full text (.pdf)
... optimized and unoptimized program. The necessary premises are obtained by inspection of the program and their validity may depend on properties of the domain of computation, but they are usually quite simple and easy to verify by inspection, since they typically only involve atomic programs and test ...
... optimized and unoptimized program. The necessary premises are obtained by inspection of the program and their validity may depend on properties of the domain of computation, but they are usually quite simple and easy to verify by inspection, since they typically only involve atomic programs and test ...
Introduction to Logic
... The term “logic” may be, very roughly and vaguely, associated with something like “correct thinking”. Aristotle defined a syllogism as “discourse in which, certain things being stated something other than what is stated follows of necessity from their being so.” And, in fact, this intuition not only ...
... The term “logic” may be, very roughly and vaguely, associated with something like “correct thinking”. Aristotle defined a syllogism as “discourse in which, certain things being stated something other than what is stated follows of necessity from their being so.” And, in fact, this intuition not only ...
On modal logics of group belief
... [59]). On the contrary, in [57] acceptance is considered to be stronger than belief. Although belief and acceptance seem very close, several authors [11, 17, 61] have argued for the importance of keeping the two notions independent. We here agree with this point of view (Section 4.3). For the aims o ...
... [59]). On the contrary, in [57] acceptance is considered to be stronger than belief. Although belief and acceptance seem very close, several authors [11, 17, 61] have argued for the importance of keeping the two notions independent. We here agree with this point of view (Section 4.3). For the aims o ...
Sets
... If P(x) is the statement “x has won a race” where the domain of discourse is all runners, then the universal quantification of P(x) is x P ( x ) , i.e., every runner has won a race. The negation of this statement is “it is not the case that every runner has won a race. Therefore there exists at l ...
... If P(x) is the statement “x has won a race” where the domain of discourse is all runners, then the universal quantification of P(x) is x P ( x ) , i.e., every runner has won a race. The negation of this statement is “it is not the case that every runner has won a race. Therefore there exists at l ...
Structural Logical Relations
... We shall prove that for every derivation of eτ , there exists a v τ , s.t. e −→∗ v and v ⇑ τ via a unary structural logical relation. The challenge, however, is the choice of a predicate P . It may come as a surprise that it is sufficient to characterize the fact that a term has a normal form withou ...
... We shall prove that for every derivation of eτ , there exists a v τ , s.t. e −→∗ v and v ⇑ τ via a unary structural logical relation. The challenge, however, is the choice of a predicate P . It may come as a surprise that it is sufficient to characterize the fact that a term has a normal form withou ...
An Introduction to Proof Theory - UCSD Mathematics
... function fA , by letting fA (x1 , ..., xk ) equal the truth value τ (A) where τ (pi ) = xi for all i. A language is a set of connectives which may be used in the formation of L-formulas. A language L is complete if and only if every Boolean function can be defined by an L-formula. Propositional logi ...
... function fA , by letting fA (x1 , ..., xk ) equal the truth value τ (A) where τ (pi ) = xi for all i. A language is a set of connectives which may be used in the formation of L-formulas. A language L is complete if and only if every Boolean function can be defined by an L-formula. Propositional logi ...
Slide 1
... Section 1.1 Proof Primer (cont) Proof By Contradiction A false statement is called a contradiction. For example, “S and not S” is a contradiction for any statement S. A truth table will show us that “if A then B,” is equivalent to “A and not B implies false.” So to prove “if A then B,” it suffices ...
... Section 1.1 Proof Primer (cont) Proof By Contradiction A false statement is called a contradiction. For example, “S and not S” is a contradiction for any statement S. A truth table will show us that “if A then B,” is equivalent to “A and not B implies false.” So to prove “if A then B,” it suffices ...
Refinement Modal Logic
... quantifier, but the two notions are incomparable, because ‘must’ is a subtype of ‘may’. We incorporate implicit quantification over informative events directly into the language using, again, a notion of refinement; also in our case a refinement is the converse of simulation. Our work is closely rel ...
... quantifier, but the two notions are incomparable, because ‘must’ is a subtype of ‘may’. We incorporate implicit quantification over informative events directly into the language using, again, a notion of refinement; also in our case a refinement is the converse of simulation. Our work is closely rel ...
Views: Compositional Reasoning for Concurrent Programs
... a typing context, which embodies the knowledge that the values of variables agree with their types, and the rights to change the state such that this typing is preserved. When views are composed, they must agree on the types of all variables they share. In a type system that permits strong (i.e. ty ...
... a typing context, which embodies the knowledge that the values of variables agree with their types, and the rights to change the state such that this typing is preserved. When views are composed, they must agree on the types of all variables they share. In a type system that permits strong (i.e. ty ...
Introduction to Logic
... without changing its value. In Aristotle this meant simply that the pairs he determined could be exchanged. The intuition might have been that they “essentially mean the same”. In a more abstract, and later formulation, one would say that “not to affect a proposition” is “not to change its truth val ...
... without changing its value. In Aristotle this meant simply that the pairs he determined could be exchanged. The intuition might have been that they “essentially mean the same”. In a more abstract, and later formulation, one would say that “not to affect a proposition” is “not to change its truth val ...
![overhead 12/proofs in predicate logic [ov]](http://s1.studyres.com/store/data/011612593_1-f0437b45ca3df4f633ddcc6197d427a6-300x300.png)
overhead 12/proofs in predicate logic [ov]
... - the rule for getting rid of the existential quantifier: Existential Instantiation (EI) (preliminary version) (x)x a - the rule for introducing the universal quantifier: Universal Generalization (UG) (preliminary version) a (x)x - these rules should seem much less intuitive than the first two- ...
... - the rule for getting rid of the existential quantifier: Existential Instantiation (EI) (preliminary version) (x)x a - the rule for introducing the universal quantifier: Universal Generalization (UG) (preliminary version) a (x)x - these rules should seem much less intuitive than the first two- ...
On presenting monotonicity and on EA=>AE (pdf file)
... But the two major books that deal with the calculational approach do a bad job of explaining how monotonicity/antimonotonicity is to be used. On page 61 of [1], Dijkstra and Scholten discuss the monotonic properties of negation and implication. But they don’t state the general theorem (5) and they d ...
... But the two major books that deal with the calculational approach do a bad job of explaining how monotonicity/antimonotonicity is to be used. On page 61 of [1], Dijkstra and Scholten discuss the monotonic properties of negation and implication. But they don’t state the general theorem (5) and they d ...
Basic Concepts of Formal Logic
... 1) Given the Law of Non-Contradiction, every proposition of the form “P and not P” must be false. Such a proposition is said to be logically false because it is false by virtue of its logical form alone. In particular, its logical form is that of a contradiction, and, given the Law of Non-Contradict ...
... 1) Given the Law of Non-Contradiction, every proposition of the form “P and not P” must be false. Such a proposition is said to be logically false because it is false by virtue of its logical form alone. In particular, its logical form is that of a contradiction, and, given the Law of Non-Contradict ...
Logic 1 Lecture Notes Part I: Propositional Logic
... as René Descartes and Immanuel Kant, thought that Aristotelian logic was in some sense complete or finished! Other philosophers, like Gottfried Leibniz, tried valiantly to introduce new ideas into logic, but these efforts led to failure, mainly because they failed to discover the central idea of qua ...
... as René Descartes and Immanuel Kant, thought that Aristotelian logic was in some sense complete or finished! Other philosophers, like Gottfried Leibniz, tried valiantly to introduce new ideas into logic, but these efforts led to failure, mainly because they failed to discover the central idea of qua ...
PPT
... Here the hypotheses P to be true but the conclusion Q false, and from this reach some type of contradiction, either contradicting the assumption P or contradicting the denial Q ...
... Here the hypotheses P to be true but the conclusion Q false, and from this reach some type of contradiction, either contradicting the assumption P or contradicting the denial Q ...
Argument construction and reinstatement in logics for
... study of interactions among competing defeasible arguments; a survey appears in Prakken and Vreewsijk (forthcoming). These argument systems are promising for several reasons. First, they often allow a more natural treatment of priorities among conflicting defeasible rules than the standard fixed-poi ...
... study of interactions among competing defeasible arguments; a survey appears in Prakken and Vreewsijk (forthcoming). These argument systems are promising for several reasons. First, they often allow a more natural treatment of priorities among conflicting defeasible rules than the standard fixed-poi ...
![Propositions as [Types] - Research Showcase @ CMU](http://s1.studyres.com/store/data/005730189_1-e85fa7d3c7cfa08d9a3b8e96a27d7888-300x300.png)
Propositions as [Types] - Research Showcase @ CMU
... The bracket types which we consider are essentially the same as the mono types of Maietti [Mai98], in a suitable setting. Palmgren [Pal01] formulated a BHK interpretation of intuitionistic logic and used image factorizations, which are used in the semantics of our bracket types, to relate the BHK in ...
... The bracket types which we consider are essentially the same as the mono types of Maietti [Mai98], in a suitable setting. Palmgren [Pal01] formulated a BHK interpretation of intuitionistic logic and used image factorizations, which are used in the semantics of our bracket types, to relate the BHK in ...
Higher Order Logic - Indiana University
... Higher order logics, long considered by many to be an esoteric subject, are increasingly recognized for their foundational importance and practical usefulness, notably in Theoretical Computer Science. In this chapter we try to present a survey of some issues and results, without any pretense of comp ...
... Higher order logics, long considered by many to be an esoteric subject, are increasingly recognized for their foundational importance and practical usefulness, notably in Theoretical Computer Science. In this chapter we try to present a survey of some issues and results, without any pretense of comp ...
Foundations of Logic Programmin:
... uniform language for data, programs, queries, views and integrity constraints has ...
... uniform language for data, programs, queries, views and integrity constraints has ...
Higher Order Logic - Theory and Logic Group
... Higher order logics, long considered by many to be an esoteric subject, are increasingly recognized for their foundational importance and practical usefulness, notably in Theoretical Computer Science. In this chapter we try to present a survey of some issues and results, without any pretense of comp ...
... Higher order logics, long considered by many to be an esoteric subject, are increasingly recognized for their foundational importance and practical usefulness, notably in Theoretical Computer Science. In this chapter we try to present a survey of some issues and results, without any pretense of comp ...
Expressiveness of Logic Programs under the General Stable Model
... different here. On the one hand, we will work on the general stable model semantics so that non-Herbrand structures will be considered. On the other hand, instead of considering query equivalence, the expressiveness in our work will be based on model equivalence. This setting is important since ASP ...
... different here. On the one hand, we will work on the general stable model semantics so that non-Herbrand structures will be considered. On the other hand, instead of considering query equivalence, the expressiveness in our work will be based on model equivalence. This setting is important since ASP ...
Pebble weighted automata and transitive - LSV
... from the grammar ϕ ::= k | α | ¬ϕ | ϕ ∨ ϕ | ϕ ∧ ϕ, with k ∈ K and α ∈ L. In particular, quantifications are only allowed in formulas α ∈ L. The following lemma shows in particular that an L-step formula assumes a finite number of values, each of which corresponds to an L-definable language. Lemma W ...
... from the grammar ϕ ::= k | α | ¬ϕ | ϕ ∨ ϕ | ϕ ∧ ϕ, with k ∈ K and α ∈ L. In particular, quantifications are only allowed in formulas α ∈ L. The following lemma shows in particular that an L-step formula assumes a finite number of values, each of which corresponds to an L-definable language. Lemma W ...