Concept
Problem of multiple generality
Overview
The problem of multiple generality names a supposed failure in traditional logic 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 traditional logic (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 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's Begriffsschrift (1879), the ancestor of modern predicate logic, 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,
∀
m
(
Mouse
(
m
)
→
∃
c
(
Cat
(
c
)
∧
Fears
(
m
,
c
)
)
)
{\displaystyle \forall m\,(\,{\text{Mouse}}(m)\rightarrow \exists c\,({\text{Cat}}(c)\land {\text{Fears}}(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.
∃
c
(
Cat
(
c
)
∧
∀
m
(
Mouse
(
m
)
→
Fears
(
m
,
c
)
)
)
{\displaystyle \exists c\,(\,{\text{Cat}}(c)\land \forall m\,({\text{Mouse}}(m)\rightarrow {\text{Fears}}(m,c))\,)}
This example illustrates the importance of specifying the scope of such quantifiers as for all and there exists.
Overview adapted from Wikipedia (CC BY-SA 4.0; see the article history for authors).
Possible passages
Found by search and not yet checked by a person; they may only mention the idea in passing.
Examples of the latter are: 'The mind is no elephant:' 'A lion is no table;' propositions which are correct but absurd, exactly like the identical propositions: 'A lion is a lion;' 'Mind is mind.' Propositions like these are undoubtedly the truth of the immediate, or, as it is called, Qualitative judgment. But they are not judgments at all, and can only occur in a subjective thought where even an untrue abstraction…
Georg Wilhelm Friedrich Hegel, The Logic of Hegel
States what Problem of multiple generality is.
Or, finally, the premises, of whichever kind they are, may never have been conceived in so distinct a manner as to produce any clear consciousness by what means they were arrived at; as in the case of what is called reasoning in a circle; and then the fallacy is one of Confusion.
John Stuart Mill, A System of Logic, Ratiocinative and Inductive
States what Problem of multiple generality is.
Moreover, we feel some quality of necessity about the proposition 'two and two are four', which is absent from even the best attested empirical generalizations. Such generalizations always remain mere facts: we feel that there might be a world in which they were false, though in the actual world they happen to be true. In any possible world, on the contrary, we feel that two and two would be four: this is not a mere…
Bertrand Russell, The Problems of Philosophy
Close in meaning to Problem of multiple generality; the term itself is not named.
Philosophers who discussed Problem of multiple generality
Related ideas
Place in the tradition
Part of Analytic philosophy.
Explore Problem of multiple generality in the interactive graph →