Modal Logic
... R' = {(Wx X Wy) | X Y and for sone w X and w' Y, wRw'} We define the new frame (W^,R^) corresponding to (W,R) as W^ = xcl Wx R^ = R' xcl Rx) We extend (W^,R^) to a model by defining V^((w,i)) = V(w) for all w W and i N. We define a relation ~ W^ x W as follows: ~ = {((w,i),w)|w ...
... R' = {(Wx X Wy) | X Y and for sone w X and w' Y, wRw'} We define the new frame (W^,R^) corresponding to (W,R) as W^ = xcl Wx R^ = R' xcl Rx) We extend (W^,R^) to a model by defining V^((w,i)) = V(w) for all w W and i N. We define a relation ~ W^ x W as follows: ~ = {((w,i),w)|w ...
Mathematical Logic Deciding logical consequence Complexity of
... A proof of a formula φ is a sequence of formulas φ1 , . . . , φn , with φn = φ, such that each φk is either an axiom or it is derived from previous formulas by reasoning rules φ is provable, in symbols ` φ, if there is a proof for φ. Deduction of φ from Γ A deduction of a formula φ from a set of for ...
... A proof of a formula φ is a sequence of formulas φ1 , . . . , φn , with φn = φ, such that each φk is either an axiom or it is derived from previous formulas by reasoning rules φ is provable, in symbols ` φ, if there is a proof for φ. Deduction of φ from Γ A deduction of a formula φ from a set of for ...
Jacques Herbrand (1908 - 1931) Principal writings in logic
... that are used in the instances allow the use of generalization, simplification, and rules of passage so as to get G back. ...
... that are used in the instances allow the use of generalization, simplification, and rules of passage so as to get G back. ...
Stephen Cook and Phuong Nguyen. Logical foundations of proof
... language for the theory. This setup has its origins in Buss’ celebrated thesis Bounded arithmetic, Bibliopolis, 1986, for complexity classes beyond PH, and following Zambella Notes on polynomially bounded arithmetic, The Journal of Symbolic Logic, vol. 61 (1996), no. 3, pp. 942–966, the authors adap ...
... language for the theory. This setup has its origins in Buss’ celebrated thesis Bounded arithmetic, Bibliopolis, 1986, for complexity classes beyond PH, and following Zambella Notes on polynomially bounded arithmetic, The Journal of Symbolic Logic, vol. 61 (1996), no. 3, pp. 942–966, the authors adap ...
Sub-Birkhoff
... name with its rule is called an axiom. Subequational logics generate subequational theories. Definition 2 For a subequational logic L = hS,Ii its theory L is generated by the following inference rules, where an inference rule (i) only applies if i ∈ I. s, t and r range over terms. `sLs ...
... name with its rule is called an axiom. Subequational logics generate subequational theories. Definition 2 For a subequational logic L = hS,Ii its theory L is generated by the following inference rules, where an inference rule (i) only applies if i ∈ I. s, t and r range over terms. `sLs ...
Propositional Logic Predicate Logic
... Name of Symbols ∀ (universal quantifier), and ∃ (existential quantifier). Definition. A formula A is valid if A is true no matter how we replace the individual constants in A with concrete individuals and the predicate variables in A with concrete predicates. Note. The set of individuals must be in ...
... Name of Symbols ∀ (universal quantifier), and ∃ (existential quantifier). Definition. A formula A is valid if A is true no matter how we replace the individual constants in A with concrete individuals and the predicate variables in A with concrete predicates. Note. The set of individuals must be in ...
REVERSE MATHEMATICS Contents 1. Introduction 1 2. Second
... the set. Thus, RCA0 cannot prove the existence of non-computable sets. The following examples are computable, hence in RCA0 . Examples 4.1. The following exist in RCA0 : (i) constant functions, (ii) function composition, and (iii) characteristic functions. Proof. (i) Define a function f = {(x, y)|ϕ( ...
... the set. Thus, RCA0 cannot prove the existence of non-computable sets. The following examples are computable, hence in RCA0 . Examples 4.1. The following exist in RCA0 : (i) constant functions, (ii) function composition, and (iii) characteristic functions. Proof. (i) Define a function f = {(x, y)|ϕ( ...
HISTORY OF LOGIC
... The Logicist Period Gottlob Frege (1848 – 1925): – Considered to be the father of Analytic Philosophy. – His Objective was demonstrating that arithmetic is identical with logic. – He invented axiomatic predicate logic and quantified variables, which solved the problem of multiple generality. ...
... The Logicist Period Gottlob Frege (1848 – 1925): – Considered to be the father of Analytic Philosophy. – His Objective was demonstrating that arithmetic is identical with logic. – He invented axiomatic predicate logic and quantified variables, which solved the problem of multiple generality. ...
Study Guide Unit Test2 with Sample Problems
... 1. Be able to translate universally and existentially quantified statements in predicate logic and find their negation 2. Be able to recognize valid and invalid arguments in predicate logic, determine the inference rule applied and the types of errors. 3. Know how to prove statements using direct pr ...
... 1. Be able to translate universally and existentially quantified statements in predicate logic and find their negation 2. Be able to recognize valid and invalid arguments in predicate logic, determine the inference rule applied and the types of errors. 3. Know how to prove statements using direct pr ...
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 ...
Section 2.6 Cantor`s Theorem and the ZFC Axioms
... and uncountable. After Cantor’s death, due to the paradoxes of Bertrand Russell and others, various logicians, such as Ernst Zermelo and Abraham Fraenkel, placed the study of sets on a firm foundation with the introduction of a set of axioms. In 1938 under the framework of these axioms, the Austrian ...
... and uncountable. After Cantor’s death, due to the paradoxes of Bertrand Russell and others, various logicians, such as Ernst Zermelo and Abraham Fraenkel, placed the study of sets on a firm foundation with the introduction of a set of axioms. In 1938 under the framework of these axioms, the Austrian ...
The Axiom of Choice
... this is a typical example of a proof using the axiom of choice (in Zorn’s lemma form). Zorn’s lemma can also be used to prove statements like this: Every vector space has a basis (specifically, a “Hamel” basis). See http://planetmath.org/encyclopedia/ EveryVectorSpaceHasABasis.html. You may have see ...
... this is a typical example of a proof using the axiom of choice (in Zorn’s lemma form). Zorn’s lemma can also be used to prove statements like this: Every vector space has a basis (specifically, a “Hamel” basis). See http://planetmath.org/encyclopedia/ EveryVectorSpaceHasABasis.html. You may have see ...
