AN EARLY HISTORY OF MATHEMATICAL LOGIC AND
... among the logicians of his time, most closely shadowed early set theory. My approach as applied to set theory begins with Bernard Bolzano and continues to treatments of Richard Dedekind and Georg Cantor. Throughout this treatment we see interaction with logic. Dedekind interacted intellectually with ...
... among the logicians of his time, most closely shadowed early set theory. My approach as applied to set theory begins with Bernard Bolzano and continues to treatments of Richard Dedekind and Georg Cantor. Throughout this treatment we see interaction with logic. Dedekind interacted intellectually with ...
REGULAR COST FUNCTIONS, PART I: LOGIC AND ALGEBRA
... doing an at most “polynomial approximation2”. On the other side, the presentation in terms of profinite languages eliminates the corresponding annoying details in the development of cost functions: namely there is no more need to control the approximation at each step. Another interesting point is t ...
... doing an at most “polynomial approximation2”. On the other side, the presentation in terms of profinite languages eliminates the corresponding annoying details in the development of cost functions: namely there is no more need to control the approximation at each step. Another interesting point is t ...
Proofs in Higher-Order Logic - ScholarlyCommons
... This dissertation is a presentation of various metatheoretical results about higher-order logic (HOL). Although many of these results should be of interest from a formal proof theory point-of-view, they were motivated by problems encountered in the construction of automatic theorem provers for this ...
... This dissertation is a presentation of various metatheoretical results about higher-order logic (HOL). Although many of these results should be of interest from a formal proof theory point-of-view, they were motivated by problems encountered in the construction of automatic theorem provers for this ...
Notions of Computability at Higher Type
... §1. Introduction. This article is essentially a survey of fifty years of research on higher type computability. It was a great privilege to present much of this material in a series of three lectures at the Paris Logic Colloquium. In elementary recursion theory, one begins with the question: what do ...
... §1. Introduction. This article is essentially a survey of fifty years of research on higher type computability. It was a great privilege to present much of this material in a series of three lectures at the Paris Logic Colloquium. In elementary recursion theory, one begins with the question: what do ...
document
... • Among these was Hope – strongly typed – polymorphism but explicit type declarations as part of all function definitions – simple module facility – user-defined concrete data types with pattern matching ...
... • Among these was Hope – strongly typed – polymorphism but explicit type declarations as part of all function definitions – simple module facility – user-defined concrete data types with pattern matching ...
a PDF file of the textbook - U of L Class Index
... For our purposes, Logic is the business of deciding whether or not a deduction is valid; that is, deciding whether or not a particular conclusion is a consequence of particular assumptions (or “hypotheses”). In this chapter, we will introduce three basic ingredients of Logic: assertions, deductions, ...
... For our purposes, Logic is the business of deciding whether or not a deduction is valid; that is, deciding whether or not a particular conclusion is a consequence of particular assumptions (or “hypotheses”). In this chapter, we will introduce three basic ingredients of Logic: assertions, deductions, ...
Cut-elimination for provability logics and some results in display logic
... Gentzen’s sequent calculus, although it should be noted that each of these systems has its own shortcomings. Examples of proof-systems that we will encounter in this thesis include the display calculus [5] and labelled sequent calculi [24, 52]. Structural proof-theory encompasses the study of these ...
... Gentzen’s sequent calculus, although it should be noted that each of these systems has its own shortcomings. Examples of proof-systems that we will encounter in this thesis include the display calculus [5] and labelled sequent calculi [24, 52]. Structural proof-theory encompasses the study of these ...
abdullah_thesis_slides.pdf
... Given d,t ∈ N, we can define the concept of type signatures of radius d with threshold t such that the values (#Type1 ,...,#Typen ) are counted only upto a threshold t and anything ≥ t is considered ∞. Two structures A and B, are said to be d-equivalent with threshold t if their type signatures with ...
... Given d,t ∈ N, we can define the concept of type signatures of radius d with threshold t such that the values (#Type1 ,...,#Typen ) are counted only upto a threshold t and anything ≥ t is considered ∞. Two structures A and B, are said to be d-equivalent with threshold t if their type signatures with ...
Declarative Programming in Escher
... equations involving boolean expressions and for Haskell one considers statements in a program as being equations involving -expressions. Having done this, programs in all three languages are equational theories in a higher-order logic and the various languages can be compared by applying the approp ...
... equations involving boolean expressions and for Haskell one considers statements in a program as being equations involving -expressions. Having done this, programs in all three languages are equational theories in a higher-order logic and the various languages can be compared by applying the approp ...
Mathematical Logic
... Definition 1.1.1. The propositional connectives are negation (¬ ), conjunction ( & ), disjunction ( ∨ ), implication ( ⇒ ), biimplication ( ⇔ ). They are read as “not”, “and”, “or”, “if-then”, “if and only if” respectively. The connectives & , ∨ , ⇒ , ⇔ are designated as binary, while ¬ is designate ...
... Definition 1.1.1. The propositional connectives are negation (¬ ), conjunction ( & ), disjunction ( ∨ ), implication ( ⇒ ), biimplication ( ⇔ ). They are read as “not”, “and”, “or”, “if-then”, “if and only if” respectively. The connectives & , ∨ , ⇒ , ⇔ are designated as binary, while ¬ is designate ...
Chiron: A Set Theory with Types, Undefinedness, Quotation, and
... Chiron is a set theory that has a much higher level of practical expressivity than traditional set theories. It is intended to be a general-purpose logic that, unlike traditional logics, is designed to be used in practice. It integrates nbg set theory, elements of type theory, a scheme for handling ...
... Chiron is a set theory that has a much higher level of practical expressivity than traditional set theories. It is intended to be a general-purpose logic that, unlike traditional logics, is designed to be used in practice. It integrates nbg set theory, elements of type theory, a scheme for handling ...
The Foundations
... occurs(happens, exists) in the situation which the proposition is intended to describe. =>Example: Since it is not raining now(the current situation), the statement It_is_raining is false (in the current situation). But if it were raining now, then I would say that It_is_raining is true. Factors a ...
... occurs(happens, exists) in the situation which the proposition is intended to describe. =>Example: Since it is not raining now(the current situation), the statement It_is_raining is false (in the current situation). But if it were raining now, then I would say that It_is_raining is true. Factors a ...
A BOUND FOR DICKSON`S LEMMA 1. Introduction Consider the
... Dickson’s lemma has many applications. For instance, it is used to prove termination of Buchberger’s algorithm for computing Gröbner bases [4], and to prove Hilbert’s basis theorem [14]. There are many other proofs of Dickson’s lemma in the literature, both with and without usage of non-constructiv ...
... Dickson’s lemma has many applications. For instance, it is used to prove termination of Buchberger’s algorithm for computing Gröbner bases [4], and to prove Hilbert’s basis theorem [14]. There are many other proofs of Dickson’s lemma in the literature, both with and without usage of non-constructiv ...
Appendix B
... Expression (let ((a 5) (b 8)) (+ a b)) is an abbreviation of the function application ((lambda (a b) (+ a b)) 5 8); Both expressions return the value 13. Also has a sequential let, called let*, that evaluates the bindings from left to right. (let* ((a 5) (b (+ a 3))) (* a b)) is equivalent to (let ( ...
... Expression (let ((a 5) (b 8)) (+ a b)) is an abbreviation of the function application ((lambda (a b) (+ a b)) 5 8); Both expressions return the value 13. Also has a sequential let, called let*, that evaluates the bindings from left to right. (let* ((a 5) (b (+ a 3))) (* a b)) is equivalent to (let ( ...
Acts of Commanding and Changing Obligations
... monadic deontic logic with a dynamic language to talk about the situations before and after the issuance of commands, and the commands that link those situations. Although the resulting language inherits various inadequacies from the language of monadic deontic logic, some interesting principles are ...
... monadic deontic logic with a dynamic language to talk about the situations before and after the issuance of commands, and the commands that link those situations. Although the resulting language inherits various inadequacies from the language of monadic deontic logic, some interesting principles are ...
View raw file - aaa
... exp : type. lam : (exp -> exp) -> exp. app : exp -> exp -> exp. check : exp -> t -> type. check/lam : check (lam M) (A arrow B) <- {x:exp} (check x A -> check (M x) B). ...
... exp : type. lam : (exp -> exp) -> exp. app : exp -> exp -> exp. check : exp -> t -> type. check/lam : check (lam M) (A arrow B) <- {x:exp} (check x A -> check (M x) B). ...
An argumentation framework in default logic
... systems have been developed [9, 22, 23]. These systems are meant to be an alternative to earlier approaches to formalize so-called nonmonotonic reasoning, in which conclusions can be invalidated by adding new information to the premises. This kind of reasoning is motivated by the fact that in real l ...
... systems have been developed [9, 22, 23]. These systems are meant to be an alternative to earlier approaches to formalize so-called nonmonotonic reasoning, in which conclusions can be invalidated by adding new information to the premises. This kind of reasoning is motivated by the fact that in real l ...
Uniform satisfiability in PSPACE for local temporal logics over
... next and (universal) until has the same expressive power as first order logic over traces [4]. Moreover, local temporal logics have usually a low complexity, i.e., satisfiability can be solved in PSPACE. We cannot expect a lower complexity since already the classical temporal logic LTL over sequence ...
... next and (universal) until has the same expressive power as first order logic over traces [4]. Moreover, local temporal logics have usually a low complexity, i.e., satisfiability can be solved in PSPACE. We cannot expect a lower complexity since already the classical temporal logic LTL over sequence ...
Curry–Howard correspondence
In programming language theory and proof theory, the Curry–Howard correspondence (also known as the Curry–Howard isomorphism or equivalence, or the proofs-as-programs and propositions- or formulae-as-types interpretation) is the direct relationship between computer programs and mathematical proofs. It is a generalization of a syntactic analogy between systems of formal logic and computational calculi that was first discovered by the American mathematician Haskell Curry and logician William Alvin Howard. It is the link between logic and computation that is usually attributed to Curry and Howard, although the idea is related to the operational interpretation of intuitionistic logic given in various formulations by L. E. J. Brouwer, Arend Heyting and Andrey Kolmogorov (see Brouwer–Heyting–Kolmogorov interpretation) and Stephen Kleene (see Realizability). The relationship has been extended to include category theory as the three-way Curry–Howard–Lambek correspondence.