Logic is a discipline that studies the principles and methods used in
... A predicate P, or propositional function, is a function that maps objects in the universe of discourse to propositions Predicates can be quantified using the universal quantifier (“for all”) ∀ or the existential quantifier (“there exists”) ∃ Quantified predicates can be negated as follows ¬∀x P(x) ...
... A predicate P, or propositional function, is a function that maps objects in the universe of discourse to propositions Predicates can be quantified using the universal quantifier (“for all”) ∀ or the existential quantifier (“there exists”) ∃ Quantified predicates can be negated as follows ¬∀x P(x) ...
2. 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 ...
Lambda Calculus as a Programming Language
... "theoretically" we should. We forbid the use of the symbol as a free variable within the λexpression itself. This is in line with Lambda calculus where λ-expressions must be textually finite and where free variables have nothing to do with defining recursive functions. The notation E1 = E2 is used t ...
... "theoretically" we should. We forbid the use of the symbol as a free variable within the λexpression itself. This is in line with Lambda calculus where λ-expressions must be textually finite and where free variables have nothing to do with defining recursive functions. The notation E1 = E2 is used t ...
ppt - Purdue College of Engineering
... Example formulas and non-formulas • “A good diet is a necessary condition for a healthy cat.” Express by implication. • C: the cat has a good diet D: the cat is healthy “B is a necessary condition for A”: A B ...
... Example formulas and non-formulas • “A good diet is a necessary condition for a healthy cat.” Express by implication. • C: the cat has a good diet D: the cat is healthy “B is a necessary condition for A”: A B ...
Ambient Logic II.fm
... this formula is meant to correspond somehow to a process of the form (νn)P where x denotes n. However, since (νn) can float, the matching of (νx) to any particular (νn) is not obvious. This means that the logical rules of our tentative (νx) quantifier are going to be fairly complex, or at least unfa ...
... this formula is meant to correspond somehow to a process of the form (νn)P where x denotes n. However, since (νn) can float, the matching of (νx) to any particular (νn) is not obvious. This means that the logical rules of our tentative (νx) quantifier are going to be fairly complex, or at least unfa ...
Intuitionistic Logic
... Negations collapse in pairs, so long as there are negations left. +++ A |=+ A. That suggests that there are three statuses: A, + A, ++ A. But A∨ + A∨ ++ A is not valid. We can have wRw0 , wRw00 such that w gives all three falsehood, w0 makes A true, and w00 makes A false. To understand this, think a ...
... Negations collapse in pairs, so long as there are negations left. +++ A |=+ A. That suggests that there are three statuses: A, + A, ++ A. But A∨ + A∨ ++ A is not valid. We can have wRw0 , wRw00 such that w gives all three falsehood, w0 makes A true, and w00 makes A false. To understand this, think a ...
proceedings version
... A here-and-there model (HT model) is made up of two sets of propositional variables H (‘here’) and T (‘there’) such that H ⊆ T . The logical language to talk about such models has connectives ⊥, ∧, ∨, and ⇒. The latter is interpreted in a non-classical way and is therefore different from the materia ...
... A here-and-there model (HT model) is made up of two sets of propositional variables H (‘here’) and T (‘there’) such that H ⊆ T . The logical language to talk about such models has connectives ⊥, ∧, ∨, and ⇒. The latter is interpreted in a non-classical way and is therefore different from the materia ...
Problem_Set_01
... 9. You have proved before that a truth table with n variables has 2n rows. a. How many different Boolean functions with n variables are there? b. For n=2, list all the functions and identify as many as you can by name. 10. Prove by induction that for n>4, 2n>n2. 11. Guess the number of different wa ...
... 9. You have proved before that a truth table with n variables has 2n rows. a. How many different Boolean functions with n variables are there? b. For n=2, list all the functions and identify as many as you can by name. 10. Prove by induction that for n>4, 2n>n2. 11. Guess the number of different wa ...
Quantified Equilibrium Logic and the First Order Logic of Here
... This report continues the work of [26] on first-order, or, as we shall say, quantified equilibrium logic (or QEL for short) and its relation to non-ground answer set programming. The report has three main contributions. First, we present a slightly different version of QEL where the so-called unique n ...
... This report continues the work of [26] on first-order, or, as we shall say, quantified equilibrium logic (or QEL for short) and its relation to non-ground answer set programming. The report has three main contributions. First, we present a slightly different version of QEL where the so-called unique n ...
4on1 - FSU Computer Science
... 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 ...
Decidable fragments of first-order logic Decidable fragments of first
... Fix any subset {b1 , b2 } of Bn of size r ∈ {1, 2} and recall that some r -table T{b1 ,b2 } from Tr is assiged to this subset. For any subset {b3 , . . . , bl+2 } of pairwise distinct elements of Bn that differ from b1 and b2 , consider the event that the table induced by b1 , . . . , bl+2 is equal ...
... Fix any subset {b1 , b2 } of Bn of size r ∈ {1, 2} and recall that some r -table T{b1 ,b2 } from Tr is assiged to this subset. For any subset {b3 , . . . , bl+2 } of pairwise distinct elements of Bn that differ from b1 and b2 , consider the event that the table induced by b1 , . . . , bl+2 is equal ...
The strong completeness of the tableau method 1 The strong
... it. Indeed, we can simply imitate the original tableau, but wherever the restricted rule has been applied with a critical variable which is free in , we use instead any other variable that is neither free in the upper nodes of the branch in question nor in ; and from that point onwards we continu ...
... it. Indeed, we can simply imitate the original tableau, but wherever the restricted rule has been applied with a critical variable which is free in , we use instead any other variable that is neither free in the upper nodes of the branch in question nor in ; and from that point onwards we continu ...
val
... • Lambda Expressions – Form is based on notation e.g., (LAMBDA (x) (* x x) x is called a bound variable • Lambda expressions can be applied to parameters e.g., ((LAMBDA (x) (* x x)) 7) • LAMBDA expressions can have any number of parameters (LAMBDA (a b x) (+ (* a x x) (* b x))) Copyright © 2012 Ad ...
... • Lambda Expressions – Form is based on notation e.g., (LAMBDA (x) (* x x) x is called a bound variable • Lambda expressions can be applied to parameters e.g., ((LAMBDA (x) (* x x)) 7) • LAMBDA expressions can have any number of parameters (LAMBDA (a b x) (+ (* a x x) (* b x))) Copyright © 2012 Ad ...
Arithmetic as a theory modulo
... about predicate logic with no axioms: analyticity and non-provability results, completeness results for proof search algorithms, decidability results for fragments, constructivity results for the intuitionistic case. . . Unfortunately, the properties of cut free proofs do not extend in the presence ...
... about predicate logic with no axioms: analyticity and non-provability results, completeness results for proof search algorithms, decidability results for fragments, constructivity results for the intuitionistic case. . . Unfortunately, the properties of cut free proofs do not extend in the presence ...
The modal logic of equilibrium models
... about the there-world: a valuation that is at least as strong as the actual valuation; and [S] allows to talk about all here-worlds that are possible if we take the actual world as a there-world: it quantifies over all valuations that are weaker than the actual world. This language is again interpr ...
... about the there-world: a valuation that is at least as strong as the actual valuation; and [S] allows to talk about all here-worlds that are possible if we take the actual world as a there-world: it quantifies over all valuations that are weaker than the actual world. This language is again interpr ...
Propositional Logic What is logic? Propositions Negation
... • This is a contentious question! We will play it safe, and stick to: – “The systematic use of symbolic techniques and mathematical methods to determine the forms of valid deductive argument.” [Shorter Oxford Dictionary]. ...
... • This is a contentious question! We will play it safe, and stick to: – “The systematic use of symbolic techniques and mathematical methods to determine the forms of valid deductive argument.” [Shorter Oxford Dictionary]. ...
review of haskell
... Lambda expressions can be used to give a formal meaning to functions defined using currying. For example: add x y = x+y ...
... Lambda expressions can be used to give a formal meaning to functions defined using currying. For example: add x y = x+y ...
(pdf)
... This definition reads much as one would expect. For a theory to be consistent it must not prove opposite things, nor things known to be completely false. As a consequence of this, T is inconsistent iff T ` ϕ for each ϕ (because ∃ ϕ ≡ 0). Finally, if T ∪ {ϕ} is inconsistent, then there is an n s.t. T ...
... This definition reads much as one would expect. For a theory to be consistent it must not prove opposite things, nor things known to be completely false. As a consequence of this, T is inconsistent iff T ` ϕ for each ϕ (because ∃ ϕ ≡ 0). Finally, if T ∪ {ϕ} is inconsistent, then there is an n s.t. T ...
Computing Default Extensions by Reductions on OR
... step. This task is trivial, since each disjunct has a unique model. In our example, Op ∨ Oq represents two distinct extensions: The extension corresponding to Op is the set of consequences of p, whereas Oq corresponds to the set of consequences of q. The logic of O R builds on the contribution to on ...
... step. This task is trivial, since each disjunct has a unique model. In our example, Op ∨ Oq represents two distinct extensions: The extension corresponding to Op is the set of consequences of p, whereas Oq corresponds to the set of consequences of q. The logic of O R builds on the contribution to on ...
Aristotle`s work on logic.
... The s-rules don’t change the copula, so if M has two negative premises, then so does si (M ). The superaltern of a negative proposition is negative and the superaltern of a positive proposition is positive. Therefore, if M has two negative premises, then so does pi (M ). The m-rule and the per-rules ...
... The s-rules don’t change the copula, so if M has two negative premises, then so does si (M ). The superaltern of a negative proposition is negative and the superaltern of a positive proposition is positive. Therefore, if M has two negative premises, then so does pi (M ). The m-rule and the per-rules ...
Functional and Logic Programming
... (see. Imperative Programming): fun sum ( a : Array [ Int ] , n : Int ) : Int var s, i : Int ...
... (see. Imperative Programming): fun sum ( a : Array [ Int ] , n : Int ) : Int var s, i : Int ...
pdf
... if ϕ ∈ L is PSPACE-hard. (Of course, if we think of a modal logic as being characterized by an axiom system, then ϕ ∈ L iff ϕ is provable from the axioms characterizing L.) We say that ϕ is consistent with L if ¬ϕ ∈ / L. Since consistency is just the dual of provability, it follows from Ladner’s res ...
... if ϕ ∈ L is PSPACE-hard. (Of course, if we think of a modal logic as being characterized by an axiom system, then ϕ ∈ L iff ϕ is provable from the axioms characterizing L.) We say that ϕ is consistent with L if ¬ϕ ∈ / L. Since consistency is just the dual of provability, it follows from Ladner’s res ...
Unification in Propositional Logic
... The substitutions θaA used in the Boolean case, contribute to the construction of minimal bases of unifiers in IP C too. θaA is indexed by a formula A ∈ F (x) and by a classical assignment a over x. How does the transformation (θaA)∗ act on a Kripke model u : P −→ P(x)? First, it does not change th ...
... The substitutions θaA used in the Boolean case, contribute to the construction of minimal bases of unifiers in IP C too. θaA is indexed by a formula A ∈ F (x) and by a classical assignment a over x. How does the transformation (θaA)∗ act on a Kripke model u : P −→ P(x)? First, it does not change th ...