Dependence Logic
... that the intersection of an empty collection of subsets of a set M is the set M itself. A consequence of this is that there are no formulas φ and ψ of dependence logic such that M |=X φ implies M 6|=X ψ, for all M and all X. Namely, letting X = ∅ would yield a contradiction. The following test is ve ...
... that the intersection of an empty collection of subsets of a set M is the set M itself. A consequence of this is that there are no formulas φ and ψ of dependence logic such that M |=X φ implies M 6|=X ψ, for all M and all X. Namely, letting X = ∅ would yield a contradiction. The following test is ve ...
Discrete Mathematics
... A methodology for objectively reasoning about their truth or falsity. It is the foundation for expressing formal proofs in all branches of mathematics. Discrete Mathematics. Spring 2009 ...
... A methodology for objectively reasoning about their truth or falsity. It is the foundation for expressing formal proofs in all branches of mathematics. Discrete Mathematics. Spring 2009 ...
Modal Logic for Artificial Intelligence
... A formula is satisfiable if there is a model in which the formula is true. A formula is a tautology if it is true in every model. A formula is contradictory if it is false in every model (hence, if it is not satisfiable). If a formula is satisfiable and its negation is also satisfiable, then we cal ...
... A formula is satisfiable if there is a model in which the formula is true. A formula is a tautology if it is true in every model. A formula is contradictory if it is false in every model (hence, if it is not satisfiable). If a formula is satisfiable and its negation is also satisfiable, then we cal ...
Chapter 9: Initial Theorems about Axiom System AS1
... the conditional connective of the object language. When we translate ‘„’ as ‘is a theorem’, we obtain: if α is a theorem, and α→β is a theorem, then β is a theorem. The informal proof of (T4) goes as follows. Proof: Suppose „α and „α→β, to show „β. Then, α and α→β are provable; so there is a proof o ...
... the conditional connective of the object language. When we translate ‘„’ as ‘is a theorem’, we obtain: if α is a theorem, and α→β is a theorem, then β is a theorem. The informal proof of (T4) goes as follows. Proof: Suppose „α and „α→β, to show „β. Then, α and α→β are provable; so there is a proof o ...
Lecture Notes
... If, assuming φ, ψ can be proved, then φ ⇒ ψ can be proved. I.e., φ ⊢ ψ implies ⊢ φ ⇒ ψ This does not hold for the logic of knowledge! ...
... If, assuming φ, ψ can be proved, then φ ⇒ ψ can be proved. I.e., φ ⊢ ψ implies ⊢ φ ⇒ ψ This does not hold for the logic of knowledge! ...
Hilbert`s Program Then and Now
... his view, highlighted in his correspondence with Frege, that consistency of an axiomatic theory guarantees the existence of the structure described, and is in this sense sufficient to justify the use of the theory. And he shared with Kronecker a recognition that elementary arithmetic has a privilege ...
... his view, highlighted in his correspondence with Frege, that consistency of an axiomatic theory guarantees the existence of the structure described, and is in this sense sufficient to justify the use of the theory. And he shared with Kronecker a recognition that elementary arithmetic has a privilege ...
Propositional logic - Cheriton School of Computer Science
... A parting of ways The meaning of proof-theoretic negation represents a major point of departure between schools of logical thought, and the choice we make fundamentally affects the properties of the resulting logic. If we believe that ¬φ means that φ is false, then we are classicists and our proof ...
... A parting of ways The meaning of proof-theoretic negation represents a major point of departure between schools of logical thought, and the choice we make fundamentally affects the properties of the resulting logic. If we believe that ¬φ means that φ is false, then we are classicists and our proof ...
Proof Theory of Finite-valued Logics
... Many-valued logic is not much younger than the whole field of symbolic logic. It was introduced in the early twenties of this century by Lukasiewicz [1920] and Post [1921] and has since developed into a very large area of research. Most of the early work done has concentrated on problems of axiomati ...
... Many-valued logic is not much younger than the whole field of symbolic logic. It was introduced in the early twenties of this century by Lukasiewicz [1920] and Post [1921] and has since developed into a very large area of research. Most of the early work done has concentrated on problems of axiomati ...
Simple multiplicative proof nets with units
... conclusion and no premise; however, ⊥-links are treated like 0-ary ?-links, i.e., they must be given a default jump. Sequentialisation is immediate. At first sight, cut elimination is unproblematic: replace a cut between conclusions 1 and ⊥ of zero-ary links with. . . nothing. But we notice a new pr ...
... conclusion and no premise; however, ⊥-links are treated like 0-ary ?-links, i.e., they must be given a default jump. Sequentialisation is immediate. At first sight, cut elimination is unproblematic: replace a cut between conclusions 1 and ⊥ of zero-ary links with. . . nothing. But we notice a new pr ...
Modal logic and the approximation induction principle
... where A denotes the set of all actions. Given a modal characterization O, we denote the sublogic of formulas in O that do not contain infinite conjunctions by OFIN and the sublogic of formulas with finite depth with OFDP . Clearly OFIN ⊆ OFDP . Using the results from Sect. 2.2, we can now prove the ...
... where A denotes the set of all actions. Given a modal characterization O, we denote the sublogic of formulas in O that do not contain infinite conjunctions by OFIN and the sublogic of formulas with finite depth with OFDP . Clearly OFIN ⊆ OFDP . Using the results from Sect. 2.2, we can now prove the ...
