Classical first-order predicate logic This is a powerful extension of
... Some other quantifiers can be expressed with these. (They can also express each other.) But quantifiers like infinitely many and more than cannot be expressed in first-order logic in general. (They can in, e.g., second-order logic.) ...
... Some other quantifiers can be expressed with these. (They can also express each other.) But quantifiers like infinitely many and more than cannot be expressed in first-order logic in general. (They can in, e.g., second-order logic.) ...
term rewriting.
... The grater-than sign > indicates starting a rewrite with the curled brackets {} showing a substitution. The variables in capitals are from the BOOL module. The less-than sign < indicates ending a rewrite replacing the LHS with the RHS. Using the commutativity property, the system may changed the ord ...
... The grater-than sign > indicates starting a rewrite with the curled brackets {} showing a substitution. The variables in capitals are from the BOOL module. The less-than sign < indicates ending a rewrite replacing the LHS with the RHS. Using the commutativity property, the system may changed the ord ...
Logic in the Finite - CIS @ UPenn
... the corresponding interval in B so as to achieve the following approximation between these distances and the corresponding distances d01 and d02 between the point she pebbles and the endpoints of her interval. Namely, for i = 1; 2 if di 2(n m) ; then di = d0i ; and if di > 2n m ; then d0i > 2n m : ...
... the corresponding interval in B so as to achieve the following approximation between these distances and the corresponding distances d01 and d02 between the point she pebbles and the endpoints of her interval. Namely, for i = 1; 2 if di 2(n m) ; then di = d0i ; and if di > 2n m ; then d0i > 2n m : ...
Fichte`s Legacy in Logic
... fixed and completed body of doctrine upon which we might rely in the transcendental investigation of the conditions of knowledge and the forms of subjectivity. For Fichte, the results of logic must remain provisional in advance of transcendental inquiry, awaiting vindication by the results of the “s ...
... fixed and completed body of doctrine upon which we might rely in the transcendental investigation of the conditions of knowledge and the forms of subjectivity. For Fichte, the results of logic must remain provisional in advance of transcendental inquiry, awaiting vindication by the results of the “s ...
ICS 353: Design and Analysis of Algorithms
... • The quantifiers and have higher precedence than all logical operators from propositional calculus. • E.g., x P(x) Q(x) • means……………………….. • does not mean …………………… ...
... • The quantifiers and have higher precedence than all logical operators from propositional calculus. • E.g., x P(x) Q(x) • means……………………….. • does not mean …………………… ...
Systems of modal logic - Department of Computing
... the intersection is closed under MP: suppose A and A → B are formulas in the intersection. Then both formulas belong to every Σi too, and since every Σi is closed under MP, B must belong to every Σi . So B is in the intersection also, so the intersection is closed under MP. A similar argument works ...
... the intersection is closed under MP: suppose A and A → B are formulas in the intersection. Then both formulas belong to every Σi too, and since every Σi is closed under MP, B must belong to every Σi . So B is in the intersection also, so the intersection is closed under MP. A similar argument works ...
AN EARLY HISTORY OF MATHEMATICAL LOGIC AND
... one strain spurs innovation in the other. This will be seen in the influence that logicism had on set theory, and the influence set theory had on the discussion of sequences in logic. The question may justly be asked: Why is a reappraisal of the development of mathematical logic and set theory impor ...
... one strain spurs innovation in the other. This will be seen in the influence that logicism had on set theory, and the influence set theory had on the discussion of sequences in logic. The question may justly be asked: Why is a reappraisal of the development of mathematical logic and set theory impor ...
A Logical Expression of Reasoning
... qualitative approaches, with more resemblance to the discipline of logic, the so called nonmonotonic logics, have been proposed to treat a form of non deductive reasoning, sometimes called “common sense reasoning”. We took this problem in AI as our initial motivation and starting point, but we have ...
... qualitative approaches, with more resemblance to the discipline of logic, the so called nonmonotonic logics, have been proposed to treat a form of non deductive reasoning, sometimes called “common sense reasoning”. We took this problem in AI as our initial motivation and starting point, but we have ...
A Conditional Logical Framework *
... Conditional Logical Framework LFK is the same exploited in [HLL07] for the General Logical Framework GLF. However, there is an important difference between the two frameworks in the definition of predicates. On one hand, predicates in [HLL07] are used both to determine whether β-reduction fires and ...
... Conditional Logical Framework LFK is the same exploited in [HLL07] for the General Logical Framework GLF. However, there is an important difference between the two frameworks in the definition of predicates. On one hand, predicates in [HLL07] are used both to determine whether β-reduction fires and ...
Proof theory of witnessed G¨odel logic: a
... supported by the FWF Austrian Science Fund projects START Y544-N23 and P22416. ...
... supported by the FWF Austrian Science Fund projects START Y544-N23 and P22416. ...
Set theory and logic
... finite sets and counting numbers - was an innovation at that time. Prejudices against this viewpoint were responsible for the rejection of his work by some mathematicians, but others reacted favorably because the theory provided a proof of the existence of transcendental numbers. Other applications ...
... finite sets and counting numbers - was an innovation at that time. Prejudices against this viewpoint were responsible for the rejection of his work by some mathematicians, but others reacted favorably because the theory provided a proof of the existence of transcendental numbers. Other applications ...
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 ...
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 ...
PhD Thesis First-Order Logic Investigation of Relativity Theory with
... Our basic concepts are explained as follows. This thesis mainly deals with the kinematics of relativity, i.e., with the motion of bodies (test-particles). However, we briefly discuss dynamics in Chap. 5, and our co-authored papers [6], [7] and [38] are fully devoted to dynamics. We represent motion ...
... Our basic concepts are explained as follows. This thesis mainly deals with the kinematics of relativity, i.e., with the motion of bodies (test-particles). However, we briefly discuss dynamics in Chap. 5, and our co-authored papers [6], [7] and [38] are fully devoted to dynamics. We represent motion ...
Classical first-order predicate logic This is a powerful extension
... Some other quantifiers can be expressed with these. (They can also express each other.) But quantifiers like infinitely many and more than cannot be expressed in first-order logic in general. (They can in, e.g., second-order logic.) ...
... Some other quantifiers can be expressed with these. (They can also express each other.) But quantifiers like infinitely many and more than cannot be expressed in first-order logic in general. (They can in, e.g., second-order logic.) ...
HONEST ELEMENTARY DEGREES AND DEGREES OF RELATIVE
... for example, Beklemishev’s work in [3–6] The degrees of relative provability and the honest α-elementary degrees are very closely related. For a theory T , let T + be the extension of T by all true Π1 sentences. Kristiansen [15] proves that PPA+ and H0 are isomorphic, and analogous results should h ...
... for example, Beklemishev’s work in [3–6] The degrees of relative provability and the honest α-elementary degrees are very closely related. For a theory T , let T + be the extension of T by all true Π1 sentences. Kristiansen [15] proves that PPA+ and H0 are isomorphic, and analogous results should h ...
Extracting Proofs from Tabled Proof Search
... II History atoms are not tabled; the table uses theories to infer additional atoms. III History atoms can be tabled; the table only infers its members. IV History atoms can be tabled; the table uses theories to infer additional atoms. The first two strategies yield proof certificates that simply us ...
... II History atoms are not tabled; the table uses theories to infer additional atoms. III History atoms can be tabled; the table only infers its members. IV History atoms can be tabled; the table uses theories to infer additional atoms. The first two strategies yield proof certificates that simply us ...
A Mathematical Introduction to Modal Logic
... Modal logic is a huge research area. Researchers from mathematics, philosophy, computer science, linguistics, political science and economics work on variety of modal logics focusing on numerous different topics with many amazingly different applications. Mathematicians approach it mostly from a mod ...
... Modal logic is a huge research area. Researchers from mathematics, philosophy, computer science, linguistics, political science and economics work on variety of modal logics focusing on numerous different topics with many amazingly different applications. Mathematicians approach it mostly from a mod ...
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 ...
Supplemental Reading 1
... of sets. In type theory we allow a dierent class of predicates | those involving predicative higher-order logic in a sense. This topic is discussed in many articles and books on type theory 33, 13, 36, 43, 12, 11] and is beyond the scope of this article, so here we will just assume that the reader ...
... of sets. In type theory we allow a dierent class of predicates | those involving predicative higher-order logic in a sense. This topic is discussed in many articles and books on type theory 33, 13, 36, 43, 12, 11] and is beyond the scope of this article, so here we will just assume that the reader ...
Truth-Functional Propositional Logic
... true (these statements are mutually inconsistent), it is possible for both of them to be false. In the standard nomenclature, contraries are mutually exclusive (mutually inconsistent) not collectively exhaustive. In contrast, contradictories are both mutually exclusive and collectively exhaustive. G ...
... true (these statements are mutually inconsistent), it is possible for both of them to be false. In the standard nomenclature, contraries are mutually exclusive (mutually inconsistent) not collectively exhaustive. In contrast, contradictories are both mutually exclusive and collectively exhaustive. G ...
Notions of Computability at Higher Type
... of higher type? If the latter, what do we mean by an “operation”? The spirit in which we are asking these questions is not to demand definitive answers to them, but to make the point that many choices are possible. Indeed, as we shall see, many different responses to the above questions are exemplif ...
... of higher type? If the latter, what do we mean by an “operation”? The spirit in which we are asking these questions is not to demand definitive answers to them, but to make the point that many choices are possible. Indeed, as we shall see, many different responses to the above questions are exemplif ...
Programming with Classical Proofs
... A fundamental result about the theory of computer programming is Rice’s theorem, which states that there is no e↵ective way of deciding whether an algorithm computes a partial recursive function with a given non-trivial property. A consequence of this is, that it is in general undecidable whether a ...
... A fundamental result about the theory of computer programming is Rice’s theorem, which states that there is no e↵ective way of deciding whether an algorithm computes a partial recursive function with a given non-trivial property. A consequence of this is, that it is in general undecidable whether a ...
Linear Contextual Modal Type Theory
... to mean that A is always available. This is only possible in the case that the derivation of A avail does not consume any resources. In compliance with the literature, we call the truth judgment a hypothetical judgment because it may rely on the assumptions that x1 : A1 true, . . . , xn : An true. ...
... to mean that A is always available. This is only possible in the case that the derivation of A avail does not consume any resources. In compliance with the literature, we call the truth judgment a hypothetical judgment because it may rely on the assumptions that x1 : A1 true, . . . , xn : An true. ...