The problem of multiple generality names a failure in traditional logic to describe certain intuitively valid inferences. For example, it is intuitively clear that if: Some cat is feared by every mouse then it follows logically that: All mice are afraid of at least one cat. The syntax of traditional logic (TL) permits exactly four sentence types: "All As are Bs", "No As are Bs", "Some As are Bs" and "Some As are not Bs". Each type is a quantified sentence containing exactly one quantifier. Since the sentences above each contain two quantifiers ('some' and 'every' in the first sentence and 'all' and 'at least one' in the second sentence), they cannot be adequately represented in TL. The best TL can do is to incorporate the second quantifier from each sentence into the second term, thus rendering the artificial-sounding terms 'feared-by-every-mouse' and 'afraid-of-at-least-one-cat'. This in effect "buries" these quantifiers, which are essential to the inference's validity, within the hyphenated terms. Hence the sentence "Some cat is feared by every mouse" is allotted the same logical form as the sentence "Some cat is hungry". And so the logical form in TL is: Some As are BsAll Cs are Ds which is clearly invalid. The first logical calculus capable of dealing with such inferences was Gottlob Frege's Begriffsschrift (1879), the ancestor of modern predicate logic, which dealt with quantifiers by means of variable bindings. Modestly, Frege did not argue that his logic was more expressive than extant logical calculi, but commentators on Frege's logic regard this as one of his key achievements. Using modern predicate calculus, we quickly discover that the statement is ambiguous. Some cat is feared by every mouse could mean (Some cat is feared) by every mouse (paraphrasable as Every mouse fears some cat), i.e. For every mouse m, there exists a cat c, such that c is feared by m, in which case the conclusion is trivial. But it could also mean Some cat is (feared by every mouse) (paraphrasable as There's a cat feared by all mice), i.e. There exists one cat c, such that for every mouse m, c is feared by m. This example illustrates the importance of specifying the scope of such quantifiers as for all and there exists. (Wikipedia).
A10 Example problem of multiplicity three
An example problem of multiplicity three.
From playlist A Second Course in Differential Equations
Multivariable Calculus | Differentiability
We give the definition of differentiability for a multivariable function and provide a few examples. http://www.michael-penn.net https://www.researchgate.net/profile/Michael_Penn5 http://www.randolphcollege.edu/mathematics/
From playlist Multivariable Calculus
15_3_3 Continuation of previous problem
In the video I continue with the previous example problem.
From playlist Advanced Calculus / Multivariable Calculus
11_3_6 Continuity and Differentiablility
Prerequisites for continuity. What criteria need to be fulfilled to call a multivariable function continuous.
From playlist Advanced Calculus / Multivariable Calculus
Solve the general solution for differentiable equation with trig
Learn how to solve the particular solution of differential equations. A differential equation is an equation that relates a function with its derivatives. The solution to a differential equation involves two parts: the general solution and the particular solution. The general solution give
From playlist Differential Equations
(PP 6.2) Multivariate Gaussian - examples and independence
Degenerate multivariate Gaussians. Some sketches of examples and non-examples of Gaussians. The components of a Gaussian are independent if and only if they are uncorrelated.
From playlist Probability Theory
In this video, I present a very classical example of a duality argument: Namely, I show that T^T is one-to-one if and only if T is onto and use that to show that T is one-to-one if and only if T^T is onto. This illustrates the beautiful interplay between a vector space and its dual space,
From playlist Dual Spaces
Solving a multi-step equation by multiplying by the denominator
👉 Learn how to solve multi-step equations with variable on both sides of the equation. An equation is a statement stating that two values are equal. A multi-step equation is an equation which can be solved by applying multiple steps of operations to get to the solution. To solve a multi-s
From playlist How to Solve Multi Step Equations with Variables on Both Sides
15_4_1 Example problem with the line integral of a multivariable functions
In this example problem I look at solving a line integral of a multivariable function with respect to a coordinate variable.
From playlist Advanced Calculus / Multivariable Calculus
On Matrix Multiplication and Polynomial Identity Testing - Robert Andrews
Computer Science/Discrete Mathematics Seminar I Topic: On Matrix Multiplication and Polynomial Identity Testing Speaker: Robert Andrews Affiliation: University of Illinois Urbana-Champaign Date: January 30, 2023 Determining the complexity of matrix multiplication is a fundamental problem
From playlist Mathematics
Cyclic Groups -- Abstract Algebra Examples 7
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From playlist Abstract Algebra
Bryna Kra : Multiple ergodic theorems: old and new - lecture 1
Abstract : The classic mean ergodic theorem has been extended in numerous ways: multiple averages, polynomial iterates, weighted averages, along with combinations of these extensions. I will give an overview of these advances and the different techniques that have been used, focusing on co
From playlist Dynamical Systems and Ordinary Differential Equations
Rasa Reading Group: On Task-Level Dialogue Composition of Generative Transformer Model
This week we'll be starting a new paper: "On Task-Level Dialogue Composition of Generative Transformer Model" by Prasanna Parthasarathi, Mila Arvind Neelakantan and Sharan Narang from the First Workshop on Insights from Negative Results in NLP. Link to paper: https://www.aclweb.org/anthol
From playlist Rasa Reading Group
Algebraic geometry 51: Bezout's theorem
This lecture is part of an online algebraic geometry course, based on chapter I of "Algebraic geometry" by Hartshorne. The lectures continue in the playlist "scheme theory". It is about Bezout's theorem and its variations, which say that under some conditions the degree of an intersecti
From playlist Algebraic geometry I: Varieties
The local Gan-Gross-Prasad conjecture for real unitary groups - Hang Xue
Joint IAS/Princeton University Number Theory Seminar Topic: The local Gan-Gross-Prasad conjecture for real unitary groups Speaker: Hang Xue Affiliation: The University of Arizona Date: March 25, 2021 For more video please visit http://video.ias.edu
From playlist Mathematics
Gravity: Newtonian, post-Newtonian, Relativistic (Lecture 2) by Clifford M Will
DATES Monday 25 Jul, 2016 - Friday 05 Aug, 2016 VENUE Madhava Lecture Hall, ICTS Bangalore APPLY Over the last three years ICTS has been organizing successful summer/winter schools on various topics of gravitational-wave (GW) physics and astronomy. Each school from this series aimed at foc
From playlist Summer School on Gravitational-Wave Astronomy
Some detail about cyclic groups and their application to cryptography, especially Diffie Hellman Key Exchange.
From playlist PubKey
Commutative algebra 27 (Associated primes)
This lecture is part of an online course on commutative algebra, following the book "Commutative algebra with a view toward algebraic geometry" by David Eisenbud. We show that every finitely generated module M over a Noetherian ring R can broken up into modules of the form R/p for p prime
From playlist Commutative algebra
Solve an equation for x by clearing fractions with multiple steps
👉 Learn how to solve multi-step equations with variable on both sides of the equation. An equation is a statement stating that two values are equal. A multi-step equation is an equation which can be solved by applying multiple steps of operations to get to the solution. To solve a multi-s
From playlist How to Solve Multi Step Equations with Variables on Both Sides
Will Sawin - Bounding the stalks of perverse sheaves in characteristic p via the (...)
The sheaf-function dictionary shows that many natural functions on the F_q-points of a variety over F_q can be obtained from l-adic sheaves on that variety. To obtain upper bounds on these functions, it is necessary to obtain upper bounds on the dimension of the stalks of these sheaves. In
From playlist Franco-Asian Summer School on Arithmetic Geometry (CIRM)