What is Quantification: Definition and 16 Discussions

In logic, a quantifier is an operator that specifies how many individuals in the domain of discourse satisfy an open formula. For instance, the universal quantifier

{\displaystyle \forall }
in the first order formula


{\displaystyle \forall xP(x)}
expresses that everything in the domain satisfies the property denoted by


{\displaystyle P}
. On the other hand, the existential quantifier

{\displaystyle \exists }
in the formula


{\displaystyle \exists xP(x)}
expresses that there is something in the domain which satisfies that property. A formula where a quantifier takes widest scope is called a quantified formula. A quantified formula must contain a bound variable and a subformula specifying a property of the referent of that variable.
The mostly commonly used quantifiers are

{\displaystyle \forall }

{\displaystyle \exists }
. These quantifiers are standardly defined as duals and are thus interdefinable using negation. They can also be used to define more complex quantifiers, as in the formula



{\displaystyle \neg \exists xP(x)}
which expresses that nothing has the property


{\displaystyle P}
. Other quantifiers are only definable within second order logic or higher order logics. Quantifiers have been generalized beginning with the work of Mostowski and Lindström.
First order quantifiers approximate the meanings of some natural language quantifiers such as "some" and "all". However, many natural language quantifiers can only be analyzed in terms of generalized quantifiers.

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  1. B

    How does a Gas Chromatograph-Mass Spectrometer detect quantities?

    So I am working on theoretical project that challenges my group to detect and quantify a certain molecule in a sample. For the detection and quantification we have chosen to work with a gas chromatography-mass spectroscopy. However I am finding difficulties in how this device is able to detect...
  2. H

    What is the International Standard for Quantifying Funk?

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  3. C

    I have a few questions about Formalization & Pseudo-code

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  4. C

    Atmospheric CO2 absorption - actual quantification?

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  5. S

    Can we derive quantification of energy mathematically?

    I'm studying for my introduction to modern physics exam, and reading about the ultraviolet catastrophe and why equipartition theorem stops working to explain blackbody emissions for high frequencies I was thinking if we could use the theorem to arrive at the quantification of energy (probably...
  6. T

    Quantification of Entropy and the 2nd Law of Thermodynamics

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  7. P

    I Could Planck have used different energy-frequency relation?

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  8. entropy1

    B Is quantification random or pseudo-random?

    Is quantification, such as for instance with the build op of an interference pattern by individual photons, the measuring of quantum spin, or the measuring of polarized photons through a polarisation filter actually, really random, or is it possible that it might be pseudo-random?
  9. C

    IOR Uncertainty Quantification for PIV/LDV Measurements

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  10. P

    Quantification of Levers and Similar Mechanical Devices

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  11. Gh778

    Force on surface with quantification

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  12. N

    Quantification logic and equivalence relations

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  13. B

    Rotor Dynamics Quantification HELP

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  14. N

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  15. M

    Proving Vacuous Quantification in First-Order Logic: A Guide

    I'm trying to prove a theorem which makes use of the identity \exists x (P) \rightarrow P (where x is not a free variable of P). Intuitively, I want to believe it, but since I'm trying to do things rigorously, I'd like to be able to justify it to myself. Can anyone offer a suggestion as to how...
  16. I

    Exploring the Quantification of 'All' with Fibonacci Numbers

    Yesterday I was enlightening a friend of mine concerning the many wonders of Fibonacci numbers and the golden ratio (let this friend be represented as X). As I was speaking with X, I learned that another friend of mine (let him be represented as V) was listening very attentively. Here is our...