The problem of multiple generality names a failure in

traditional logic In logic and formal semantics, term logic, also known as traditional logic, syllogistic logic or Aristotelian logic, is a loose name for an approach to formal logic that began with Aristotle and was developed further in ancient history mostly by ...

to describe valid inferences that involves multiple quantifiers. 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''. The syntax of

(TL) permits exactly one quantifier, i.e. there are four sentence types: "All A's are B's", "No A's are B's", "Some A's are B's" and "Some A's are not B's". Since the sentences above each contain two quantifiers ('some' and 'every' in the first sentence and 'all' and 'at least one' in the second sentence), they cannot be adequately represented in TL. The best TL can do is to incorporate the second quantifier from each sentence into the second term, thus rendering the artificial-sounding terms 'feared-by-every-mouse' and 'afraid-of-at-least-one-cat'. This in effect "buries" these quantifiers, which are essential to the inference's validity, within the hyphenated terms. Hence the sentence "Some cat is feared by every mouse" is allotted the same

logical form In logic, the logical form of a statement is a precisely specified semantic version of that statement in a formal system. Informally, the logical form attempts to formalize a possibly ambiguous statement into a statement with a precise, unamb ...

as the sentence "Some cat is hungry". And so the logical form in TL is: :''Some A's are B's'' :''All C's are D's'' which is clearly invalid. The first logical calculus capable of dealing with such inferences was

Gottlob Frege Friedrich Ludwig Gottlob Frege (; ; 8 November 1848 – 26 July 1925) was a German philosopher, logician, and mathematician. He was a mathematics professor at the University of Jena, and is understood by many to be the father of analytic philos ...

's ''

Begriffsschrift ''Begriffsschrift'' (German for, roughly, "concept-writing") is a book on logic by Gottlob Frege, published in 1879, and the formal system set out in that book. ''Begriffsschrift'' is usually translated as ''concept writing'' or ''concept notati ...

'' (1879), the ancestor of modern

predicate logic First-order logic, also called predicate logic, predicate calculus, or quantificational logic, is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantified variables ove ...

, which dealt with quantifiers by means of variable bindings. Modestly, Frege did not argue that his logic was more expressive than extant logical calculi, but commentators on Frege's logic regard this as one of his key achievements. Using modern predicate calculus, we quickly discover that the statement is ambiguous. :''Some cat is feared by every mouse'' could mean ''(Some cat is feared) by every mouse'' (paraphrasable as ''Every mouse fears some cat''), i.e. :''For every mouse m, there exists a cat c, such that c is feared by m,'' :

\forall m \, (\, \text(m) \rightarrow \exists c \, (\text(c) \land \text(m,c)) \, )

in which case the conclusion is trivial. But it could also mean ''Some cat is (feared by every mouse)'' (paraphrasable as '' There's a cat feared by all mice''), i.e. :''There exists one cat c, such that for every mouse m, c is feared by m.'' :

\exists c \, ( \, \text(c) \land \forall m \, (\text(m) \rightarrow \text(m,c)) \, )

This example illustrates the importance of specifying the scope of such quantifiers as ''for all'' and ''there exists''.

Further reading