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. B

    Interpretation of induction in terms of symmetry transformations

    It just occurred to me that induction can be seen as a statement quite analogous to that of "a function whose derivative is 0 on an interval is constant on that interval". Suppose there is a property P about the natural numbers that we want to prove. Then let P: N -> {0, 1} be a function for...
  2. M

    Physical Interpretation of Coordinates in GR

    What is the relationship between the differentiable manifold that is space-time and the physical space around us? How does one relate the three seemingly Cartesian coordinates around us, those which we can measure out with a ruler, to the coordinates of the Lorentzian manifold? If i say, measure...
  3. Telemachus

    Continuum mechanics, physical interpretation of terms in balance eq.

    Hi there. I'm reading Gurtin's 'the mechanics and thermodynamics of continua', and working some exercises of his book. In the section 21: 'The first law: balance of energy', after the derivation of the balance equation, he uses an identity to rewrite the balance of energy. The balance of energy...
  4. microsansfil

    Wave propagation interpretation

    Hello, We read very often that wave propagation électro-magnetism is the result of an interaction between electric field and magnetic field. However, Maxwell's equations are not causal. These equations have no delay between electric field and magnetic field, they simply pose relations...
  5. arpansen

    Interpretation of the rating of a real transformer

    The rating of a real transformer is 15 kVA , 2300/230-V then can we say that the primary rms is 2300 V or can we say that the secondary rms is 230 V or both. To clarify further, suppose to solve for certain characteristics of this transformer like primary current and secondary current i use...
  6. WannabeNewton

    Interpretation of vorticity with non-vanishing strain tensor

    Hi guys. Let me just say at the outset that I know very little fluid mechanics but I keep coming back to the same issue over and over in a general relativity related problem so I figured I'd just ask the fluid mechanics question here. In countless places the interpretation of the vorticity...
  7. J

    Question Interpretation: "Describe the energy conversions in spring"

    Homework Statement Hi, I have a question in interpreting the following question: "Describe the energy conversions in a spring undergoing simple harmonic motion as it moves from the point of maximum compression to maximum stretch in a frictionless environment. Be sure to point out the...
  8. N

    Interpretation of the Van der Waals Equation

    There is a silly detail about the interpretation of the Van der Waals (VDW) equation that I cannot fully understand. Say we have the Van der Waals equation for one mole: (P + a / V^2 ) (V - b) = RT The usual interpretation is that if you start from the ideal gas law PV=RT, you have to...
  9. HakimPhilo

    What's the physical interpretation of ##\mu_0## and ##\varepsilon_0##?

    What are the physical interpretations of ##\mu_0## and ##\varepsilon_0##, the magnetic permeability and electric permittivity of vacuum? Can these be directly measured? How?
  10. T

    Looking for a hard determinism type of QM interpretation

    Hi all! I am wondering if there is an interpretation of QM where the future is "set in stone" (for lack of a better phrase). It can be unknowable (the future)...but it shouldn't be random in any way. Edit: basically I'm looking for a hard determinism type of QM interpretation.
  11. davidbenari

    How Should We Interpret Integrals: Sums or Antiderivatives?

    So I'm struggling if I should interpret the integral as a sum of infinitesimally small quantities or as just the antiderivative. I know these two things are equivalent but I can't think of those two things at the same time when I'm doing an integral. The reason I find this troublesome is...
  12. bcrowell

    How to rule out a classical interpretation for negative-energy states?

    The relativistic mass-energy-momentum relation m^2=E^2-p^2 predates quantum mechanics by a couple of decades. It allows a particle such as an electron to have a negative mass-energy. If it's 1906, and you're shown this equation, do you have any way to show that the negative-energy solutions...
  13. afcsimoes

    Exploring the Redshift Phenomenon: Beyond the Expansion of the Universe

    When a photon loses a little bit of energy, its frequency lows a little. That is, it suffers a slight redshift. So, why to interpret the redshift as being only due to the expansion of the Universe?
  14. J

    Oscillating E Field Interpretation of Light

    I don't know how to explain the classical view of an EM wave quantum mechanically. First off, when you see the graphical representation, there is an E field (x-axis) and a B field (y-axis) oscillating in parallel. Parallel to both of those is the direction of motion (z-axis)...
  15. J

    Static Friction Experiment: Help Result interpretation

    Hello, I hope someone can help me with this question. "Design a simple experiment that you could carry out in your home to i) determine the coefficient of static friction between an object and a surface. ii) prove that the coefficient of static friction is dependent only on the surfaces in...
  16. jedishrfu

    Revisiting an Old Interpretation of QM

    I found this article recently on Wired.com and thought others might like to see it. Basically its revisiting Bohmian Mechanics in the context of fluidics and how everyday objects can have quantum-like behavior explained via pilot waves: Fluid Experiments Support Deterministic...
  17. B

    Physical Interpretation of a Vector Quantity

    Homework Statement If \mathbf{r}, \mathbf{v},\mathbf{a} denote the position, velocity, and acceleration of a particle, prove that \frac{d}{dt} [\mathbf{a} \cdot (\mathbf{v} \times \mathbf{r})] = \dot{ \mathbf{a}} \cdot (\mathbf{v} \times \mathbf{r})Homework Equations The Attempt at a Solution...
  18. B

    Geometrical interpretation of this property

    Hi there! I have the following property: If x(t) is a solution of \left\{ \begin{array}{l} \dot{x} = f(x) \\ x(t_0) = x_0 \end{array} \right. then the function y(t) = x(t+t_0) is a solution of the equation with initial data y(0) = x_0 . How could it be interpreted geometrically? Thanks!
  19. J

    Geometric interpretation for d²f/dxdy

    If the following integral: $$\\ \iint\limits_{a\;c}^{b\;d} f(x,y) dxdy$$ represents: So which is the geometric interpretation for ##f_{xy}(x_0, y_0)## ?
  20. J

    Interpretation for voltage and flux linkage

    The interpretation for charge and current through of a circuit is known... "the current is the motion of the charges, electrons, that is repulsed by negative pole and attracted by positive pole through of a conducting wire...". Ok... But what is the explanation for flux linkage and voltage...
  21. C

    Did Einstein have a Physical Interpretation of Frames?

    By a frame I mean four orthogonal vectors for which one is timelike and the others are spacelike. In a bypassing some time ago I read something about the basis vectors in a frame representing sticks and and a clock. However, the author also noted that this interpretation was problematic. I...
  22. P

    Interpretation of the first passage time

    Hi all, Suppose I have a kinetic model for a stochastic system of three states, as shown in the attachment. I solve for the probability distribution of the first passage time from A to B and I get the distribution shown on the right hand side. I can understand that if there is a peak in the...
  23. J

    Geometrical interpretation of this coordinate transformation

    How can I geometrically interpret this coordinate transformation (from x,y space to \check{x},\check{y} space)? x = \check{x}cos(β) - \check{y}sin(β) y = \frac{1}{2}(\check{x}2 -\check{y}2)sin(2β) -\check{x}\check{y}cos (2β)
  24. Math Amateur

    Simple question of interpretation - D&F Ch 10 - Proposition 30

    I am reading Dummit and Foote, Section 10.5 : Exact Sequences - Projective, Injective and Flat Modules. I am studying Proposition 30 (D&F, page 389) I need some help in order to interpret one of the statements in Proposition 30. Proposition 29 and its proof (Ch 10, D&F page 388) reads...
  25. Math Amateur

    MHB Simple question of interpretation - D&F Ch 10 - Proposition 30

    I am reading Dummit and Foote, Section 10.5 : Exact Sequences - Projective, Injective and Flat Modules. I am studying Proposition 30 (D&F, page 389) I need some help in order to interpret one of the statements in Proposition 30. Proposition 30 reads as follows: Statement (3) of...
  26. B

    Interpretation to ARPES measured Fermi surface

    For any system, different experimental tools are able to measure the Fermi surface or electron dispersion. Then, are these Fermi surface or dispersion the ultimate outcome of the combined effects from the existing interactions? In other words, should the detected motion of electrons have been...
  27. A

    Probability interpretation question

    If a have the following formula ψ*(r1,...rn,σ1,...σn)ψ(r1,...rn,σ1,...σn) how can interpret it? the probability of finding a particle in a set of coordinates r1,...rn, with spin σ1,...σn or the probability of finding a particle in a set of coordinates r1,...rn,σ1,...σn ? I...
  28. P

    Rotating frame: a question of interpretation

    I'm reading Lanczos: 'The variational Principles of Mechanics'. I need help resolving a paradox - which is probably trivial… Lanczos (page 100 Dover edition) introduces a system, S', rotating at angular velocity \vec \Omega about an axis through a fixed point with respect to inertial system...
  29. H

    Interpretation verification: Partition function vs. number of states

    Greetings, I have been studying stat mech lately, and while I have gotten good at using partition functions to solve problems, I wanted to check my interpretation of what a partition function is, and especially to contrast it with the number of states. So, I'm just looking for a yes or no to...
  30. B

    Physical Interpretation of EM Field Lagrangian

    Using differential forms and their picture interpretations, I wonder if it's possible to give a nice geometric & physical motivation for the form of the Electromagnetic Lagrangian density? The Lagrangian for the electromagnetic field without current sources in terms of differential forms is F...
  31. L

    Generalized statistical interpretation

    I red griffiths many times but even now there is something I can't understand. It's about statistical interpretation. In his book chapter 3.4 he says "If you measure an observable Q(x,p) on a particle in the state ψ(x,t), you are certain to get one of the eigenvalues of the hermitian operator...
  32. Math Amateur

    Interpretation of 'homologous to' - Munkres analysis of homology grou

    I am reading James Munkres' book, Elements of Algebraic Topology. Theorem 6.2 on page 35 concerns the homology groups of the 2-dimensional torus. I would appreciate some help with interpreting the term 'homologous to' as it relates to a part of the proof of Munkres' Theorem 6-2 concerning...
  33. D

    Matrix notation for Lorentz transformations

    I'm having some confusion with index notation and how it works with contravariance/covariance. (v_{new})^i=\frac{\partial (x_{new})^i}{\partial (x_{old})^j}(v_{old})^j (v_{new})^i=J^i_{\ j}(v_{old})^j (v_{new})_i=\frac{\partial (x_{old})^j}{\partial (x_{new})^i}(v_{old})_j...
  34. N

    Geometric Interpretation of VSEPR Theory

    I am making a Geometric model for VESPR theory, which states that valence electron pairs are mutually repulsive, and therefore adopt a position which minimizes this, which is the position at which they are farthest apart, still in their orbitals. For example, the 2 electron pairs on either side...
  35. L

    Interpretation of Multiplicity Function

    Hi. I am trying to understand the multiplicity function Ω!/L!(Ω-L)! where Ω= number of boxes and L= number of distinguishable boxes. I just want a simple intuitive explanation. I have seen a couple of these but none of them ever stick. The term that confuses me the most is the (Ω-L)! Any help...
  36. E

    What is the geometric interpretation of the vector triple product?

    The interpretation of the vector product is the area of the parallelogram with sides made up of a and b and the scalar triple product is the volume of the parallelpiped with sides a, b, and c, but what is the interpretation of the vector triple product. Is it just simply the area of the...
  37. M

    Interpretation of Quantum Mechanics

    I want to ask you guys about the interpretation of Quantum Mechanics. I am using Griffiths' Introduction To Quantum Mechanics as a textbook. In this book, on chapter 1.3, there is an explanation about 3 different views of quantum mechanics, realistic, orthodox, agnostic. My...
  38. MexChemE

    Geometric interpretation of partial derivatives

    Good afternoon guys! I have some doubts about partial derivatives. The other day, my analytic geometry professor told us that slopes do not exist in three-dimensional space. If that's the case, then what does a partial derivative represent? Given that the derivative of a function with respect to...
  39. MexChemE

    Mathematical interpretation of work done by a gas

    Hello everybody! First of all, I would like to say that this is my first post in this forum, even though I've occasionally read some posts before. I'm a ChemE major from Mexico! I am currently taking Thermodynamics I, and I have trouble figuring out the expression: W = \int_{V_1}^{V_2} PdV I...
  40. U

    Is something wrong with statistical interpretation of QM?

    The probabilistic nature of QM seems apparent(in theoretical formulation as well as experimentally), then why do most of the physicists give the credit of in-determinism to the nature itself and look for many worlds, abruptly collapsing observer conscious experiment setups, and the likes...
  41. D

    Interpretation of Hodge Dual of Antisymmetric Tensors in GR

    Hiya, I am a grad student who has had a couple semesters of GR. I am currently perusing a book about Two Spinors in Spacetime by Penrose and Rindler, as background for an essay on Spinor Methods in GR. My question relates to the concept of taking the Hodge Dual of a antisymmetric tensor. I...
  42. P

    A question regarding the Copenhagen interpretation.

    hi guys, i just have a quick question concerning the Copenhagen interpretation of things. say there is a star, very far away (perhaps 1 billion light-years from Earth) that for the sake of this idea, has never been observed, by any intelligent being, anywhere. we happen to turn our telescope...
  43. A

    Inserted mass on a rotating ring, problem with solution interpretation

    Homework Statement Its about a system like in picture. I have it solved alredy BUT I still have a problem After getting the dynamic equation by Newtonian and Lagrangian methods I solved one of them making the hipótesis of Little oscillations. This diferential equation has three solutions...
  44. P

    Deriving the statistical interpretation from Schrodinger's equation?

    So, there are two things in Quantum Mechanics that I understand are axioms: the first is the schrodinger equation, which cannot be derived. Okay fine, we have to start somewhere. The second axiom is that the integral from a to b of the wavefunction-mod-squared gives the probability of finding...
  45. A

    Bogoliubov transformation / Interpretation of diagonalized Hamiltonian

    Hey, I consider a diagonalized Hamiltonian: H=\sum\limits_{k} \underbrace{ (\epsilon_{k} u_{k}^2 -\epsilon_{k} v_{k}^2 -2\Delta u_{k} v_{k} )}_{E_{k}}(d_{k \uparrow}^{\dagger}d_{k \uparrow} + d_{k \downarrow}^{\dagger}d_{k \downarrow}) +const with fermionic creation and annihilation...
  46. morrobay

    Interpretation Of Statistic: smoker mortality/total smokers = 1/5 ?

    Is this a correct conclusion from these three facts ? One in five cigarette smokers dies from smoking. There are about 45, 000,000 smokers in the U.S. There are about 450,000 smoker deaths/year from their smoking habit. So 450, 000/45,000,000 = 1% . So if the number of smokers (45,000,000/yr)...
  47. N

    Integration and interpretation of constant of acceleration-answer chec

    Homework Statement Derive equation 2.10 by integration equation 2.7 over time. You'll have to interpret the constant of integration. Homework Equations The Attempt at a Solution
  48. L

    Where Can I Find Comprehensive Notes on Quantum Mechanics Interpretations?

    Does anyone know good notes or a book on quantum mechanics that covers well the interpretational issues? Especially, which deals also with the last fifty or sixty years, i.e. that has Bell, decoherence, GHZ, Aspect experiment, Mach-Zehnder interferometer, delayed-choice, mesocopic Schrödinger...
  49. J

    Graphic Interpretation for Æ©f(x)Îx

    Graphic Interpretation for Ʃf(x)Δx Hello! I was trying to understand what means: \sum_{x_0}^{x_1}f(x)\Delta x (when Δx = 1 and x ∈ Z, ie, a "discrete integration", topic very comun in discrete calculus). I computed the result so: \sum_{1}^{4}x^2\Delta x=\left...
  50. K

    The interpretation of noether currents in scalar QED

    In scalar QED, there are two noether currents ##J_{global}## and ##J_{local}##corresponding to the global and local gauge transformations respectively. In QED, the two currents are exactly the same. But in scalar QED, they are totally different. $$J_{global}^\mu=i e (\phi^\dagger...
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