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. L

    Interpretation of Curvature

    On an oriented surface, the integral of the Gauss curvature over a smooth triangle can be interpreted as the angle of rotation of a vector that is parallel translated once around the three bounding edges. How does one interpret the integral of the Gauss curvature of an arbitrary SO(20 bundle...
  2. C

    What is the physical interpretation of this calculus of variations problem?

    It seems like a problem that a physicist would need to solve, but I can't wrap my head around the physical interpretation of it. http://exampleproblems.com/wiki/index.php/CoV7 Also, why do they use u=c*x2? What is c in this case? It says "classical" so it can't be the speed of light, right?
  3. Y

    Quantum Gravity & the Many-Worlds Interpretation

    I have read that MWI predicts that gravity is quantised. Is this prediction unique to MWI? If for example, someone discovered a quantum theory of gravity that was correct, that MWI would be declared the only valid interpretation of QM and that interpretations such the Copenhagen interpretation...
  4. Simfish

    Remote Sensing/Image Interpretation

    Some more info about it is here: http://rst.gsfc.nasa.gov/Front/tofc.html http://en.wikipedia.org/wiki/Remote_sensing Anyways, does anyone here know about remote sensing careers, and what it takes to get into one?
  5. K

    Complex numbers physical interpretation

    Hi guys. What is really the need for complex numbers? Is there any physical meaning associated with it?
  6. G

    On the interpretation of a spacetime diagram

    Below is a spacetime diagram which depicts the movement of a rod, four units long, relative to another rod of equal length. The red lines are the "moving" rod; the purple lines are the "stationary" rod. The gray lines are the world lines of the unit marks (0, 1, 2, 3, 4) on each rod. As I...
  7. A

    Interpretation and satisfaction

    Can anyone help me with the following? satisfaction and models: The sentence α : ∀x(odd(x) ⊃ even(s(x))) is not logically implied by the set S of 4 sentences: even(0) odd(s(0)) ∀x(even(x) ⊃ even(s(s(x)))) ∀x(even(s(x)) ⊃ odd(x)) Prove that this is so by identifying an interpretation...
  8. E

    Minimum uncertainty wave packets energy interpretation

    I am having trouble understanding (not for homework) what a wave packet is in terms of the correspondence of the idea of a wave packet to a "point like" particle. I'd like to focus on the 1d wave packet ultimately, but in order to describe my consternation -- let me detour to a well defined...
  9. N

    Interpretation of tensor indices

    Hi I am trying to figure out the physical interpretation of tensor indices. E.g., if we have an anisotropic material, then the tensor element of the susceptibility xx,y gives the response in the x-direction of an electric field in the y-direction. Lets not go up one dimension and look at...
  10. L

    Finding Solutions for the Intersection of a Plane and Line

    Homework Statement hi, i have been given a question and would appreciate help in interpreting it. given a plane, state the problem of finding points on the intersection of the plane and the line in the form Ax=d, where A is a 3by3 matrix, x and d are vectors. Homework Equations The...
  11. Pythagorean

    Cell tomography interpretation

    how would you learn to interpret the components of something like this? Am i seeing the whole cross-sectional area of a cell? http://learn.genetics.utah.edu/content/begin/cells/membranes/images/tomography.jpg (insulin-producing pancreas cell...
  12. A

    Is this interpretation logical? (Perturbation theory)

    Hi all. Just have a quick question on perturbation theory. Let's consider a molecule in ground electronic state. If a time-independent external perturbation acts on the molecule, the average electronic energy is going to change. From time-independent perturbation theory, we know that <E> =...
  13. I

    Possible argument in favor of many worlds interpretation

    Please forgive me if I am hideously wrong in any of my points. I am not terribly well-versed in quantum physics. Alright. This all began when I was thinking of a somewhat speculative and arguably philosophical idea that probably wouldn't have much of a place on this forum. My train of...
  14. D

    Interpretation of power rule for integration applied to 1/x

    We all know \int \frac{1}{x} dx = ln(x) + c but if you try to apply the power rule for integration: \int x^n dx = \frac{x^{n+1}}{n+1} + c you get \int x^{-1} dx = \frac{x^0}{0} What can you learn from this/what does this mean? David
  15. G

    C13 NMR Interpretation Homework - C,H,O 116g/mol

    Homework Statement I'm having a bit of difficulty interpreting the spectrum. I've given it a shot but I think I might be missing something. The compound contains only C,H,O and any assistance would be appreciated. The molecular weight is 116g/mol. Homework Equations none The Attempt...
  16. A

    Interpretation of Moment of Inertia Tensor in generalized coordinates

    I've been wondering what the interpretation of the moment of inertia tensor in generalized coordinates is, and whether there is a way to derive it from first principles, similar to the integration we do in a Cartesian coordinate system. Specifically, I've been given the inertia matrix for a...
  17. A

    Interpretation of P(A and B|C)

    Hello everybody! Should I interpret the expression P(A and B|C) as P((A and B)|C) or as P(A and (B|C))? Or the last expression has no meaning because P(B|C) is a ratio and I can't make an intersection with a ratio? Thanks, Alexei.
  18. T

    Special relativity adandons Maxwell's mechanical interpretation of EM?

    page 61 in Introduction To Electrodynamics (3rd ed, Griffiths) writes: " What exactly is an electric field? ... I encourage you to think of the field as a "real" physical entity, filling the space in the neighborhood of any electric charge. Maxwell himself came to believe that electric and...
  19. N

    Exploring Quantum Theory: Schrödinger's Cat & the Many-Worlds Interpretation

    Hi I was wondering if someone could better explain this. I do not quite understand the implications. Thanks for any input. "Unlike the Schrödinger's cat thought experiment which used poison gas and a radioactive decay trigger, this version involves a lethal weapon and a device that...
  20. F

    Geometric interpretation of the spacetime invariant

    For a euclidean space, the interval between 2 events (one at the origin) is defined by the equation: L^2=x^2 + y^2 The graph of this equation is a circle for which all points on the circle are separated by the distance L from the origin. For space-time, the interval between 2 events is...
  21. 1

    Velocity vs. Time Graphical Interpretation

    Homework Statement 1) Time = 0 to 2.5s a. What is the velocity during this time? b. Determine the displacement traveled during this time. c. What is the acceleration during this time? 2) Time = 5.5 to 10s a. What is the velocity during this time? b. Determine the...
  22. S

    Questions about Einstein SR interpretation

    Three simple experiments put to question Einstein's Special Relativity interpretation. It has nothing to do with the formulas. Formulas work pretty well; the BIG problem is interpretation! 1.The most recent particle collisions at CERN: Two Proton Beams at 3.5 TeV, traveling in opposite...
  23. R

    Penrose interpretation of quantum gravity

    hi, what do you think of the Penrose interpretation of quantum gravity, that gravity is responsible of the transitions between the quantum world and the macroscopic world?
  24. C

    Interpretation of Complex numbers - confused

    I know that generally complex numbers are represented in a two-dimensional plane with one real and one imaginary dimension. I also know that we have the quaternions, consisting of one real number and three imaginary numbers. The imaginary axis is always perpendicular to the real axis. The...
  25. P

    Does Pauli Exclusion still make sense when the particle interpretation fails?

    Pauli exclusion says no two particles should occupy the same state. Alternatively, it says that exchanging two particles generate a factor of -1. This is a basic fact about a spinor field, as a result of anticommutation relations. However, I hear that in strongly curved space time, QFTs have no...
  26. J

    The probability interpretation of the wavefunction?

    Hi, I'm about to go into my 4th and final year of my undergraduate physics degree and after all the quantum mechanics we've done so far, I still get this nagging feeling that I'm answering homework and exam problems blindly. Apologies for the length of this question by the way. For example...
  27. O

    Boltman interpretation of entropy

    we have been learning ststisticsl physics for the last month and the lecturer has still not explained what entropy is.Other than say "this is the boltman interpretation of entropy" all the info i find on the web seems to say different things for different situations so i was hoping someone...
  28. T

    A question on the interpretation of randomness in algorithmic information theory

    Hi everyone, I'd like to start by saying that I have no background or knowledge of maths other than high school. As a consequence, I have no idea if this question is stupid, trivial, nonsensical or whatever. I also don't even know if it's in the correct thread (I made the best guess I could). So...
  29. J

    Physical interpretation of operators in QM?

    hi, I am a novice to quantum mechanics and get a lot of troubles with operators. I cannot explain why: - why QM uses operators for observables such as position, momentum, energy, ..ect, but classical physics does not? - what are physical interpretations of operators? - why are operators needed...
  30. T

    What is the Combinatorial Interpretation of this Sum?

    Is there a combinatorial interpretation of the sum \sum_{k=0}^{n}\binom{k}{r}\binom{n-k}{m-r} ? If it were a sum over r instead, it would be n C m. But I don't know about this one.
  31. X

    Physical interpretation of a determinate?

    I just finished my first Linear Algebra class and loved it. There is one thing we didn't go over much though. What exactly IS a determinate? Is there a physical interpretation? Or is it just an operator that has these special properties? Thanks! ~Matt
  32. Y

    Second opinion on ODE interpretation sought.

    Hi, this is from a physics subforum of physicsforums: My calculus is not very good, but the above does not strike me as true and I would like a second opinion. If it is true, then my calculus is even worse than I thought and I need someone to explain to me how dr/dt=0 always implies that...
  33. S

    Error in CMB Interpretation? Implications?

    As noted in the thread "Dark Matter, On the Ropes?" there is disagreement with the observed velocity profile of spiral galaxies, the size of spiral galaxies' bulge, and the spiral galaxies' halo when compared to what theory predicts and what simulations with dark matter indicate...
  34. M

    A physical interpretation of classical electrodynamics

    I was wondering whether any members of this PF sub-forum would help me towards some physical interpretation of Maxwell’s time-dependent equations, which must ultimately underpin any classical description of EM wave propagation. I know that some might simply suggest reading a good textbook, but I...
  35. R

    Minkowsky force and physical interpretation.

    Homework Statement Compute the \mu=0 component of the Minkowski force law K^\mu=q\eta_\nu F^{\mu\nu}. (Einstien summation convention applies.) Homework Equations \eta_\nu=\frac{1}{\sqrt{1-u^2/c^2}}(-c,u_x,u_y,u_z) F^{\mu\nu} is the field tensor where...
  36. G

    Books on philosophy and interpretation of quantum mechanics

    Dear all, could you please suggest me books on philosophy and interpretation of quantum mechanics?? Kindest regards. Goldbeetle
  37. C

    What are the differences between the old and current Copenhagen Interpretation?

    Hi all, I have read in several places about an "old Copenhagen Interpretation" and I wonder what are the differences between the old one and the current one. I read several times Wikipedia article about CI but had no success finding any references about the old one. I guess that the old...
  38. D

    Confusing Interpretation of reality

    Mathematical constructs may or may not agree with reality (that is observed by us through our senses or high-tech instruments). What’s the touchstone that a mathematical construct for a theory is the right one and would not be violated in future? For example the mathematics that was used...
  39. FeDeX_LaTeX

    Vectors and Interpretation in Physics Calculations

    Hello; I have never really come across vectors before in physics, so could someone help me out? For example, what would be an example question and a calculation involving this; \phi_m = \int \vec{B} \cdot d \vec{A} Correct me if I'm wrong but is this saying that the magnetic flux is equal to...
  40. N

    Understanding the Propagator in Quantum Field Theory

    Hello PF :) Let me for the moment consider just <0|\varphi(y)\varphi(x)|0> as a propagator (instead of commutator of the fields)... and so in this expression evolves only <0|aa^{+}|0> part. Now my question is: 1) We can consider this expression as <0|a vector multiplied by a^{+}|0> which...
  41. W

    Breaking Causality: Physical Interpretation

    I understand that certain vectors in space-time (spacelike vectors) present a non-causal situation where E2 which takes place at a time after E1 in the rest frame cannot physically be a consequence in a different frame at sufficient speed. but there is also the relationship that arises from the...
  42. M

    A quick question on notation interpretation

    Hello, How is the following notation read? (a_n) is a bounded sequence. Let y_n = sup(a_k : k\geq n) What is meant by k\geq n ? Thank you for your help. M
  43. I

    Interpretation of the Debye length

    What exactly is the Debye length? From what I understand, when a charge q>0 is shielded by an electron cloud, then the Debye length is the distance the potential of the charge q falls by 1/e. Is it that simple? Furthermore, how do you go about deriving the expression...
  44. R

    Interpretation of the covariant derivative

    Is this the right way to think about the covariant derivative, and if not, what improvements would you suggest to visualize the meaning of the covariant derivative? (1)\mbox{ }\vec{e}_i(x')=\vec{e}_i(x)+\frac{\partial \vec{e}_i(x)}{\partial x^j}dx^j...
  45. icystrike

    Geometric interpretation and vector problem

    Homework Statement They were asking for the geometric interpretation and the says its triangular prism with infinite right angles. I don't understand what they mean by that. Homework Equations The Attempt at a Solution
  46. N

    Physical interpretation of Parity?

    I would like to know how can parity be physically interpreted :confused:?
  47. marcus

    Beyond the standard glosses of QM (the realistic clock interpretation)

    We've been watching this development taking shape ever since the tentative Gambini Pullin papers about it in 2006. It represents a fundamentally new way to address the problem of interpreting Quantum Mechanics. One person who worked on this back in 2006, with Gambini and Pullin, used to post...
  48. P

    Geometric interpretation of a given Alexandrov compactification

    What is the Alexandrov compactification of the following set and give the geometric interpretation of it: [(x,y): x^2-y^2>=1, x>0] that is, the right part of the hyperbola along with the point in it. This is a question from my todays exam in topology. I wrote that the given set is...
  49. R

    Questions re Cramer's Transactional Interpretation?

    Hello Forum Users, I am currently writing up a general discussion of John Cramer's Transactional Interpretation (TI) , intended to address questions, confusions, or concerns about TI. I have noticed occasional comments here and there regarding TI on this forum, both pro and con...
  50. S

    How Do IR Spectra Differ Between 4-Methylcyclohexene and 4-Methylcyclohexanol?

    1. Compare and interpret the infared spectra of 4-methylcyclohexene and 4-methylcyclohexanol. 2. Identify the C-H out-of-plane bending vibrations in the infared spectrum of 4-methylcyclohexene. What structural information can be obtained from these bands?
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