Problem of multiple generality: Difference between revisions

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The{{Short '''problem of multiple generality''' names a failuredescription|Failure in [[traditional logic]] to describe certain intuitively valid inferences. For example, it is intuitively clear that if:}}

The '''problem of multiple generality''' names a failure in [[term logic|traditional logic]] to describe [[validity (logic)|valid]] inferences that involves multiple [[Quantifiers (logic)|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 AsA's are BsB's", "No AsA's are BsB's", "Some AsA's are BsB's" and "Some AsA's are not BsB's". 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 allotedallotted the same [[logical form]] as the sentence "Some cat is hungry". And so the logical form in TL is:

:''Some AsA's are BsB's''

:''All CsC's are DsD'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 [[first-order logic|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 [[first-order logic|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,''

:<math>\forall m. \, (\, \text{Mouse}(m) \rightarrow \exists c. \, (\text{Cat}(c) \land \text{Fears}(m,c)) \, )</math>

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.''

:<math>\exists c. \, ( \, \text{Cat}(c) \land \forall m. \, (\text{Mouse}(m) \rightarrow \text{Fears}(m,c)) \, )</math>

This example illustrates the importance of specifying the [[Scope (logic)#Quantifiers|scope]] of such quantifiers as ''for all'' and ''there exists''.

==Further reading==

* [[Patrick Suppes]], ''Introduction to Logic'', D. Van Nostrand, 1957, {{ISBN |978-0-422442-08072-7 {{Please check ISBN|reason=Check digit (7) does not correspond to calculated figure.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}}.

{{Aristotelian logic}}

{{Classical logic}}

[[Category:Term logic]]

[[Category:Classical logic]]

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