In mathematics, this usually refers to some form of the principle, going back to Leibniz, that two mathematical objects are equal if there is no test to distinguish them. For example, given two mathematical functionsf and g, we can say that they are equal if
f(x) = g(x)
for all x in the common function domainX. This extensional equality is the usual definition if the function rangeY is also common to the two functions. If, on the other hand, we distinguish functions by the data attached to them in the type theory sense, so that we could for example choose a larger set Z as range for one of them, that equality is not the same sense extensional. That is one sense in which extensionality may fail. Another one is that consideration of the process by which a function is computed, if taken into account, will usually contradict extensionality.
In axiomatic set theory, extensionality is expressed in the axiom of extension, which states that two sets are equal if and only if they contain the same elements. In lambda calculus, extensionality is expressed by the eta-conversion rule, which allows conversion between any two expressions that denote the same function.
Instead, the criticism is intended to show that it is not in virtue of its being an instantiation of the property of weighing less than 180 pounds that the event causes the scales to tip; and that its being a `having' of this property is therefore causally irrelevant to that effect.
After all, given that the extensional definition contains no mention of the causal connection to breaking when struck, and given that none of the properties that it lists (F1-Fn) is essentially a typical cause of breaking when struck, it is hard to see why fragility should be thought to be so.
Unfortunately, once this parallel is noticed, it becomes evident that even though the strategy of extensional definition effects a tighter connection between dispositions and their causally relevant first-order bases, dispositions thus conceived do not inherit the causal powers of their first-order realizers.
Extensionalequality captures the mathematical notion of the equality of functions: that two functions are 'equal' if they always produce the same results for the same arguments.
In contrast, the terms themselves capture the notion of intensional equality of functions: that two functions are 'equal' only if they have identical implemenations.
It is a perhaps astonishing fact that S and K can be composed to produce combinators that are extensionallyequal to any lambda term, and therefore, by Church's thesis, to any computable function whatsoever.
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