Propositional Logic: Part I - Semantics
... “If pigs could fly then I’d enjoy brussel sprouts!” p : Pigs fly; b : Enjoy sprouts This (p |= b) is an invalid argument. Why use it? The real argument is: p, ¬p |= b which is a valid argument. Why is it valid? There is no counter example where p ∧ ¬p is true and b is false. Ex falso quod libet! i. ...
... “If pigs could fly then I’d enjoy brussel sprouts!” p : Pigs fly; b : Enjoy sprouts This (p |= b) is an invalid argument. Why use it? The real argument is: p, ¬p |= b which is a valid argument. Why is it valid? There is no counter example where p ∧ ¬p is true and b is false. Ex falso quod libet! i. ...
Discrete Structure
... equivalences instead. They provide a pattern or template that can be used to match all or part of a much more complicated proposition and to find an equivalence for it. ...
... equivalences instead. They provide a pattern or template that can be used to match all or part of a much more complicated proposition and to find an equivalence for it. ...
Relating Infinite Set Theory to Other Branches of Mathematics
... Let f be an axiom that asserts the existence of an infinite cardinal C at some level of the hierarchy of infinites. Then there exists an integer polynomial p(x1 . . . xn) such that *I am grateful to Martin Davis for discussions on this point. ...
... Let f be an axiom that asserts the existence of an infinite cardinal C at some level of the hierarchy of infinites. Then there exists an integer polynomial p(x1 . . . xn) such that *I am grateful to Martin Davis for discussions on this point. ...
Set theory, by Thomas Jech, Academic Press, New York, 1978, xii +
... false. Another front is large cardinals, or cardinals whose size transcends those which can be produced on the basis of the ZFC axioms alone. The smallest large cardinal is an inaccessible cardinal. If K is inaccessible, then, among other things, K > expa(Ko) for any finite or countable a, or for an ...
... false. Another front is large cardinals, or cardinals whose size transcends those which can be produced on the basis of the ZFC axioms alone. The smallest large cardinal is an inaccessible cardinal. If K is inaccessible, then, among other things, K > expa(Ko) for any finite or countable a, or for an ...
HISTORY OF LOGIC
... • Bertrand Russell (1872- 1970): – He is widely held to be one of the 20th century's premier logicians. ...
... • Bertrand Russell (1872- 1970): – He is widely held to be one of the 20th century's premier logicians. ...
Chapter 1: The Foundations: Logic and Proofs
... 1.4 Nested Quantifiers 1.5 Rules of Inference 1.6 Introduction to Proofs 1.7 Proof Methods and Strategy ...
... 1.4 Nested Quantifiers 1.5 Rules of Inference 1.6 Introduction to Proofs 1.7 Proof Methods and Strategy ...
22.1 Representability of Functions in a Formal Theory
... p(x)=y iff x=0 and y=0 or if x=y+1. Based on this analysis we define the predicate R p as Rp (x,y) ≡ (x=0 ∧ y=0) ∨ y+1=x. What remains to show is that this representation is in fact correct, that is that p(x)=y implies the validity of Rp (x,y) in Peano Arithmetic and that p(x)6=y implies the validit ...
... p(x)=y iff x=0 and y=0 or if x=y+1. Based on this analysis we define the predicate R p as Rp (x,y) ≡ (x=0 ∧ y=0) ∨ y+1=x. What remains to show is that this representation is in fact correct, that is that p(x)=y implies the validity of Rp (x,y) in Peano Arithmetic and that p(x)6=y implies the validit ...
College Geometry University of Memphis MATH 3581 Mathematical
... Proposition: Technically, any statement which has one of two values, True or False. However, the term “proposition” is also used to refer to a theorem (see below). Propositions may be thought of as the preliminary theory which follows from the axioms and postulates and are used to create more compli ...
... Proposition: Technically, any statement which has one of two values, True or False. However, the term “proposition” is also used to refer to a theorem (see below). Propositions may be thought of as the preliminary theory which follows from the axioms and postulates and are used to create more compli ...
Continuous Model Theory - Math @ McMaster University
... Fix a language L and fix a tuple of variables x from a sequence of sorts S. We define a pseudo-metric on the formulas with free variables x as follows: we define the distance between ϕ(x) and ψ(x) to be sup{|ϕM (a) − ψ M (a)| : M, an L-structure, and a ∈ M} We will call this space FS . This can also ...
... Fix a language L and fix a tuple of variables x from a sequence of sorts S. We define a pseudo-metric on the formulas with free variables x as follows: we define the distance between ϕ(x) and ψ(x) to be sup{|ϕM (a) − ψ M (a)| : M, an L-structure, and a ∈ M} We will call this space FS . This can also ...
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.