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False equivalence is a common result when an anecdotal similarity is pointed out as equal, but the claim of equivalence does not bear scrutiny because the similarity is based on oversimplification or ignorance of additional factors. The pattern of the fallacy is often as such:
In breaking it up, he segregates those disturbing causes, whose wanderings happen to be inconvenient, for the time in a pound called Ceteris Paribus. The study of some group of tendencies is isolated by the assumption other things being equal: the existence of other tendencies is not denied, but their disturbing effect is neglected for a time.
[3] That is the meaning intended by statisticians when they say causation is not certain. Indeed, p implies q has the technical meaning of the material conditional: if p then q symbolized as p → q. That is, "if circumstance p is true, then q follows." In that sense, it is always correct to say "Correlation does not imply causation."
The first use of an equals sign, equivalent to + = in modern notation. From The Whetstone of Witte (1557) by Robert Recorde. Recorde's introduction of =."And to avoid the tedious repetition of these words: "is equal to" I will set as I do often in work use, a pair of parallels, or twin lines of one [the same] length, thus: ==, because no 2 things can be more equal." [5]
Another way to think of some of the cases is in terms of sampling, in statistics. Imagine a population of X items (or people), of which we choose N. Two different schemes are normally described, known as "sampling with replacement" and "sampling without replacement". In the former case (sampling with replacement), once we've chosen an item, we ...
In algebraic expressions, equal variables may be substituted for one another, a facility that is not available for equivalence related variables. The equivalence classes of an equivalence relation can substitute for one another, but not individuals within a class. A strict partial order is irreflexive, transitive, and asymmetric.
Or put another way, x is applied to y inside the environment z. Given S and K, I itself is unnecessary, since it can be built from the other two: ((S K K) x) = (S K K x) = (K x (K x)) = x. for any term x. Note that although ((S K K) x) = (I x) for any x, (S K K) itself is not equal to I. We say the terms are extensionally equal.
As another example, the statement "the solution to an indefinite integral is sin(x), up to addition of a constant" tacitly employs the equivalence relation R between functions, defined by fRg if the difference f−g is a constant function, and means that the solution and the function sin(x) are equal up to this R.