Problem of multiple generality
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The problem of multiple generality names a failure in traditional logic to describe certain intuitively valid inferences. For example, it is intuitively clear that if:
- Some cat is feared by every mouse
then it follows logically that:
- All mice are afraid of at least one cat
However, it is not possible to express this inference in the traditional system, because the manner we should represent the first term in the classical subject-predicate form, ensures that the second term our predicate will be "X is feared by every mouse", which places the quantifier "every" out of reach of the syllogisms available in the theory.
When mediæval logicians recognised this problem, they saw that it was possible to add further, more complex syllogisms to the theory to allow such inferences, but all attempts to add such inferences still left other intuitively valid inferences unaccounted for, that arose from a similar pattern.
The first logical calculus capable of dealing with such inferences was Gottlob Frege's Begriffsschrift, the ancestor of modern predicate logic, which dealt with quantifiers by means of variable bindings. Rather oddly, Frege did not argue that his logic was more expressive than extant logical calculi, but commentators of Frege's logic regard this as one of his key achievements.