Normal modal logics (Syntactic characterisations)
... (iii) We have to show that ΣF contains PL and is closed under MP and US. The first two are straightforward and are left as an exercise (tutorial sheet). To show closure under US is not difficult but is rather long and fiddly so details omitted here. The basic idea is simple enough. Blackburn et al p ...
... (iii) We have to show that ΣF contains PL and is closed under MP and US. The first two are straightforward and are left as an exercise (tutorial sheet). To show closure under US is not difficult but is rather long and fiddly so details omitted here. The basic idea is simple enough. Blackburn et al p ...
Section.8.3
... A logic is higher-order if it allows predicate names or function names to be quantified or to be arguments of a predicate. Example. The sentence, “There is a function with a fixed point.” can be formalized as f x p(ƒ(x), x), where p denotes equality. Example. Given the sentence, “ Every binary rel ...
... A logic is higher-order if it allows predicate names or function names to be quantified or to be arguments of a predicate. Example. The sentence, “There is a function with a fixed point.” can be formalized as f x p(ƒ(x), x), where p denotes equality. Example. Given the sentence, “ Every binary rel ...
The Foundations: Logic and Proofs - UTH e
... Two compound propositions p and q are logically equivalent if p↔q is a tautology. We write this as p⇔q or as p≡q where p and q are compound propositions. Two compound propositions p and q are equivalent if and only if the columns in a truth table giving their truth values agree. This truth table sho ...
... Two compound propositions p and q are logically equivalent if p↔q is a tautology. We write this as p⇔q or as p≡q where p and q are compound propositions. Two compound propositions p and q are equivalent if and only if the columns in a truth table giving their truth values agree. This truth table sho ...
A(x)
... raise a question: having a formula , does the calculus decide ? In other words, is there an algorithm that would answer in a finite number of steps Yes or No, having as input and answering the question whether is logically valid or no? If there is such an algorithm, then the calculus is decida ...
... raise a question: having a formula , does the calculus decide ? In other words, is there an algorithm that would answer in a finite number of steps Yes or No, having as input and answering the question whether is logically valid or no? If there is such an algorithm, then the calculus is decida ...
A(x)
... raise a question: having a formula , does the calculus decide ? In other words, is there an algorithm that would answer in a finite number of steps Yes or No, having as input and answering the question whether is logically valid or no? If there is such an algorithm, then the calculus is decida ...
... raise a question: having a formula , does the calculus decide ? In other words, is there an algorithm that would answer in a finite number of steps Yes or No, having as input and answering the question whether is logically valid or no? If there is such an algorithm, then the calculus is decida ...
ON PRESERVING 1. Introduction The
... transmuted into single formula truth. It may be more correct to say that, but it makes the whole paradigm somewhat less forceful or even less appealing. What we need in order to rescue the very idea of preservationism is to talk entirely about sets. So we shall have to replace the arbitrary conclusi ...
... transmuted into single formula truth. It may be more correct to say that, but it makes the whole paradigm somewhat less forceful or even less appealing. What we need in order to rescue the very idea of preservationism is to talk entirely about sets. So we shall have to replace the arbitrary conclusi ...
Modal Logic and Model Theory
... if A is atomic, then xA = A ; if A = JBvO, then x{BvG) = rJSvrO; if A =BaC9 then x{BaG) = tJBatC; if A = lxhB(xh), then r(3%JB(%)) = lxhxB(xh); ...
... if A is atomic, then xA = A ; if A = JBvO, then x{BvG) = rJSvrO; if A =BaC9 then x{BaG) = tJBatC; if A = lxhB(xh), then r(3%JB(%)) = lxhxB(xh); ...
Computational lambda calculus: A combination of functional and
... Computational lambda calculus: A combination of functional and imperative programming. ...
... Computational lambda calculus: A combination of functional and imperative programming. ...
PowerPoint file for CSL 02, Edinburgh, UK
... Thus, when recursive functions are replaced by D0n-functions, a new mathematics is created. ...
... Thus, when recursive functions are replaced by D0n-functions, a new mathematics is created. ...
Basic Logic and Fregean Set Theory - MSCS
... Arithmetic may be considered a proof interpretation in which there is a universe of ‘proofs’ encoded in numbers. However, a formula of first-order arithmetic is realized by a number if and only if that formula is derivable from HA plus a formalized version of Church’s Thesis [20, page 196]. It appe ...
... Arithmetic may be considered a proof interpretation in which there is a universe of ‘proofs’ encoded in numbers. However, a formula of first-order arithmetic is realized by a number if and only if that formula is derivable from HA plus a formalized version of Church’s Thesis [20, page 196]. It appe ...
A(x)
... question: having a formula , does the calculus decide ? In other words, is there an algorithm that would answer Yes or No, having as input and answering the question whether is logically valid or no? If there is such an algorithm, then the calculus is decidable. If the calculus is complete, th ...
... question: having a formula , does the calculus decide ? In other words, is there an algorithm that would answer Yes or No, having as input and answering the question whether is logically valid or no? If there is such an algorithm, then the calculus is decidable. If the calculus is complete, th ...
Propositional Dynamic Logic of Regular Programs*+
... The definition of n@(Q) depends only on @(Y) which has been already well defined previously. The new structure 9 is over @i and & , where A,, = CD;- CD,,is the set of Q-variables introduced in taking the closure of F. At this point it is helpful to explain the role of the Q-variables in the proof. F ...
... The definition of n@(Q) depends only on @(Y) which has been already well defined previously. The new structure 9 is over @i and & , where A,, = CD;- CD,,is the set of Q-variables introduced in taking the closure of F. At this point it is helpful to explain the role of the Q-variables in the proof. F ...
Functional_Programming
... Origin of Functional Programming Church’s thesis: All models of computation are equally powerful and can compute any function Turing’s model of computation: Turing machine Reading/writing of values on an infinite tape by a finite state machine Church’s model of computation: lambda calculus This insp ...
... Origin of Functional Programming Church’s thesis: All models of computation are equally powerful and can compute any function Turing’s model of computation: Turing machine Reading/writing of values on an infinite tape by a finite state machine Church’s model of computation: lambda calculus This insp ...
A Brief Introduction to the Intuitionistic Propositional Calculus
... constructions. Intuitionistic epistemology centers on proof, rather than truth. Thus, intuitionists analyze propositional combinations of mathematical statements in terms of what it takes to prove them, and a proof of ϕ ∧ ψ consists of a proof of ϕ together with a proof of ψ, a proof of ϕ ∨ ψ consis ...
... constructions. Intuitionistic epistemology centers on proof, rather than truth. Thus, intuitionists analyze propositional combinations of mathematical statements in terms of what it takes to prove them, and a proof of ϕ ∧ ψ consists of a proof of ϕ together with a proof of ψ, a proof of ϕ ∨ ψ consis ...
Chapter 3
... • In functional programming, functions are firstclass data values – Functions can be computed by other functions – Functions can be parameters to other functions ...
... • In functional programming, functions are firstclass data values – Functions can be computed by other functions – Functions can be parameters to other functions ...
Document
... directly on the von Neumann architecture - Efficiency is the primary concern, rather than the suitability of the language for software development - The design of the functional languages is based on mathematical functions - A solid theoretical basis that is also closer to the user, but relatively u ...
... directly on the von Neumann architecture - Efficiency is the primary concern, rather than the suitability of the language for software development - The design of the functional languages is based on mathematical functions - A solid theoretical basis that is also closer to the user, but relatively u ...
Constructive logic and type theory (lecture notes 2009)
... Constructive logic is based on the leading principle that (I) proofs are programs. In constructive type theory it has later been joined with the further principle that (II) propositions are data types. The latter is also called the propositions-as-types principle. The meaning of these principles is ...
... Constructive logic is based on the leading principle that (I) proofs are programs. In constructive type theory it has later been joined with the further principle that (II) propositions are data types. The latter is also called the propositions-as-types principle. The meaning of these principles is ...
An Introduction to Modal Logic VII The finite model property
... Relation between the two properties A normal modal logic Λ has the finite model property if and only if it has the finite frame property. Clearly the f.f.p. implies the f.m.p. On the other hand, suppose now that Λ has the f.m.p. and let ϕ∈ / Λ; by f.m.p., there is a finite model M where ϕ is not val ...
... Relation between the two properties A normal modal logic Λ has the finite model property if and only if it has the finite frame property. Clearly the f.f.p. implies the f.m.p. On the other hand, suppose now that Λ has the f.m.p. and let ϕ∈ / Λ; by f.m.p., there is a finite model M where ϕ is not val ...
A Simple Exposition of Gödel`s Theorem
... against materialism. Nine years later I was able to go to Princeton to study mathematical logic properly, and on my return tried out my argument on colleagues at Cambridge, then in a paper in 1959 to the Oxford Philosophical Society, which was finally published in 1961. G del's argument is self-refe ...
... against materialism. Nine years later I was able to go to Princeton to study mathematical logic properly, and on my return tried out my argument on colleagues at Cambridge, then in a paper in 1959 to the Oxford Philosophical Society, which was finally published in 1961. G del's argument is self-refe ...
slides
... a list with the parameters as elements - Predicate Functions: (#T and () are true and false) 1. EQ? takes two symbolic parameters; it returns #T if both parameters are atoms and the two are the same e.g., (EQ? 'A 'A) yields #T (EQ? 'A '(A B)) yields () Note that if EQ? is called with list parameters ...
... a list with the parameters as elements - Predicate Functions: (#T and () are true and false) 1. EQ? takes two symbolic parameters; it returns #T if both parameters are atoms and the two are the same e.g., (EQ? 'A 'A) yields #T (EQ? 'A '(A B)) yields () Note that if EQ? is called with list parameters ...