What is Interpretation: Definition and 831 Discussions

An interpretation is an assignment of meaning to the symbols of a formal language. Many formal languages used in mathematics, logic, and theoretical computer science are defined in solely syntactic terms, and as such do not have any meaning until they are given some interpretation. The general study of interpretations of formal languages is called formal semantics.
The most commonly studied formal logics are propositional logic, predicate logic and their modal analogs, and for these there are standard ways of presenting an interpretation. In these contexts an interpretation is a function that provides the extension of symbols and strings of symbols of an object language. For example, an interpretation function could take the predicate T (for "tall") and assign it the extension {a} (for "Abraham Lincoln"). Note that all our interpretation does is assign the extension {a} to the non-logical constant T, and does not make a claim about whether T is to stand for tall and 'a' for Abraham Lincoln. Nor does logical interpretation have anything to say about logical connectives like 'and', 'or' and 'not'. Though we may take these symbols to stand for certain things or concepts, this is not determined by the interpretation function.
An interpretation often (but not always) provides a way to determine the truth values of sentences in a language. If a given interpretation assigns the value True to a sentence or theory, the interpretation is called a model of that sentence or theory.

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  1. H

    Correct quantum numbers interpretation?

    Greetings everyone, I'm going through Griffiths QM, and just wanted to make sure I understood the roles (where they come from, etc.) of the quantum numbers n, l, and m. What is written below is basically an outline of my current understanding, of which I am seeking either a confirmation or...
  2. R

    Arrow of time in copenhagen interpretation.

    Most physicists don't draw a distinction between past, present and future.. this is called the arrow of time. Physicists such as Sean caroll, Paul Davies e.t.c But if all events exist in some sense then the future is <not> open. If the future is not open then determinism is true.. thus...
  3. Q

    Overlapping vs Non-overlapping worlds in Many Worlds/Everettian Interpretation

    As the poll on this forum from 2011 that has recently been resurrected shows, there are a portion of the users here who adhere to the modern reading of the Everettian interpretation. What I have never seen discussed here though is the issue of whether the worlds split(overlap) or...
  4. M

    What is the qm interpretation of moseley's law?

    moseley did experiments on various elements and showed that the frequency of the k-alpha emission was proportional to the square of the proton no. The formula is (z-1)^2 proportional to the square root of frequency, with the -1 thrown in for the screening effect. The bohr model can explain...
  5. S

    What is the Born Interpretation in Quantum Mechanics?

    Hi, I've started to watch some lectures on quantum mechanics & they're going well except for the fact that some of it makes no sense. Basically I just don't see how |ψ|²dτ represents the probability of finding a particle described by ψ in the volume element dτ. Most likely it's due to me having...
  6. A

    What is the physical interpretation of zero divergence?

    When a vector field representing a physical quantity (e.g. B) has ∇\cdotB = 0 what is then the physical interpretation of this? Some people have said that the field doesn't diverge away from anything, but as far as I can tell magnetic field can easily get weaker and weaker the further you go...
  7. C

    Many worlds interpretation, probability distribution

    Where exactly do the probability distributions we observe come from under the many worlds interpretation? I know it has something to do with being the only stable configurations in Hilbert space but I don't understand why. This is my attempt to explain it in my own way, please tell me if it...
  8. S

    Heiseinberg's Microscope - Trigonometric and Interpretation questions

    Homework Statement The problem along with its solution is attached as ProblemSolution.jpg. Homework Equations Δx = λ/sinθ (Eq. 1) Δp_x = (h/λ)(sinε) (Eq. 2) The Attempt at a Solution In Wikipedia, I found this article: http://en.wikipedia.org/wiki/Heisenberg's_microscope which uses...
  9. T

    Interpretation of quantum fields and their ground state energies.

    Quantum field theory predicts a value for the cosmological constant that is 123 orders of magnitude larger than the observed value (if one assumes the Standard Model to be correct up to the Planck scale of 10^19 GeV)! To theoretically predict the value of the cosmological constant, one must, I...
  10. C

    Vector Mechanics, Interpretation of Problem

    Hi there. I am an Aerospave Engineering student, revisiting Vector Mechanics. My course recommended textbook is "Vector Mechanics of Enginners, from Beer and Johnston". However, I have been complemeting with another book - " Methods of Analytical Dynamics, from Meirovitch". This book, treats...
  11. A

    Modal Analysis - Interpretation of Stability Diagrams

    Hello! I'm doing a modal analysis and I'm trying to extract modal parameters from my aqcuired data. I'm using the Pulse Reflex Software. I got to the point where the Software provides me with a MIF and a stability diagram and I am to choose the parameters. However, I am unsure how...
  12. jfy4

    SL(3,R) Generators and their Interpretation

    Hi, I would like to study SL(3,\mathbb{R}) a little. I was motivated to look into them a bit because of their volume-preserving nature. Now this group has n^2-1=9-1=8 generators and I found out through some reading that three of them are the generators for rotation (duh Judah, volume...
  13. I

    Interpretation of torsion vs riemann tensor

    Hi all, I am working through Visser's notes http://msor.victoria.ac.nz/twiki/pub/Courses/MATH465_2012T1/WebHome/notes-464-2011.pdf section 3.5 onward. I am trying to differentiate between the torsion and the Riemann curvature tensor in a heuristic manner. It appears from "Geometric...
  14. S

    What is the interpretation of the vacuum correlation function in QFT

    I'm currently trying to make some intuitive sense out of Quantum field theory, but I'm not really understanding the vacuum. Consider a real (or complex, with + in the right places) scalar particle (a Klein-Gordon field). Now consider the propagator (or correlation function) G(x-y)=...
  15. A

    Frightening Double Slit Experiment Interpretation

    I noticed a youtube video, where Thomas Warren Campbell, a physicist and author of the book trilogy My big TOE the last standing for Theory of Everything and apparently he also works for NASA, deducts some very disturbing things from a certain double slit experiment. Here it is, I marked the...
  16. K

    How can a basis in a vector space be used to determine linear independence?

    From what I understand, a basis is essentially a subset of a vector space over a given field. Now what I'm not so sure of is the linearly independence part. If the basis has two linearly independent vectors, then than means they aren't collinear: rather, they wouldn't have the same slope...
  17. C

    Statistical interpretation (intro to quantum)

    When we calculate the average of anything: we add up (or integrate) the sum{ [all the things were taking the average of] * [the probability of getting that thing ]}. The thing about the average I'm that curious about is for example: the average height of 5 people turns out to be say 6...
  18. S

    Ohmic Resistor modelling ?(Check my interpretation of question)

    http://dl.dropbox.com/u/33103477/Resistor.png For the first bit first I solved the equation \frac{dI}{dt}=\frac{U-RI}{L} I(t)=\frac{U-C_2e^{\frac{-t}{L}}}{R} Now I put in T(0)=0 to work out the constant and I got I(t)=\frac{U-Ue^{\frac{-t}{L}}}{R} Now here's the dodgy bit, I did...
  19. F

    Diracs delta equation - general interpretation

    Im really just searching for a general explanation! If you are solving a pretty standard left hand side differential equation, but a diracs delta function on the right hand side. I am abit confused about how to interpret this. If this is the case for the right hand side: r(t) = Diracs (t)...
  20. K

    T( period) Vs. L (length) Interpretation

    Homework Statement Now , analyze the data with a power function least square fit. Let the x values be the length and the y value be the period. From the results of the power fit, compare the experimental value of the period to that calculated USING THE POWER FIT YOU JUST DETERMINED...
  21. S

    Help with interpretation of question. (Differential equations)

    http://dl.dropbox.com/u/33103477/Linear%20oscilator.png I am having trouble understand the question, what I have done its solve the equation using the substitution x=e^{rx} Then, I have the solution given by: x(t)=c_1 e^{t(\sqrt{\gamma^2 - \omega^2 })} + c_2e^{-t(\sqrt{\gamma^2 -...
  22. W

    Possible 3rd Interpretation of the Michelson-Morley Experiment

    I am a programmer by trade, so I am more adept at logic than mathematics. One thing that has always perplexed me about the Michelson-Morley experiment was how it is used to reach the conclusion that there is no ether. Just given basic logic, there is a third alternative that I have never seen...
  23. T

    Physical Interpretation of Spin

    We are starting to learn about spin in my introductory quantum mechanics course, and I was wondering if anyone could provide a physical interpretation of an electron's spin. I understand its a form of angular momentum which has nothing to do with the motion of the electron in space, but since an...
  24. S

    Physical Interpretation of Reynolds Number

    Wikipedia: "Reynolds Number is a dimensionless number that gives a measure of the ratio of inertial forces to viscous forces." What is meant by the inertial forces? I have a much better intuition of viscosity. But it seems to be like the Reynolds Number predicts turbulent flow for a...
  25. M

    Interpretation of Confidence Intervals and the like

    Statistics question: I'm having trouble understanding my statistics professor's objection to this interpretation of confidence intervals. If you have some distribution with parameter x: X = Dist(x), and you perform a random experiment drawing N random variables from it, and derive from those...
  26. Physics Monkey

    Does quantum mechanics partially supply its own interpretation?

    I will try to keep things brief. I am trying to understand what is considered interpretation. Imagine we have a totally isolated and complex quantum system in a fixed initial state which we can prepare with essentially perfect fidelity. Suppose that this system is macroscopic in which small...
  27. E

    Quick question about interpretation of contravariant and covariant components

    Can the covariant components of a vector, v, be thought of as v multiplied by a matrix of linearly independent vectors that span the vector space, and the contravariant components of the same vector, v, the vector v multiplied by the *inverse* of that same matrix? thinking about it like that...
  28. P

    Reasoning and Writing Question, what's your interpretation?

    Homework Statement A student takes a 30-question mathematics exam on which she got n questions correct and m questions incorrect and left k questions blank. If correct answers are worth 4 points, incorrect answers are worth -1 points, and blank answers are worth 0 points, write an equation to...
  29. P

    Interpretation of homomorphism question

    Hey, I had this question I thought i answered but now I'm questioning if the group operation is supposed to be addition and not multiplication. The question \varphi :\mathbb{Z}\to {{\mathbb{Z}}_{n}}\,\,\,\,\,where\,\,\varphi (a)=\text{Remainder}\left( \frac{a}{n} \right) When I...
  30. J

    Interpretation of Q.M. (with more options)

    I know, I know, it might be a little bit spam-y to post another poll, but since the option 'Other' was gaining so much attention, I decided to create a new poll with more options so that people could be more accurate in their description. Since I support the MWI, I will post a few links with...
  31. J

    Poll: Interpretations of Quantum Mechanics

    I was wondering who/how many people in the forum subscribe to which interpretation, so I decided to make a poll :) Also anyone is welcome to post content about each interpretation to illuminate people who may not know the difference and/or definition of an interpretation.
  32. E

    Relativistic interpretation question?

    Two individuals A and B are traveling through space at a velocity in the "x" direction at and are next to each other so they can synchronize their clocks. Since there is no other available reference they believe they are not moving. Both have a duplicate shape charge which as they are in the...
  33. A

    Interpretation of projective space

    Hi, I am am currently taking a second course in geometry, the first part of the course concerns projective geometry, and I feel I'm not getting the picture. I would like to know what the motivation for the development of projective geometry is. What picture you guys have in your head of...
  34. S

    Understanding the copenhagen interpretation

    This is an extract from the lecture notes I took for the 'Foundations of QM' third year course. Copenhagen QM - classical-quantum division State: wavefunction ψ(x); (ψ,\varphi) = \int d^{3}r ψ^{*}(r)\varphi(r) Evolution: TDSE Observables (\hat{x},\hat{p},\hat{H}): A = A-dagger...
  35. M

    Interpretation of dx as the differential of x for Indefinite Integrals

    Interpretation of "dx" as the differential of x for Indefinite Integrals This question is concept-as-opposed-to-calculation based. I understand that when one sees the integral sign, followed by f(x)dx, that we can think of this as the indefinite integral, or antiderivative of f(x), with...
  36. A

    Second Law's Interpretation

    I have this physical chemistry this semester with a very small section on thermodynamics--namely the Second Law and its applications to feasibility of reactions.Because our class is not very familiar with the Second Law(this is our second sem and we were taught only the First Law in school)our...
  37. G

    Finite element analysis results interpretation

    Hello, I have been following this forum for some time now, but this is the first time I participate. I would appreciate if you could give me a hand in understanding the results of a finite element analysis of an automotive component we have designed. The analysis results indicate that the...
  38. X

    Why is there no 'classical' interpretation of movements in quantum mechanics?

    There are tons of interpretations of quantum mechanics, but I'm unaware of any that are 'classical' as in being local, having realism and determinism. There is Bells famous work proving that that can't be true for certain aspects of quantum mechanics, but why aren't there any classical...
  39. J

    Holographic Universe Interpretation

    I was watching "The Fabric of the Cosmos: What is Space?" on pbs.com, and I couldn't help but wonder a few things after looking up some information about it. I find it funny that people have been using String Theory with only two dimensions (since they are supposed to keep adding dimensions...
  40. F

    Interpretation of Gyroscopic effect and rotation

    hello forum, we all know that a rotation wheel, a rotating football, etc.. maintain their direction of motion with more stability than if they were not spinning... A bicycle wheel is suspended from one of end of its axie by a rope, and spun up by hand. The wheel's axle is then placed...
  41. D

    Interpretation of Quantum Mechanics - which one?

    Hello, Before I ask, I'd just to say that I know this has been heavily discussed before, but I am asking for peoples' insight directly as this is for a report I'm currently working on, hence why I'm not just looking through old posts using the search option. So I'm sorry if this may be of an...
  42. J

    Simplified Afshar experiment: many-worlds interpretation

    Hi, I would be interested in what people think of the following experiment. Imagine a laser beam is split into two coherent beams A and B that are made to cross each other at right-angles. Beyond the point of intersection let us assume that there is a detector A in the path of beam A and...
  43. A

    Geometric Interpretation of √(x^2+y^2+z^2) and its Derivative

    Let r = √(x^2+y^2+z^2) One can easily show that \nablar= \vec{r}/r. But I'm having a hard time understanding what this means geometrically - who can help? :)
  44. M

    Interpretation of a rare event - lucky draw

    Hi, I was at a party where there was a lucky draw. There were about 200 people and each of them had put a chit with their name on it, in a transparent bowl. There were 10 prizes. The announcer chose a person at random to pick a chit for each prize. For one of the prizes, the picker picked his...
  45. A

    Mathematical Logic, Interpretation, Satisfiable, Consequence relation

    Theorem : let A be a set of formulas, a be a formula For all A and all a, Every interpretation which is a model of A is also a model of a iff not (Sat A) U {~a} Proof Every interpretation which is a model of A is also a model of a iff(1) there is no interpretation which is a model of A but...
  46. N

    Many world interpretation on large scale

    can many world interpretation be valid for large scale collapses that brings out a single reality?
  47. P

    Interpreting a PDE for Heat/Diffusion with Vanishing Neumann Boundary Conditions

    Given \triangle u = f(x,y,z) on a given body with vanishing neumann boundary conditions. I'm asked to interpret it in terms of heat and diffusion. Since heat/diffusion take the form u_t = k \triangle u, I am a little confused as I there is no time term here. I think the answer is that u...
  48. G

    Which Quantum Interpretation could make a difference?

    Is there an interpretation of quantum mechanics, which could make a difference to future calculations and results? Maybe an interpretation which could rewrite and simplify the algebra? Or one which has new testible results which when correct would change the quantum mechanical equations to make...
  49. R

    Physical interpretation of conductivity with electromagnetic waves?

    Hi all, I'm trying to understand exactly what the physical meaning of conductivity/current is in relation to waves. if we have a wave traveling through a conductor, we find that it decays exponentially, i.e. e^{-\alpha z} where \alpha=imag(k)=\omega\sqrt{\frac{\epsilon\mu}{2}}...
  50. M

    An interpretation of quantum mechanics in terms of Brownian–like motion.

    Edward Nelson, a Princeton mathematician showed that quantum mechanics could be derived from the principle that elementary particles are subjected to a universal jiggling of an unspecified cause. As I understand the obstacle to use this mathematical model as a physical one, is that dynamics of...
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