formulasearchengine - User contributions [en]

Music without sound

2014-12-14T09:22:39Z

130.64.160.71: changed from gendered (only male - he) to neutral. Not share if there's a general she/he preference to neutral pronoun policy but srsly little things count

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Asymptotic theory

2014-10-20T17:48:11Z

130.64.35.137: /* Asymptotic distribution */

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Benzene (data page)

2014-10-02T19:53:05Z

130.64.82.132:

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Cartan decomposition

2014-09-19T19:22:00Z

130.64.79.215: Started disambiguating a distinct notion of Cartan pair

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Self-financing portfolio

2014-06-24T19:28:53Z

130.64.26.145:

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Rough set

2014-01-30T19:39:38Z

130.64.16.30: /* Objective analysis */ etal -> et al.

A '''free logic''' is a [[logic]] with fewer [[existential clause|existential]] presuppositions than classical logic. Free logics may allow for [[Term (first-order logic)|terms]] that do not denote any object. Free logics may also allow [[structure (mathematical logic)|models]] that have an [[empty domain]]. A free logic with the latter property is an '''inclusive logic'''.

== Explanation ==

In [[classical logic]] there are theorems that clearly presuppose that there is something in the [[domain of discourse]]. Consider the following classically valid theorems.

:1. <math> \forall xA \rightarrow \exists xA</math>
:2. <math> \forall xA \rightarrow A(r/x)</math> (where r does not occur free for x in A and A(r/x) is the result of substituting r for all free occurrences of x in A)
:3. <math> Ar \rightarrow \exists xAx</math> (where r does not occur free for x in A)

A valid scheme in the theory of [[First-order_logic#Equality_and_its_axioms|equality]] which exhibits the same feature is

:4. <math> \forall x(Fx \rightarrow Gx) \land \exists xFx \rightarrow \exists x(Fx \land Gx)</math>

Informally, if F is '=y', G is 'is Pegasus', and we substitute 'Pegasus' for y, then (4) appears to allow us to infer from 'everything identical with Pegasus is Pegasus' that something is identical with Pegasus. The problem comes from substituting nondesignating constants for variables: in fact, we cannot do this in standard formulations of [[first-order logic]], since there are no nondesignating constants. Classically, ∃x(x=y) is deducible from the open equality axiom y=y by particularization (i.e. (3) above).

In free logic, (1) is replaced with

:1b. <math> \forall xA \land E!t \rightarrow \exists xA</math>, where E! is an existence predicate (in some but not all formulations of free logic, E!t can be defined as ∃y(y=t))

Similar modifications are made to other theorems with existential import (e.g. the Rule of Particularization becomes (Ar → (E!r → ∃xAx)).

[[Axiom]]atizations of free-logic are given in [[Jaakko Hintikka|Hintikka]] (1959),<ref>Jaako Hintikka (1959). Existential Presuppositions and Existential Commitments. Journal of Philosophy 56 (3):125-137.</ref> [[Karel Lambert|Lambert]] (1967), [[Theodore Hailperin|Hailperin]] (1957), and [[Richard L. Mendelsohn|Mendelsohn]] (1989).

==Interpretation==

[[Karel Lambert]] wrote in 1967:<ref>''Free Logic and the Concept of Existence'' by Karel Lambert, Notre Dame Journal of Formal Logic, V.III, numbers 1 and 2, April 1967</ref>

"In fact, one may regard free logic... literally as a theory about singular existence, in the sense that it lays down certain minimum conditions for that concept." The question that concerned the rest of his paper was then a description of the theory, and to inquire whether it gives a necessary and sufficient condition for existence statements.

Lambert notes the irony in that [[Willard Van Orman Quine]] so vigorously defended a form of logic that only accommodates his famous dictum, "To be is to be the value of a variable," when the logic is supplemented with [[Bertrand Russell|Russellian]] assumptions of [[Theory of Descriptions|description theory]]. He criticizes this approach because it puts too much ideology into a logic, which is supposed to be philosophically neutral. Rather, he points out, not only does free logic provide for Quine's criterion—it even proves it! This is done by brute force, though, since he takes as axioms <math> \exists xFx \rightarrow (\exists x(E!Fx))</math> and <math>Fy \rightarrow (E!y \rightarrow \exists xFx)</math>, which neatly formalizes Quine's dictum. So, Lambert argues, to reject his construction of free logic requires you to reject Quine's philosophy, which requires some argument and also means that whatever logic you develop is always accompanied by the stipulation that you must reject Quine to accept the logic. Likewise, if you reject Quine then you must reject free logic. This amounts to the contribution that free logic makes to ontology.

The point of free logic, though, is to have a formalism that implies no particular ontology, but that merely makes an interpretation of Quine both formally possible and simple. An advantage of this is that formalizing theories of singular existence in free logic brings out their implications for easy analysis. Lambert takes the example of the theory proposed by [[Wesley C. Salmon]] and [[George Nahknikian]],<ref>George Nahknikian and Wesley C. Salmon, "'Exists' as a Predicate" ''Philosophical Review'' Vol. 66: 1957 pp. 535-542</ref> which is that to exist is to be self-identical.

==See also==
* [[Square of opposition]]
* [[Table of logic symbols]]

==References==

{{reflist}}

* Lambert, Karel, 2003. ''Free logic: Selected essays.'' Cambridge Univ. Press.
*-------, 2001, "Free Logics," in Goble, Lou, ed., ''The Blackwell Guide to Philosophical Logic''. Blackwell.
* ------, 1997. ''Free logics: Their foundations, character, and some applications thereof.'' Sankt Augustin: Academia.
* ------, ed. 1991. ''Philosophical applications of free logic.'' Oxford Univ. Press.
* Morscher, Edgar, and Hieke, Alexander, 2001. ''New essays in free logic.'' Dordrecht: Kluwer.

== External links ==
*{{sep entry|logic-free|Free logic|John Nolt}}

{{DEFAULTSORT:Free Logic}}
[[Category:Non-classical logic]]

CM-field

2013-07-04T18:01:03Z

130.64.79.183: Silly typo

The '''hot chocolate effect''', also known as the '''allassonic effect''', is a phenomenon of [[Wave equation|wave mechanics]] first documented in 1982 by Frank Crawford, where the pitch heard from tapping a cup of hot liquid rises after the addition of a soluble powder.<ref name=hotchocolate>Frank S. Crawford, May 1982, "The hot chocolate effect", ''American Journal of Physics'', Volume 50, Issue 5, pp. 398-404, [[doi:10.1119/1.13080]] (Abstract only)</ref><ref>Frank S. Crawford, November 1990, "Hot water, fresh beer, and salt", ''American Journal of Physics'', Volume 58, Issue 11, pp. 1033-1036, [[doi:10.1119/1.16268]] (Abstract only)</ref> It was first observed in the making of [[hot chocolate]] or [[instant coffee]], but also occurs in other situations such as adding salt to [[Supersaturation|supersaturated]] hot water or cold beer. Recent research has found many more substances which create the effect, even in initially non-supersaturated liquids.<ref name=BARDS>D. Fitzpatrick ''et al.'', March 2012, "Principles and Applications of Broadband Acoustic Resonance Dissolution Spectroscopy (BARDS): A Sound Approach for the Analysis of Compounds", ''Analytical Chemistry'', Volume 84, Issue 5, pp. 2202-2210, [[doi:10.1021/ac202509s]]</ref>

It can be observed by pouring hot milk into a mug, stirring in chocolate powder, and tapping the bottom of the mug with a spoon while the milk is still in motion. The pitch of the taps will increase progressively with no relation to the speed or force of tapping. Subsequent stirring will gradually decrease the pitch again.
Upon initial stirring, entrained gas bubbles reduce the speed of sound in the liquid, lowering the frequency. As the bubbles clear, sound travels faster in the liquid and the frequency increases.

== Origin of the phenomenon ==
The phenomenon is explained by the effect of bubble density on the [[speed of sound]] in the liquid. The note heard is the [[frequency]] of a [[standing wave]] where a quarter [[wavelength]] is the distance between the base of the mug and the liquid surface. This frequency ''f'' is equal to the [[speed]] ''v'' of the [[wave]] [[division (mathematics)|divided]] by four times the height of the water column h:
:<math>
f = 0.25\frac{v}{h}
</math>

The speed of sound in a homogeneous liquid or gas is dependent on the fluid's mass density (<math>\rho</math>) and compressibility (<math>K</math>), according to the equation:
:<math>
v = \frac{1}{\sqrt{K\rho}}
</math>

Water is approximately 800 times denser than air, and air is approximately 15,000 times more compressible than water. When water is filled with air bubbles, however, the fluid's density is very close to the density of water, but the compressibility will be the compressibility of air. This greatly reduces the speed of sound in the liquid. Wavelength is constant for a given volume of fluid, therefore the frequency (pitch) of the sound will decrease as long as gas bubbles are present.<ref name=hotchocolate />

Different rates of bubble formation will generate different acoustic profiles, allowing differentiation of the added solutes.<ref name=BARDS /><ref>D. Fitzpatrick ''et al.'', 2012, "Blend uniformity analysis of pharmaceutical products by Broadband Acoustic Resonance Dissolution Spectroscopy (BARDS)", ''International Journal of Pharmaceutics'', Volume 438, Issue 1-2, pp. 134-139, [[doi:10.1016/j.ijpharm.2012.07.073]]</ref><ref>D. Fitzpatrick ''et al.'', 2013, "The relationship between dissolution, gas oversaturation and outgassing of solutions determined by Broadband Acoustic Resonance Dissolution Spectroscopy (BARDS)", ''Analyst'', Volume 138, Issue 17, pp. 5005-5010, [[doi:10.1039/C3AN36838F]]</ref>

==See also==
*[[Broadband Acoustic Resonance Dissolution Spectroscopy]], a spectroscopic technique that uses the Hot Chocolate Effect as its fundamental principle.

==References==
<references />

==External links==
*[http://www.acoustics.org/press/143rd/Rossing.html Sound of a Cup With and Without Instant Coffee: A Foam-Filled Acoustics Demonstration], Andrew Morrison and Thomas D. Rossing, 143rd ASA Meeting, Pittsburgh.
*[http://www.kilty.com/coffee.htm The cheap instant coffee effect] Account by Kevin T Kilty
*[http://physics.weber.edu/carroll/SciMethodAAS.ppt The hot chocolate effect: introducing the scientific method] PowerPoint presentation by Bradley Carroll and Michelle More
*[http://mup.piettes.com/Hot_Chocolate/index.htm Hot Chocolate Experiment] In an educational format by Michael Piette. Includes recorded sounds.

[[Category:Wave mechanics]]
[[Category:Acoustics]]
[[Category:Physics experiments]]

Isobutane (data page)

2012-02-18T19:19:55Z

130.64.138.130: Undid revision 465301678 by 212.14.228.250 (talk)

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