Problem of multiple generality: Difference between revisions

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The{{Short '''problem of multiple generality''' names a failuredescription|Failure in [[term logic|traditional logic]] to describe certain intuitively [[validity (logic)|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 allotted 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.

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:''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.

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:''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==

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[[Category:Term logic]]

[[Category:Classical logic]]

{{logic-stub}}