Indirect self-reference describes an object referring to itself indirectly. For example, define the function f such that f(x) = x(x). Any function passed as an argument to f is invoked with itself as an argument, and thus in any use of that argument is indirectly referring to itself. This example is similar to the Scheme expression "((lambda(x)(x x)) (lambda(x)(x x)))", which is expanded to itself by beta reduction, and so its evaluation loops indefinitely despite the lack of explicit looping constructs. An equivalent example can be formulated in lambda calculus. Indirect self-reference is special in that its self-referential quality is not explicit, as it is in the sentence "this sentence is false." The phrase "this sentence" refers directly to the sentence as a whole. An indirectly self-referential sentence would replace the phrase "this sentence" with an expression that effectively still referred to the sentence, but did not use the pronoun "this." An example will help to explain this. Suppose we define the quine of a phrase to be the quotation of the phrase followed by the phrase itself. So, the quine of: is a sentence fragment would be: "is a sentence fragment" is a sentence fragment which, incidentally, is a true statement. Now consider the sentence: "when quined, makes quite a statement" when quined, makes quite a statement The quotation here, plus the phrase "when quined," indirectly refers to the entire sentence. The importance of this fact is that the remainder of the sentence, the phrase "makes quite a statement," can now make a statement about the sentence as a whole. If we had used a pronoun for this, we could have written something like "this sentence makes quite a statement." It seems silly to go through this trouble when pronouns will suffice (and when they make more sense to the casual reader), but in systems of mathematical logic, there is generally no analog of the pronoun. It is somewhat surprising, in fact, that self-reference can be achieved at all in these systems. Upon closer inspection, it can be seen that in fact, the Scheme example above uses a quine, and f(x) is actually the quine function itself. Indirect self-reference was studied in great depth by W. V. Quine (after whom the operation above is named), and occupies a central place in the proof of GΓΆdel's incompleteness theorem. Among the paradoxical statements developed by Quine is the following: "yields a false statement when preceded by its quotation" yields a false statement when preceded by its quotation (Wikipedia).
This video explains and provides examples of direct and inverse or indirect variation. http://mathispower4u.wordpress.com/
From playlist Linear Equations and Lines
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π Learn how to evaluate inverse trigonometric functions. The inverse trigonometric functions are used to obtain theta, the angle which yielded the trigonometric function value. It is usually helpful to use the calculator to calculate the inverse trigonometric functions, especially for non-
From playlist Evaluate Inverse Trigonometric Functions
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π Learn how to evaluate inverse trigonometric functions. The inverse trigonometric functions are used to obtain theta, the angle which yielded the trigonometric function value. It is usually helpful to use the calculator to calculate the inverse trigonometric functions, especially for non-
From playlist Evaluate Inverse Trigonometric Functions
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This video provides two examples of how to determine the inverse function of a one-to-one function. A graph is used to verify the inverse function was found correctly. Library: http://mathispower4u.com Search: http://mathispower4u.wordpress.com
From playlist Determining Inverse Functions
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π Learn how to evaluate inverse trigonometric functions. The inverse trigonometric functions are used to obtain theta, the angle which yielded the trigonometric function value. It is usually helpful to use the calculator to calculate the inverse trigonometric functions, especially for non-
From playlist Evaluate Inverse Trigonometric Functions
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π Learn how to evaluate inverse trigonometric functions. The inverse trigonometric functions are used to obtain theta, the angle which yielded the trigonometric function value. It is usually helpful to use the calculator to calculate the inverse trigonometric functions, especially for non-
From playlist Evaluate Inverse Trigonometric Functions
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From playlist Functions
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From playlist Microsoft Excel Tutorial Videos π₯[2022 Updated]
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Subscribe! http://www.freemathvideos.com Want more math video lessons? Visit my website to view all of my math videos organized by course, chapter and section. The purpose of posting my free video tutorials is to not only help students but allow teachers the resources to flip their classro
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From playlist Introduction to Programming with Rust
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From playlist Crust of Rust
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This video provides two examples of how to determine the inverse function of a one-to-one function. A graph is used to verify the inverse function was found correctly. Library: http://mathispower4u.com Search: http://mathispower4u.wordpress.com
From playlist Determining Inverse Functions
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From playlist Rust Ports
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From playlist Introduction to Programming with Rust
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From playlist CS1: Higher Computing - Richard Buckland UNSW
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From playlist AgileRoots2010
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π Learn how to evaluate inverse trigonometric functions. The inverse trigonometric functions are used to obtain theta, the angle which yielded the trigonometric function value. It is usually helpful to use the calculator to calculate the inverse trigonometric functions, especially for non-
From playlist Evaluate Inverse Trigonometric Functions
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