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In classical symbolic logic, Can we conclude "Some are Italian" from "All are Italian"?

All are Italian.
Therefore, Some are Italian.

Apparently, Modern logicians argue that it is invalid since we cannot drop "all" to from "All are Italian" to derive "Someone is Italian". I'm looking to know the views of Classical and modern logic on this if it is true.

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Where do you have the "Apparently, Modern logicians argue that it is invalid since we cannot drop 'all' to ..." from? – Thomas Nov 13 '12 at 18:45
@Thomas: It's from my book that I read. – user121314 Nov 13 '12 at 19:14
@user121314: most modern logic texts use definitions such that $(\forall x)\phi$ implies $(\exists x)\phi$ in first order logic. One keyword to search for is "existential fallacy". – Carl Mummert Nov 14 '12 at 3:04
up vote 15 down vote accepted

Consider the following statements:

All unicorns are pink.

Some unicorns are pink.

The former is true, since there are no unicorns that are not pink. The latter is false, since there is no unicorn that is pink.

The classical conception of logic apparently operated on the assumption that we'd only ever logically quantify over meaningful subjects--that is, that we'd never have a vacuously true statement like the first one above. For more detail about the relationships between quantified statements in classical (Aristotelean) logic, look at this article on the so-called "square of opposition" (in particular, up through the "Modern Squares of Opposition" section).

In your case, you could drop the "all" down to "some", but only if you knew that you were quantifying over a non-empty collection of individuals. For example, we couldn't do this if we were talking about leprechauns. However, if we were talking about guys named Vito, and we also had the statement "Some guy is named Vito," then we could drop the "all" down to "some" as described. In other words, the following would be a valid argument:

  1. All guys named Vito are Italian.

  2. Some guy is named Vito.

  3. Therefore, some guy is Italian.

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I think in your first statement it should be "there is no unicorn that is NOT pink" , – Theorem Nov 13 '12 at 18:55
@Theorem: Let's say that $U$ is the set of unicorns and that $p(x)$ is the statement "$x$ is pink." In symbolic form, then, my first statement is: $$\forall x\in U,p(x)$$ Your suggested alteration is: $$\neg\exists x\in U,\neg p(x)$$ This is equivalent to: $$\forall x\in U,\neg\bigl(\neg p(x)\bigr)$$ Can you see that this is equivalent to my statement? – Cameron Buie Nov 13 '12 at 19:01
So it would be true if I had, 1. All are Italian 2. There is at least one Italian 3. Therefore, Some are Italian I suppose this what you meant by non-empty collection? – user121314 Nov 13 '12 at 19:11
@user121314: Your statement 2 is exactly that "Some are Italian." I rewrote my answer slightly to clarify what I mean by quantifying over a non-empty collection. Dan's answer discusses this, too, under the assumption that we're talking about persons. (I didn't want to assume that. We could as sensibly been talking about salad dressings.) – Cameron Buie Nov 14 '12 at 18:36
To say that (A) "All unicorns are pink" is true because (B) "there are no unicorns that are not pink" is true of course flatly begs the question at stake here, which is whether (A) is indeed equivalent to (B)! A traditional line on the existential import of universals will of course deny the equivalence! – Peter Smith Nov 17 '12 at 8:35

By "all are Italian," we really mean all persons are Italian. By "some are Italian," we mean at least one person is Italian.

Suppose all persons really are Italian.

If there exists at least one person, then we could conclude that at least one person is Italian.

If there are no persons, we could not conclude this.

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There is an absolutely masterly treatment of the relation between Aristotelian logic and modern quantificational logic by Timothy Smiley in his paper "Syllogism and Quantification" Journal of Symbolic Logic 1962, pp. 58-72. This now classic paper is still not as well-known as it should be: it is a must-read for anyone who wants to know about the issue and really should be on any student reading-list.

Take a many-sorted predicate calculus, i.e. with sorted variables which run over different domains of quantification -- and NB, as with the standard modern single-sorted calculus, domains are non-empty. And now add sortal predicates corresponding to the domains. Thus corresponding to the sorted variable $a$ there is a sortal $A$ such that $\forall aAa$. Then Smiley's headline news is that the traditional syllogistic translates neatly into this many sorted framework -- preserving the traditional rule that All A are B implies Some A are B, for this goes over to $\forall aBa$ implies $\exists aBa$. For all the details, see Smiley.

What this shows is that the issue about existential import has little to do with the traditional vs post-Fregean quantifier/variable treatment of quantification, and a lot more to do with a decision early on the development of modern logic to privilege single-sorted calculi, and then implement sorted quantifiers artificially by the use of restricting predicates. (This decision was for logicians' convenience rather than for mathematical utility -- mathematicians use sorted informal quantifiers all the time.)

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(+1): Very interesting. Peter. I'd never heard of many sorted predicate calculus before. – Cameron Buie Dec 27 '12 at 14:54

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