Traditional Logic on the Problem of Multiple Generality

Some say that some statements are not addressed well by traditional logic.

e.g.

Some cat is feared by every mouse.

and

All mice are afraid of at least one cat.

 

Now I could not find this specific controversy on http://plato.stanford.edu/, so I only have the wiki, http://en.wikipedia.org/wiki/Problem_of_multiple_generality.

 

The converse of these propositions are both treated in traditional logic. The first is a singular proposition, and yes, singular propositions exist in traditional logic. The Wikipedia article is wrong when it says, “The syntax of traditional logic (TL) permits exactly four sentence types.”

 

Traditional logic also permits indesignate types and singular types. The converse of the singular proposition,

Some cat is feared by every mouse.

becomes,

One feared by every mouse is some cat.

That was easy!

 

The second is a general proposition.

All mice are afraid of at least one cat.

becomes,

Some afraid of at least one cat are mice.

 

The subject and predicate do not depend on the structure of the sentence. Not every colloquial sentence is formed as subject copula predicate. In all singular propositions, the subject is always the singular individual. Sometimes in propositions, you may reform it if you want. “All mice” is a generality. “At least one cat” is a generality. “Every mouse” is a generality. “Some cat” is a singular individual. In addition, we are dealing with the predicate as either “afraid of” or “feared by”. We can change a subject into a predicate through the use of these principles. We may change actions into passions. However, by themselves, the sentences are in logical form and need no more additions. If we ever wish to reform propositions, it sometimes depends on the context. The main objective is to always try and convey the real intended meaning. Sometimes, its not even possible. We can change the first sentence from

Some cat is feared by every mouse.

to

Every mouse is afraid of some cat.

We have now changed the logical subject, but there is no change in meaning. We can do this in these instances because actions can be converted to passions and vice versa. The converse of this is,

Some afraid of some cat is every mouse.

 

The way you decide to present the proposition in logical form shows how you think these are logically divided. Sometimes logical divisions are natural, and sometimes logical divisions are artificial. Dividing a category of “afraid of some cat” into a subset of “every mouse” is an artificial division, for example. “Some cat is feared by every mouse” is a singular proposition, so its not based on any division. This is the type of insight that traditional logic gives, a way to organize knowledge in the mind.

 

According to the Wiki:

“Using modern predicate calculus, we quickly discover that the statement is ambiguous.”

However, the the two propositions were not the same propositions. Its not possible to convert one into the other. “Modern predicate calculus” gives us,

“For every mouse m, there exists a cat c, such that c is feared by m”

or

“There exists one cat c, such that for every mouse m, c is feared by m.

These, for some reason, are both formed from the original proposition. Logically, the proposition was always in the form,

Some cat is (feared by every mouse)

Where the object in parenthesis is the predicate.

(Some cat is feared) by every mouse

Here, what are they designating with the parenthesis? Its not the predicate, so I have no idea.

 

All in all, once I comprehend the appeal of this system, I will blog about my giant revelation.

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