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.
The syntax of traditional logic (TL) permits exactly four sentence types: "All As are Bs", "No As are Bs", "Some As are Bs" and "Some As are not Bs". Each type is a quantified sentence containing exactly one quantifier. 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 As are Bs
- All Cs are Ds
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,
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.
This example illustrates the importance of specifying the scope of such quantifiers as for all and there exists.
Further reading
- Patrick Suppes, Introduction to Logic, D. Van Nostrand, 1957, ISBN 978-0-442-08072-3.
- A. G. Hamilton, Logic for Mathematicians, Cambridge University Press, 1978, ISBN 0-521-29291-3.
- Paul Halmos and Steven Givant, Logic as Algebra, MAA, 1998, ISBN 0-88385-327-2.
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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 The syntax of traditional logic TL permits exactly four sentence types All As are Bs No As are Bs Some As are Bs and Some As are not Bs Each type is a quantified sentence containing exactly one quantifier 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 As are Bs All Cs are Ds 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 Further readingPatrick Suppes Introduction to Logic D Van Nostrand 1957 ISBN 978 0 442 08072 3 A G Hamilton Logic for Mathematicians Cambridge University Press 1978 ISBN 0 521 29291 3 Paul Halmos and Steven Givant Logic as Algebra MAA 1998 ISBN 0 88385 327 2 This logic related article is a stub You can help Wikipedia by expanding it vte
