Basic Terms in Logic - Law, Politics, and Philosophy
... The truth value of a statement is not proven by logicians but of empirical scientists, researchers and private detectives. Logicians only study the reasoning found on statements and not the question of their truth values. ...
... The truth value of a statement is not proven by logicians but of empirical scientists, researchers and private detectives. Logicians only study the reasoning found on statements and not the question of their truth values. ...
Ch1 - COW :: Ceng
... Propositional logic is one of the simplest logics Propositional logic has direct applications e.g. circuit design There are efficient algorithms for reasoning in propositional logic Propositional logic is a foundation for most of the more expressive logics ...
... Propositional logic is one of the simplest logics Propositional logic has direct applications e.g. circuit design There are efficient algorithms for reasoning in propositional logic Propositional logic is a foundation for most of the more expressive logics ...
Syntax of first order logic.
... Syntax of first order logic. A first-order language L is a set {f˙i ; i ∈ I} ∪ {R˙j ; j ∈ J} of function symbols and relation symbols together with a signature σ : I ∪ J → N. In addition to the symbols from L, we shall be using the logical symbols ∀, ∃, ∧, ∨, →, ¬, ↔, equality =, and a set of variab ...
... Syntax of first order logic. A first-order language L is a set {f˙i ; i ∈ I} ∪ {R˙j ; j ∈ J} of function symbols and relation symbols together with a signature σ : I ∪ J → N. In addition to the symbols from L, we shall be using the logical symbols ∀, ∃, ∧, ∨, →, ¬, ↔, equality =, and a set of variab ...
λ Calculus - Computer Science at RPI
... expression evaluation terminate. Otherwise, the answer is no for expressions whose evaluation does not terminate. Consider the expression: (λx.(x x) λx.(x x)). It is easy to see that reducing this expression gives the same expression back, creating an infinite loop. If we consider the expanded expre ...
... expression evaluation terminate. Otherwise, the answer is no for expressions whose evaluation does not terminate. Consider the expression: (λx.(x x) λx.(x x)). It is easy to see that reducing this expression gives the same expression back, creating an infinite loop. If we consider the expanded expre ...
Notes Predicate Logic
... ∀ x, ∃y, P( x, y) 6≡ ∃ x, ∀y, P( x, y). Quantifiers are applied following a left to right precedence. That is, each quantifier applies to the statement to its right. Thus ∀ x, ∃y, P( x, y) asserts that for each x, it is true that there exists a y, which may depend on x, for which P( x, y) is true. O ...
... ∀ x, ∃y, P( x, y) 6≡ ∃ x, ∀y, P( x, y). Quantifiers are applied following a left to right precedence. That is, each quantifier applies to the statement to its right. Thus ∀ x, ∃y, P( x, y) asserts that for each x, it is true that there exists a y, which may depend on x, for which P( x, y) is true. O ...
the common rules of binary connectives are finitely based
... τ (p, q, r, s) = qq 2 (s2 s2 )p3 r3 (qq 2 (s2 s2 )p3 )3 as is shown by straight-forward calculation. Theorem 2. If `1 , . . . , `n are independent and f.b. then `1 ∩ . . . ∩ `n is f.b. Example 2. As is well known, |=→ , |=← , |=↔ , |=↑ are f.b. Since these logics are independent according to the abo ...
... τ (p, q, r, s) = qq 2 (s2 s2 )p3 r3 (qq 2 (s2 s2 )p3 )3 as is shown by straight-forward calculation. Theorem 2. If `1 , . . . , `n are independent and f.b. then `1 ∩ . . . ∩ `n is f.b. Example 2. As is well known, |=→ , |=← , |=↔ , |=↑ are f.b. Since these logics are independent according to the abo ...
Functional Programming
... (equal? x1 x2) tests whether x1 and x2 are structurally equivalent (equal? 'a 'a) #t (equal? '(a b) '(a b)) #t (member x xs) returns the sublist of xs that starts with x, or returns () (member 5 '(a b)) () (member 5 '(1 2 3 4 5 6)) (5 6) ...
... (equal? x1 x2) tests whether x1 and x2 are structurally equivalent (equal? 'a 'a) #t (equal? '(a b) '(a b)) #t (member x xs) returns the sublist of xs that starts with x, or returns () (member 5 '(a b)) () (member 5 '(1 2 3 4 5 6)) (5 6) ...
Joy: Forth`s Functional Cousin
... unordered collection of zero or more small integers, 0.. 31. Literals of type set are written inside curly braces, like this {3 7 21}. The usual set operations are available. A string is an ordered sequence of zero or more characters. Literals of this type string are written inside double quotes, li ...
... unordered collection of zero or more small integers, 0.. 31. Literals of type set are written inside curly braces, like this {3 7 21}. The usual set operations are available. A string is an ordered sequence of zero or more characters. Literals of this type string are written inside double quotes, li ...
19th Century Logic and 21st Century Computing
... the more motivated reader to puzzle out. But, remarkably, from modus ponens and the second and third axioms one can derive every truth about implication, including the first axiom. In addition to A implies B, written A → B, logicians considered other logical connectives, such as A and B written A ∧ ...
... the more motivated reader to puzzle out. But, remarkably, from modus ponens and the second and third axioms one can derive every truth about implication, including the first axiom. In addition to A implies B, written A → B, logicians considered other logical connectives, such as A and B written A ∧ ...
.pdf
... Substitution A|pB is the replacement of all occurrences of the variable p in A by the formula B. There are a few issues, however, that one needs to be aware of. Variables that are bound by a quantifier, must not be replaced, as this would change the meaning. ((∃p)(p⊃∼q))|qp should not result in ((∃p ...
... Substitution A|pB is the replacement of all occurrences of the variable p in A by the formula B. There are a few issues, however, that one needs to be aware of. Variables that are bound by a quantifier, must not be replaced, as this would change the meaning. ((∃p)(p⊃∼q))|qp should not result in ((∃p ...
notes
... exactly to the proof rules of propositional intuitionistic logic. Intuitionistic logic is the basis of constructive mathematics. Constructive mathematics takes a much more conservative view of truth than classical mathematics. It is concerned less with truth than with provability. Its main proponent ...
... exactly to the proof rules of propositional intuitionistic logic. Intuitionistic logic is the basis of constructive mathematics. Constructive mathematics takes a much more conservative view of truth than classical mathematics. It is concerned less with truth than with provability. Its main proponent ...
Lecture Slides
... definition, but generally speaking: Functional programming is style of programming in which the basic method of computation is the application of functions to arguments; ...
... definition, but generally speaking: Functional programming is style of programming in which the basic method of computation is the application of functions to arguments; ...
Sample Exam 1 - Moodle
... CSC 4-151 Discrete Mathematics for Computer Science Exam 1 May 7, 2017 ____________________ name For credit on these problems, you must show your work. On this exam, take the natural numbers to be N = {0,1,2,3, …}. 1. (6 pts.) State and prove one of DeMorgan’s Laws for propositional logic, using a t ...
... CSC 4-151 Discrete Mathematics for Computer Science Exam 1 May 7, 2017 ____________________ name For credit on these problems, you must show your work. On this exam, take the natural numbers to be N = {0,1,2,3, …}. 1. (6 pts.) State and prove one of DeMorgan’s Laws for propositional logic, using a t ...
Full version - Villanova Computer Science
... They can be divided into two major classes: Hilbert-style and Gentzen-style. Hilbert-style systems are axiom-based while Gentzen-style systems are rule-based. Gentzen-style systems have a number of advantages, including existence of straightforward proof search algorithms. In this course we will dea ...
... They can be divided into two major classes: Hilbert-style and Gentzen-style. Hilbert-style systems are axiom-based while Gentzen-style systems are rule-based. Gentzen-style systems have a number of advantages, including existence of straightforward proof search algorithms. In this course we will dea ...
22.1 Representability of Functions in a Formal Theory
... If y6=n, then p(x)6=y and ∼Rp (x,y) = ∼((n+1=0 ∧ y=0) ∨ y+1=n+1). The left disjunct in this formula is false because of the axiom non-surjective. Since y6=n, y must have a form different from n, which means that the right disjunct either has the form n+i+1+1=n+1 for some number i if y>n or the form ...
... If y6=n, then p(x)6=y and ∼Rp (x,y) = ∼((n+1=0 ∧ y=0) ∨ y+1=n+1). The left disjunct in this formula is false because of the axiom non-surjective. Since y6=n, y must have a form different from n, which means that the right disjunct either has the form n+i+1+1=n+1 for some number i if y>n or the form ...
Philosophy 120 Symbolic Logic I H. Hamner Hill
... • A truth-functional statement is a theorem if and only if it can be derived without using any premises. • The proof for a theorem must be either a CP or an RAA. ...
... • A truth-functional statement is a theorem if and only if it can be derived without using any premises. • The proof for a theorem must be either a CP or an RAA. ...
Lecture 11 Artificial Intelligence Predicate Logic
... appealing because you can derive new knowledge from old mathematical deduction. • In this formalism you can conclude that a new statement is true if by proving that it follows from the statement that are already known. • It provides a way of deducing new statements from old ones. ...
... appealing because you can derive new knowledge from old mathematical deduction. • In this formalism you can conclude that a new statement is true if by proving that it follows from the statement that are already known. • It provides a way of deducing new statements from old ones. ...
Chapter 11 slides
... • Church’s model of computing is called the lambda calculus – based on the notion of parameterized expressions (with each parameter introduced by an occurrence of the letter λ—hence the notation’s name. – Lambda calculus was the inspiration for functional programming – one uses it to compute by subs ...
... • Church’s model of computing is called the lambda calculus – based on the notion of parameterized expressions (with each parameter introduced by an occurrence of the letter λ—hence the notation’s name. – Lambda calculus was the inspiration for functional programming – one uses it to compute by subs ...
Slides - Chapter 10
... • Logic programming is tied to the notion of constructive proofs, but at a more abstract level: – the logic programmer writes a set of axioms that allow the computer to discover a constructive proof for each particular set of inputs Copyright © 2005 Elsevier ...
... • Logic programming is tied to the notion of constructive proofs, but at a more abstract level: – the logic programmer writes a set of axioms that allow the computer to discover a constructive proof for each particular set of inputs Copyright © 2005 Elsevier ...