Interpretation Definition and 834 Threads

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. Quotidian

    The Many-Worlds Interpretation of QM

    Greetings all. I'm a new poster here but have spent some time on philosophy forums previously. I subscribe to New Scientist and try and stay reasonably current with science, on the popular level at least. This week they have had a feature about physical cosmologies and 'theories of...
  2. kai_sikorski

    Probobalistic interpretation of a PDE

    Consider the following PDE. A lot of this is from "Numerical Analysis of an Elliptic-Parabolic Partial Differential Equation" by J. Franklin and E. Rodemich. \frac{1}{2} \frac{\partial^2 T}{\partial y^2} + y \frac{\partial T}{\partial x} = -1 With |x|<1, |y| < \infty and we require...
  3. Jim Kata

    Can Topos Theory Resolve Quantum Mechanics Paradoxes?

    I'm kind of rambling here since I'm not sure how to formulate what I'm trying to say. Is there an interpretation of quantum mechanics that uses topos theory? As in is there a way to fix some of the logical paradoxes of QM interpretations, by using a many valued logic system?
  4. ShayanJ

    Interpretation of temperature for negative ones

    In statistical mechanics and kinetic theory,it is shown that temperature can be interpreted as a measure of the energy of particles.But such an interpretation seems to need refinement because of introduction of negative absolute temperatures. Can we still interpret temperature as a quantity...
  5. N

    Geometrical Interpretation of V in Span S

    What does it mean geometrically to say that V lies in span S, in other words that V is in a linear combination of S?
  6. J

    De Broglie Bohm interpretation & virtual particles

    I sincerely apologise if this has been asked previously, I searched via Google and have been unable to find an answer I understand. How does the de Broglie-Bohm, or Pilot Wave, interpretation, well interpret virtual particles. The beauty of the deBB interpretation seems to be the unity of the...
  7. C

    Interpretation of Linear Variable Differential Transformer (LVDT) voltage output

    Good day to all Physics Forums members, I am currently undertaking a project which involves the use of a Linear Variable Differential Transformer (LVDT) to measure the distance an object makes between two points. My setup is as follows: LVDT Type: Solartron DC50 with sensitivity of 6.158...
  8. atyy

    Interpretation of quantum mechanics in cosmology

    In Mukhanov's Physical Foundations of Cosmology p348, he writes "Decoherence is a necessary condition for the emergence of classical inhomogeneities and can easily be justified for amplified cosmological perturbations. However, decoherence is not sufficient to explain the breaking of...
  9. snoopies622

    Interpretation of Klein-Gordon equation

    Comparing the Klein-Gordon equation to the equation of motion for a classical harmonic oscillator, I notice that for a particle of mass m, \frac {mc^2}{\hbar} is a frequency. Does this frequency have a physical meaning?
  10. M

    QM Interpretation & Young's Exp: Electron Position Defined?

    According to the Copenhagen Interpretation, and considering Young's double slit experiment: Is what interferes the mathematical wave function flowing out the two slits but the electron itself takes a well defined path? Anyhow, I need a clearer idea about the position definition or...
  11. T

    Historical Lorentz Interpretation and SR

    Is there anyone here who could help me understand the state of physicists understanding just prior to 1905 and Einstein's SR paper? I keep running into length contraction, considerations of simultaneity and its non-universal nature, Lorentz transformations, time dilation and other concepts that...
  12. A

    Physical Interpretation of point transformation invariance of the Lagrangian

    Homework Statement The problem asked us to show that the Euler-Lagrange's equations are invariant under a point transformation q_{i}=q_{i}(s_{1},...,s_{n},t), i=1...n. Give a physical interpretation. Homework Equations \frac{d}{dt}(\frac{\partial L}{\partial \dot{s_{j}}})=\frac{\partial...
  13. bohm2

    New Quantum Interpretation Poll

    A Snapshot of Foundational Attitudes Toward Quantum Mechanics http://lanl.arxiv.org/pdf/1301.1069.pdf
  14. P

    Question about Schrödinger equation interpretation

    Homework Statement Im reading and thinking about the interpretation about Schrödinger equation s solutions (wave functions) - what they really mean. What does the amplitude of wave function correspond to? Does it mean that if amplitude is greater then energy of particle is greater as well ...
  15. I

    Transactional Interpretation Absorber?

    I'm just curious, but I've been reading about Transactional Interpretation For some time now and I know about the law of causality, Wave function collapse, etc. My question is How would I know when a Absorber (object) emits a Confirmation wave to my emitter (laser) which offers an offer wave...
  16. L

    How Does Holonomy Relate to Curvature in Higher Dimensional Principal Bundles?

    On an SO(2) bundle over a smooth manifold the holonomy around a closed curve that bounds a disk equals the integral of the curvature 2 form over the interior of the disk. So holonomy measures curvature and visa vera. More generally if two closed curves are homologous then the difference...
  17. shounakbhatta

    Orthogonal set - Geometric interpretation

    Orthogonal set -- Geometric interpretation Hello, If we have two vectors u,v then in an inner product space, they are said to be orthogonal if <u,v>=0. Well, orthogonal means perpendicular in Euclidean space, i.e. 90 degrees. How <u,v> becomes zero. Secondly, if I have three vectors...
  18. Z

    Classical vs. Quantum interpretation of spin 4-vector

    I have a few basic questions about the Pauli-Lubanski spin 4-vector S. 1. I've used it in quantum mechanical calculations as an operator, that is to say each of the components of S is a matrix operator that operates on an eigenvector or eigenspinor. But my question is about the utility of S...
  19. Barnak

    Interpretation of the integration domain symbol

    I'm having an interpretation problem with the notation used in physics, under the integration sign. What is the proper interpretation of the domain of integration symbol, on the integration sign ? To be more precise, consider a function F(x) of one or several variables. Its integral on a...
  20. F

    Interpretation of finite element analysis results

    Hi, Recently started with FEA - loving it, at least the modelling / load application part. Interpreting the results is tricky - particularly around where loads are applied. Got a project (no pics sorry) which has a drilled hole in a plate, and have applied the load as a a pressure...
  21. F

    Physical interpretation of j in E&M field equations

    Hi all. I've been trying to study microwave and electromagnetic engineering . I'm not sure how I should interpret j in some of the field equations. For example, for the field equations for a rectangular waveguide resonant cavity are: E_{y} = E_{0} sin\frac{\pi x }{a} sin \frac{l \pi z}{a}...
  22. C

    Interpretation on the meaning of some graph theory statements

    Hello everyone, I'm studying basic graph theory, and my instructor gives me these statements to translate into pictures. I don't quite understand the meanings of the statements, but I have some thoughts about them. 1/ "Any two vertices a, b are connected by at least 2 distinct paths of...
  23. J

    Correct interpretation of terms in proper velocity expression?

    The "cosmological" proper distance from the origin, D(t), to an object at radial co-ordinate r at cosmological time t is given by D(t) = a(t) r(t) The corresponding "cosmological" proper velocity v of the object is given by v = \frac{dD}{dt} = \frac{da}{dt} r(t) + a(t) \frac{dr}{dt} Using...
  24. S

    What is the Significance of Subspaces in Linear Algebra?

    Ok, so I understand that a vector space is basically the span of a set of vectors (i.e.) all the possible linear combination vectors of the set of vectors... I don't understand the concept behind a subspace or why it's useful. I know the conditions are: 1. 0 vector must exist in the set...
  25. mnb96

    Geometric interpretation of metric tensor

    Hello, can anyone suggest a geometric interpretation of the metric tensor? I am also interested to know how we could "derive" the metric tensor (i.e. the matrix <ai,aj>) from some geometric considerations that we impose.
  26. P

    Visualizing Integration as Summing Over a Variable

    Hello all, I am sorry for this really basic question but am having trouble with visualizing something in my head... I read up on interpretation of integration as area under the curve by splitting it in strips with Δx as the length of the strip and we have the integral in the limit as Δx→0...
  27. R

    Damped oscillator- graphical interpretation

    Hi! The damped oscillator equation is as follows: x(t)= A exp(γt/2) cos(ωt) where ω= √( (w0)^2 + (γ^2)/4 ) I have attached a graph of a damped oscillator. The question is if I use graph to measure angular frequency, will it be w0 or ω? It should be w0 because if I put γ=0, I should...
  28. S

    Physical interpretation of one of Hamilton's equations

    Homework Statement I've attached a picture from a passage of my book (Liboff, Quantum Mechanics) with which I am having difficulty. Specifically, equation 1.25 claims to possesses a certripetal force factor (in the text underneith) and a moment arm factor. I see both of these terms present...
  29. C

    Full Width at Half Maximum (FWHM) Interpretation

    Very commonly in thin film articles/journals, I find the full width at half maximum of X-ray rocking curves associated with the quality (order) of the surface. For example, if the same material is grown on two different substrates, A and B, with FWHM values of 0.11° for A and 0.6° for B, why do...
  30. W

    Interpretation of deformation electron density maps (for Al alloys)

    I am posting this from a friend's account since I've been unable to register for a while. Brace yourselves for this is going to be a long post. ----------------------- TLDR: I am trying to figure out the reason for AlSi's lower than expected from atomic misfit solid solution hardening...
  31. R

    Operators interpretation (Dirac notation)

    Hi all! If you are given an operator such that A|1> = √(1/3) |1> +√(2/3) |2>, how do we interpret it? I do know that 1/3 and 2/3 are probabilities but is this operator application on state one suggesting that this state in state 1 and 2 with probabilities 1/3 and /3 respectively? Thank you!
  32. N

    What is the Geometrical Interpretation of line integral?

    So I'm able to calculate them no problem. But the problem is, I don't really understand what they mean. For example, W=∫Fdr I understand that a vector field is something that defines a vector at every point. Then if we pick a curve in the field, we are integrating along that curve. So does...
  33. T

    Triangle on a Sphere Question Interpretation

    Homework Statement Consider a tiling of the unit sphere in ##\mathbb R^3## by equilateral triangles so that the triangles meet full edge to full edge (and vertex to vertex). Suppose n such triangles meet an one vertex. Show that the only possibilities for n are ## n=3 ##, ##n = 4##, or...
  34. U

    Physical interpretation for this? (dynamics of Newton's method)

    Hello! I'm a math student, currently trying to write my diploma thesis. My field of study is complex dynamics (iteration of holomorphic/meromorphic functions, Julia sets and stuff). It's a farfetched idea, but currently I'm curious about a potential physical interpretation of the things I'm...
  35. K

    The commutator of position and momentum - interpretation?

    Hello friends! Im trying to get an geometric interpretation of QM and am now confused about the commutation relation between operators. Lets take momentum and position... sure, the fact that they don't commute show that we can not diagonalize them simultanesly. But what is the interpretation of...
  36. A

    Do you think this theory/website REALLY prove Many World Interpretation of QM?

    Hello all, On one forum I found this link: http://oyhus.no/QM_explaining_many-worlds.html claiming to demonstrate correctness of Many Wordls interpretation of QM. On the one hand - seems like well presented arguments. On the other hand: it claimes to be discovered in 1990,and it seems...
  37. A

    Interpretation of grand potential.

    Hello! I understand Gibbs energy and Helmholtz energy, but how is grand potential to be interpreted and where is it useful? For instance, Gibbs energy is the maximum amount of energy that can be used to do work at constant pressure-temperature while Helmholtz energy is the is the maximum...
  38. D

    Dot Product/Cross Product Interpretation, Geometric Construction

    Homework Statement Given the nonzero vector a ε ℝ3, a\dot{}x = b ε ℝ, and a × x = c ε ℝ3, can you determine the vector x ε ℝ3? If so, give a geometric construction for x. Homework Equations a\dot{}x = ||a||||x||cos\Theta The Attempt at a Solution I'm not really certain what it is...
  39. harrylin

    Interpretation and application of Poynting's theorem?

    In several recent threads the Poynting's theorem was brought up, and the discussion there became a distraction from the questions at hand without really being solved: https://www.physicsforums.com/showthread.php?t=621018 (starting #74) https://www.physicsforums.com/showthread.php?t=628896 (this...
  40. U

    MHB Interpretation of estimated coefficients

    http://img502.imageshack.us/img502/402/question2s.jpg
  41. A

    Black Hole Time Dilation: Light-ray Interpretation

    I know that a black hole creates infinite curvature in spacetime and hence infinite time dilation. I was wondering though, if I could think of this stopping of time due to the fact that a light ray moving radially towards the centre of a black hole would have to travel infinitely far along the...
  42. T

    Molecular interpretation of temperature

    i'm trying to do my ap physics summer work, but I've been having a hard time trying to teach this to myself. i do really well in class, but i need a little bit of guidance to help explain how to use the equations the right way. i'm currently stuck on a question about vrms an it's relation to...
  43. A

    Copenhagen interpretation of Big Bang

    How old was the universe when the first wave function collapsed? Just wondering. I've checked all the FAQs on this, and the one hit in Google doesn't really cover it from a cosmological angle. I see there is a "Participatory Anthropic Principle" but my question is really more about, well, how...
  44. I

    A. Neumaier's interpretation of quantum mechanics.

    I'd like to find out more about A. Neumaier's interpretation of quantum mechanics. How?
  45. D

    Interpretation of the determinant of an operator in complex vector space

    There's a geometric interpretation of the determinant of an operator in a real vector space that I've always found intuitive. Suppose we have a n-dimensional real-valued vector space. We can plot n vectors in an n-dimensional Cartesian coordinate system, and in general we'll have an...
  46. E

    Interpretation of FFT time frequency analysis diagram

    Hi, I was told that in order to analyze cycles, wave patterns etc in empirical data, the time frequency analysis using the discrete Fourier transform (or the fast Fourier transform) are most appropriate (instead of say the autocorrelation spectrum). Using the Python scipy.fftpack as...
  47. A

    Classical Charged Particle backward in time/Feynman Stueckelberg interpretation

    Hello Friends, I was trying to illustrate intuitively (rather than rigorously) the equivalence of negative energy solutions moving backward in time to be equal to a particle of opposite sign a la Feynman. I begin with a quote from the maestro himself "A backwards-moving electron when...
  48. I

    On the interpretation of a correlator with different definite states

    Hello everyone, I was reading Ashos Das book on field theory, chapter 4.3, and I had this question. This expression: \begin{equation} \left< \psi_f | \psi_i \right> \end{equation} is the transition amplitude of two states. This expression: \begin{equation} \frac{ \left<...
  49. A

    Wrong interpretation of physics

    I need help with this, please. Name three moments in the terminator 1 in which the physics was wrong. what was wrong with the physics, and what should have happened? I need it urgently and it is not a homework problem, but it is very urgent. thanks, i appreciate it
  50. T

    Exploring the Many Worlds Interpretation of Quantum Theory

    I have a question regarding the Many Worlds Interpretation(MWI) of quantum theory. Is there any proposed thought experiment that can/does provide some kind of solid/good proof for the MWI? I have an idea, an experiment and I am very curious about it's validity/soundness and the...
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