) is true. It follows that ∀xP (x) is true and ∀xQ(x) is true. So, if a is in the domain, then P (a) is true and Q(a) is true. It follows that if a is in the domai ...
Introduction to Computational Logic
... and interactive theorem proving with the proof assistant Coq. At Saarland University the course is taught in this format since 2010. Students are expected to be familiar with basic functional programming and the structure of mathematical definitions and proofs. Talented students at Saarland Universi ...
... and interactive theorem proving with the proof assistant Coq. At Saarland University the course is taught in this format since 2010. Students are expected to be familiar with basic functional programming and the structure of mathematical definitions and proofs. Talented students at Saarland Universi ...
PowerPoint
... • Reasoning with ground clauses identical to reasoning with propositional clauses (treating each term like a proposition). • An interpretation is defined to be a model for a non-ground clause if it is a model for every ground instance. • So to show M = {obliged_to(compaq, fred), customer_of(fred, co ...
... • Reasoning with ground clauses identical to reasoning with propositional clauses (treating each term like a proposition). • An interpretation is defined to be a model for a non-ground clause if it is a model for every ground instance. • So to show M = {obliged_to(compaq, fred), customer_of(fred, co ...
Assumption-Based Argumentation with Preferences
... [13] for a recent overview). Whereas in AA arguments are atomic, structured argumentation formalisms usually specify the internal structure of arguments and/or attacks. Assumption-Based Argumentation (ABA) [15, 35, 33, 34, 58, 59, 23] is one particular structured argumentation formalism, where knowl ...
... [13] for a recent overview). Whereas in AA arguments are atomic, structured argumentation formalisms usually specify the internal structure of arguments and/or attacks. Assumption-Based Argumentation (ABA) [15, 35, 33, 34, 58, 59, 23] is one particular structured argumentation formalism, where knowl ...
The Liar Paradox: A Consistent and Semantically Closed Solution
... This thesis develops a new approach to the formal de nition of a truth predicate that allows a consistent, semantically closed de nition within classical logic. The approach is built on an analysis of structural properties of languages that make Liar Sentences and the paradoxical argument possible. ...
... This thesis develops a new approach to the formal de nition of a truth predicate that allows a consistent, semantically closed de nition within classical logic. The approach is built on an analysis of structural properties of languages that make Liar Sentences and the paradoxical argument possible. ...
Bridge to Abstract Mathematics: Mathematical Proof and
... Such discovery devices as examples, pictures, analogies, and counterexamples are brought into play. Rhetorical questions are employed often in this chapter to instill in the student the habit of thinking aggressively, of looking for questions as well as answers. Also, a case is made at this stage fo ...
... Such discovery devices as examples, pictures, analogies, and counterexamples are brought into play. Rhetorical questions are employed often in this chapter to instill in the student the habit of thinking aggressively, of looking for questions as well as answers. Also, a case is made at this stage fo ...
Simply Logical: Intelligent Reasoning by Example
... What can and cannot be found in this book The book consists of three parts. Part I presents the necessary material on Logic and Logic Programming. In an introductory chapter, the main concepts in Logic Programming are introduced, such as program clauses, query answering, proof trees, and recursive d ...
... What can and cannot be found in this book The book consists of three parts. Part I presents the necessary material on Logic and Logic Programming. In an introductory chapter, the main concepts in Logic Programming are introduced, such as program clauses, query answering, proof trees, and recursive d ...
On Weak Ground
... characterization is intended to be a definition of weak ground in terms of strict ground. Nevertheless, the question of whether any of these characterizations can be used to specify the notion is worth exploring. The arguments that follow, if sound, provide reasons not only for rejecting the full ex ...
... characterization is intended to be a definition of weak ground in terms of strict ground. Nevertheless, the question of whether any of these characterizations can be used to specify the notion is worth exploring. The arguments that follow, if sound, provide reasons not only for rejecting the full ex ...
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 ...
Functional Dependencies in a Relational Database and
... directly converting results in that area into results about functional dependencies. In this paper, we refer to functional dependencies by the name “dependency statements.” This is done for several reasons. T h e first is to emphasize the analogy with implicational statements, defined soon. T h e se ...
... directly converting results in that area into results about functional dependencies. In this paper, we refer to functional dependencies by the name “dependency statements.” This is done for several reasons. T h e first is to emphasize the analogy with implicational statements, defined soon. T h e se ...
Intuitionistic and Modal Logic
... • Platonism. Most famous modern representatives: Frege, Gödel. View that mathematical objects have independent existence outside of spacetime, that mathematical truths are independent of us. At the time mixed with logicism, Frege’s idea that mathematics is no more than logic, since mathematics can ...
... • Platonism. Most famous modern representatives: Frege, Gödel. View that mathematical objects have independent existence outside of spacetime, that mathematical truths are independent of us. At the time mixed with logicism, Frege’s idea that mathematics is no more than logic, since mathematics can ...
A survey on Interactive Theorem Proving
... In 1998 Hales claimed to have a proof of Kepler conjecture (the optimal way of packing spheres is the way one normally stacks oranges) that however relied on a huge number of inequalities checked by means of a computer program. The work was rejected in the Annals of Mathematics and Hales decided to ...
... In 1998 Hales claimed to have a proof of Kepler conjecture (the optimal way of packing spheres is the way one normally stacks oranges) that however relied on a huge number of inequalities checked by means of a computer program. The work was rejected in the Annals of Mathematics and Hales decided to ...
Classical Propositional Logic
... A valid inference is truth-preserving: If the premises are true, the conclusion is true. ...
... A valid inference is truth-preserving: If the premises are true, the conclusion is true. ...
Relevant and Substructural Logics
... ponens is written in the form using a turnstile to echo the general definition of logical consequence in a Hilbert system. Given a set X of formulas, and a single formula A, we say that A can be proved from X (which I write “X ⇒ A”) if and only if there is a proof in the Hilbert system with A as the ...
... ponens is written in the form using a turnstile to echo the general definition of logical consequence in a Hilbert system. Given a set X of formulas, and a single formula A, we say that A can be proved from X (which I write “X ⇒ A”) if and only if there is a proof in the Hilbert system with A as the ...
Everything Else Being Equal: A Modal Logic for Ceteris Paribus
... the modality 3< ϕ is not definable in terms of 3≤ ϕ – even though the strict relation ≺ is first-order defined in terms of . It is easy to draw two pointed models which are bisimilar in the above sense with respect to the clauses for the weak order, while the truth value of some formula 3< ϕ differ ...
... the modality 3< ϕ is not definable in terms of 3≤ ϕ – even though the strict relation ≺ is first-order defined in terms of . It is easy to draw two pointed models which are bisimilar in the above sense with respect to the clauses for the weak order, while the truth value of some formula 3< ϕ differ ...
Understanding SPKI/SDSI Using First-Order Logic
... intuition behind SPKI/SDSI. Third, the entailment should provide a basis for comparing the deduction mechanisms associated with SPKI/SDSI and evaluating them. Fourth, the entailment provided by the semantics should be well-understood and well-studied. A good semantics may also allow techniques devel ...
... intuition behind SPKI/SDSI. Third, the entailment should provide a basis for comparing the deduction mechanisms associated with SPKI/SDSI and evaluating them. Fourth, the entailment provided by the semantics should be well-understood and well-studied. A good semantics may also allow techniques devel ...
Logic in Nonmonotonic Reasoning
... wholesale theories about the world and acting in accordance with them. Both commonsense and nonmonotonic reasoning are just special forms of a general scientific methodology in this sense. The way of thinking in partially known circumstances suggested by nonmonotonic reasoning consists in using rea ...
... wholesale theories about the world and acting in accordance with them. Both commonsense and nonmonotonic reasoning are just special forms of a general scientific methodology in this sense. The way of thinking in partially known circumstances suggested by nonmonotonic reasoning consists in using rea ...
Problems on Discrete Mathematics1 (Part I)
... Of course, we will not step into the dark side of the road in this book. In stead, we will study some naive concepts of sets; most of them are intuitively understandable from our daily-life experiences. For example, all students of Syracuse University is a set; all students in the United State of Am ...
... Of course, we will not step into the dark side of the road in this book. In stead, we will study some naive concepts of sets; most of them are intuitively understandable from our daily-life experiences. For example, all students of Syracuse University is a set; all students in the United State of Am ...
Logic and Proof - Numeracy Workshop
... The truth or falsity of a converse can not be inferred from the truth or falsity of the original statement. For example, x = 2 ⇒ x2 = 4 is true, but . . . its converse x2 = 4 ⇒ x = 2 is false, because x could be equal to −2. ...
... The truth or falsity of a converse can not be inferred from the truth or falsity of the original statement. For example, x = 2 ⇒ x2 = 4 is true, but . . . its converse x2 = 4 ⇒ x = 2 is false, because x could be equal to −2. ...
Many-Valued Logic
... It is normal in the sense that it agrees with two-valued logic on the values assigned all combinations of 1s and 0s, and it is uniform in the sense that it maintains that, in defining the connectives, if a compound has the same value whether a component is true or false, it also has that value if th ...
... It is normal in the sense that it agrees with two-valued logic on the values assigned all combinations of 1s and 0s, and it is uniform in the sense that it maintains that, in defining the connectives, if a compound has the same value whether a component is true or false, it also has that value if th ...
Formale Methoden der Softwaretechnik Formal methods of software
... To prove S from P1 ∨ . . . ∨ Pn , prove S from each of P1 , . . . , Pn . Claim: there are irrational numbers b and c such that b c is ...
... To prove S from P1 ∨ . . . ∨ Pn , prove S from each of P1 , . . . , Pn . Claim: there are irrational numbers b and c such that b c is ...
Introduction to Linear Logic
... The main concern of this report is to give an introduction to Linear Logic. For pedagogical purposes we shall also have a look at Classical Logic as well as Intuitionistic Logic. Linear Logic was introduced by J.-Y. Girard in 1987 and it has attracted much attention from computer scientists, as it i ...
... The main concern of this report is to give an introduction to Linear Logic. For pedagogical purposes we shall also have a look at Classical Logic as well as Intuitionistic Logic. Linear Logic was introduced by J.-Y. Girard in 1987 and it has attracted much attention from computer scientists, as it i ...
Constraint propagation
... SO: if F follows from T, then we find a proof, else it is possible that the procedure doesn’t terminate. ...
... SO: if F follows from T, then we find a proof, else it is possible that the procedure doesn’t terminate. ...