Notes on Propositional Logic
... In propositional logic, we would like to apply operators not only to atomic propositions, but also to the result of applying other operators. This means that our language of well-formed formulas in propositional logic should be inductively defined as follows. Definition 1. For a given set A of propo ...
... In propositional logic, we would like to apply operators not only to atomic propositions, but also to the result of applying other operators. This means that our language of well-formed formulas in propositional logic should be inductively defined as follows. Definition 1. For a given set A of propo ...
Second order logic or set theory?
... • A second order sentence is complete if it has a model and for any second order sentence logically implies the sentence or its nega;on. • Categorical sentence are complete. • Ajtai: Axiom of Construc ...
... • A second order sentence is complete if it has a model and for any second order sentence logically implies the sentence or its nega;on. • Categorical sentence are complete. • Ajtai: Axiom of Construc ...
1.3 Functions, Continued
... Polynomials are used to model many situations in which change occurs at different rates. For example, the polynomial graphed on the right might represent the total cost of manufacturing x units of a product. At first, costs rise steeply because of high start-up expenses, then more slowly as the econ ...
... Polynomials are used to model many situations in which change occurs at different rates. For example, the polynomial graphed on the right might represent the total cost of manufacturing x units of a product. At first, costs rise steeply because of high start-up expenses, then more slowly as the econ ...
Introduction to Theoretical Computer Science, lesson 3
... Formula A is satisfiable in interpretation I, if there exists valuation v of variables that |=I A[v]. Formula A is true in interpretation I, |=I A, if for all possible valuations v holds that |=I A[v]. Model of a formula A is an interpretation I, in which A is true (that means for all valuations of ...
... Formula A is satisfiable in interpretation I, if there exists valuation v of variables that |=I A[v]. Formula A is true in interpretation I, |=I A, if for all possible valuations v holds that |=I A[v]. Model of a formula A is an interpretation I, in which A is true (that means for all valuations of ...
Jordan University of Science and Technology
... The Output is as the following: Please insert two numbers : 4.5 2 Their product is 9 ...
... The Output is as the following: Please insert two numbers : 4.5 2 Their product is 9 ...
Graph Sketcher Suggestions
... Using different numbers, substitute (x*number) for x in the function formula. Graph several such functions on the same picture, for example, (x*3)**2, (x*5)**2, (x*.1)**2 What do you see? Why does it happen? Try it with several different functions. What is different? What is the same? ...
... Using different numbers, substitute (x*number) for x in the function formula. Graph several such functions on the same picture, for example, (x*3)**2, (x*5)**2, (x*.1)**2 What do you see? Why does it happen? Try it with several different functions. What is different? What is the same? ...
Indexed Collections Let I be a set (finite of infinite). If for each
... Let I be a set (finite of infinite). If for each element i of I there is an associated object xi (exactly one for each i) then we say the collection xi , i ∈ I is indexed by I. Each element i ∈ I is called and index and I is called the index set for this collection. The definition is quite general. Usu ...
... Let I be a set (finite of infinite). If for each element i of I there is an associated object xi (exactly one for each i) then we say the collection xi , i ∈ I is indexed by I. Each element i ∈ I is called and index and I is called the index set for this collection. The definition is quite general. Usu ...
Data Types and Arithmetic Operations
... The data values must agree in type with the variables. The data values can be separated by blanks or ...
... The data values must agree in type with the variables. The data values can be separated by blanks or ...
Principia Mathematica
The Principia Mathematica is a three-volume work on the foundations of mathematics, written by Alfred North Whitehead and Bertrand Russell and published in 1910, 1912, and 1913. In 1927, it appeared in a second edition with an important Introduction To the Second Edition, an Appendix A that replaced ✸9 and an all-new Appendix C.PM, as it is often abbreviated, was an attempt to describe a set of axioms and inference rules in symbolic logic from which all mathematical truths could in principle be proven. As such, this ambitious project is of great importance in the history of mathematics and philosophy, being one of the foremost products of the belief that such an undertaking may be achievable. However, in 1931, Gödel's incompleteness theorem proved definitively that PM, and in fact any other attempt, could never achieve this lofty goal; that is, for any set of axioms and inference rules proposed to encapsulate mathematics, either the system must be inconsistent, or there must in fact be some truths of mathematics which could not be deduced from them.One of the main inspirations and motivations for PM was the earlier work of Gottlob Frege on logic, which Russell discovered allowed for the construction of paradoxical sets. PM sought to avoid this problem by ruling out the unrestricted creation of arbitrary sets. This was achieved by replacing the notion of a general set with the notion of a hierarchy of sets of different 'types', a set of a certain type only allowed to contain sets of strictly lower types. Contemporary mathematics, however, avoids paradoxes such as Russell's in less unwieldy ways, such as the system of Zermelo–Fraenkel set theory.PM is not to be confused with Russell's 1903 Principles of Mathematics. PM states: ""The present work was originally intended by us to be comprised in a second volume of Principles of Mathematics... But as we advanced, it became increasingly evident that the subject is a very much larger one than we had supposed; moreover on many fundamental questions which had been left obscure and doubtful in the former work, we have now arrived at what we believe to be satisfactory solutions.""The Modern Library placed it 23rd in a list of the top 100 English-language nonfiction books of the twentieth century.