ex-xian

This little problem came up in a discussion between my philosophy prof, math prof, and myself.

According to categorical logic, the statement "All S are not P" is logically equivalent to "No S are P." For example, to say "All cows are not fish" means "No cows are fish." Another example with a false statement, "All cows are not brown" is "No cows are brown."

The problem arises from the use of english (or any I guess, I use english because it's what I speak ). Consider a reporter making the statement "All politicians are not dishonest."
The logically equivalent statement is "No policitians are dishonest." Of course no one would seriously say this!

When spoken with an emphasis on "all," it sounds as if she means "Not all politicians are dishonest." And this is how most people would understand the sentence.

Does anyone have any comments on this?

Is there any way to incorporate stress and tense into categorical or predicate logic?

tk

There are many ambiguities in the English language, but this doesn't seem to be one of them. I parsed "All politicians are not dishonest" as "[All politicians] are [not dishonest]", i.e. "All politicians are honest".

Shadowy Man

What is ambiguous about the difference between "All" and "Not all"??

ex-xian

Right, that's what the statement means when examined analytically. My point was that people frequently use that form "All S are not P" and mean "Not all S are not P." The difference can't be appreciated by reading, it must be heard. If you were to say "All policitians aren't dishonest" a certain way, it sounds as if you mean "Not all policitians are dishonest."

Maybe this is pointless to try to explain with the written (typed) word. Perhaps it has to be heard.

Shadowy Man

Oh! Yes, I see what you mean now. I misread what you had before. Yes, the multiple negatives gets confusing.

tk

I see it now too... the parse is "All politicians [are not] dishonest", where "are not" is taken as a phrase, even if it's not really a single component in the English grammar.

Yet another problem for NLP researchers... *grrr*

John Page

The truth functionality of this proposition is clear. However, what it might mean in "real life" is another issue. Does it mean that all politicians are dishonest all the time? Continuously dishonest? That they have been dishonest at least once in their lives or careers? As to your English usage issue, does the speaker mean that politicians are not really dishonest, just a little bit dishonest?

To me, it is absurd to suggest that:

1. Anyone (let alone a politician) can be telling lies (literally) all the time.
2. We can really test the truth of the claim (as opposed to determine is truth functionality a la categorical logic). The extension of the quality dishonest over all politicians would be hard to prove.

A tenable claim writ in full might be "Not all the politicians that I know appear to be dishonest". In "real life" the statement "All politicians are not dishonest" would seem less verbose and convey the meaning just as well.

It is arguable, therefore, that formal logic cannot always encapsulate the "true meaning" conveyed by natural language. On the other hand, logic can encourage one to be much clearer about what you really do mean.

Cheers, John

lisarea

The difference, which is conveyed tonally (and by context and probability) in spoken English, is between the existential and the universal quantifier.

So, "I don't like all olives" technically could mean either of the following:

Universal: I consistently don't like olives, encompassing all varieties.

Existential: I don't consistently like olives regardless of variety. This carries the assumption that I do like some varieties, just not all of them. (Note that, if we move that rephrase to read "I consistently don't like olives..." it goes back to the existential meaning, even without the quantifier 'all.' Right there is another discrete rule that would need to be articulated in order to represent natural language in logical terms.)

In casual, spoken language, of course, the existential meaning is assumed. When you take what could be argued is a logical rephrasing, "I dislike all olives," it takes on the universal meaning.

This is a good example of the difficulties in representing and parsing natural language artificially. It's not that the individual rules are so complex they're not understandable. It's just that there are so damned many of them, all tangled up together, that it'd take forever to get them all down. Look at a linguistics journal sometime. It's almost scary how much unique information there is about tiny little words and concepts we just take for granted. I once made the mistake of looking for information on the word 'on.' Yow.

Fortunately, though, we have an innate understanding of these rules, so we can communicate effectively among other native speakers. (Often, people who acquired a language post-childhood have some of the same difficulties in understanding the rules as do machines. Essentially, the subtleties of natural language are almost impossible to understand consciously.)

ex-xian

My study of logic has only been theoretical, but I've interested in AI research. That was one of the first things I thought about when this issue was brought up.

It would be hard enough to program a computer to understand the english language, much less all the subtle variations.

Of course this presupposes that AI necessarily converts language into logical symbolism to interpret it.

anonymousj

In the treatment of categorical logic/categorical propositions in Copi and Cohen and in Hurley (popular college-level logic textbooks in the United States), "All S are not P" is not equivalent to "No S are P". In fact, "All S are not P" is not a categorical proposition. The obverse of "No S are P" is "All S are non-P" where "non-P" names the complement set to "P". The obverse of any categorical proposition an equivalent catergorical proposition. This will remove some of the ambiguity won't it? "All cows are non-brown" isn't ambiguous in the way that "All cows are not brown" is, is it?

anonymousj