Problem of multiple generality
Encyclopedia
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:
then it follows logically that:
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 alloted the same logical form
as the sentence "Some cat is hungry". And so the logical form in TL is:
which is clearly invalid.
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. 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.
could mean (Some cat is feared) by every mouse, i.e.
in which case the conclusion is trivial.
But it could also mean Some cat is (feared by every mouse), i.e.
This example illustrates the importance of specifying the scope of quantifiers as for all and there exists.
- 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 alloted the same logical form
Logical form
In logic, the logical form of a sentence or set of sentences is the form obtained by abstracting from the subject matter of its content terms or by regarding the content terms as mere placeholders or blanks on a 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
Gottlob Frege
Friedrich Ludwig Gottlob Frege was a German mathematician, logician and philosopher. He is considered to be one of the founders of modern logic, and made major contributions to the foundations of mathematics. He is generally considered to be the father of analytic philosophy, for his writings on...
's Begriffsschrift
Begriffsschrift
Begriffsschrift is a book on logic by Gottlob Frege, published in 1879, and the formal system set out in that book...
, the ancestor of modern predicate logic
Predicate logic
In mathematical logic, predicate logic is the generic term for symbolic formal systems like first-order logic, second-order logic, many-sorted logic or infinitary logic. This formal system is distinguished from other systems in that its formulae contain variables which can be quantified...
, 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, 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), 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 quantifiers as for all and there exists.
Further reading
- Patrick Suppes, Introduction to Logic, D. Van Nostrand, 1957, ISBN 0-422-08072-7.
- 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.