# Negation of a relational quantified term?

Formally, when you negate a quantified term which is attached to a relation, is a distributed term always made undistributed and vice-versa?

For example, consider this statement :

Anyone who assists a criminal is immoral.

Where "anyone" is taken to mean "all persons" and "a criminal" is taken to mean "some criminal or other" (ie, an undistributed term). The justification for this interpretation is that, intuitively, we mean that anyone who assists even one criminal is immoral.

Now the contraposition of the proposition is :

Any moral person doesn't assist criminals.

Intuitively, we mean all criminals here (it would seem absurd to suggest that we're only talking about some criminals). So what's happened is that the relation "assist" has been negated (it becomes "doesn't assist"), and its object "criminal" has been transformed into a distributed term.

The above seems correct but I have a doubt whether it applies in general. This could be because the quantity "any" is ambiguous; sometimes it can mean "all" and at other times "some".

ie the above proposition could have been expressed as :

Anyone who assists any criminal is immoral, where "any" means "some criminal or other".

and its contraposition could be expressed as :

Any moral person doesn't assist any criminal, where in this case, "any" (criminal) means "all".

However, it's certainly true that not(all) = some and not(some) = all, so I guess it doesn't matter about this ambiguity, as long as you're clear about its meaning in any particular case.

Well I think maybe I've just answered my own question, but would appreciate some feedback. Thanks.

• Comments are not for extended discussion; this conversation has been moved to chat. Commented Jul 18, 2019 at 20:13

Anyone who assists a criminal is immoral,

will be symbolized with : ∀x [ ∃y(Assist(x,y) ∧ Crim(y)) → Immoral(x)].

Thus, its negation will be :

∃x [∃y(Assist(x,y) ∧ Crim(y)) ∧ Moral(x)],

assuming the abbreviation : Moral(x) for not-Immoral(x).

If instead we want to express it eqivalently used contraposition, what we get will be :

∀x [ Moral (x) → ¬∃y(Assist(x,y) ∧ Crim(y))].

• Thanks Mauro. I'm not a fan of the predicate calculus but it has answered my question. Commented Jul 11, 2019 at 9:41

If you want to avoid the ambiguities of words like 'any' and 'some,' think in terms of categories, instances, and properties. In this case we have two categories: People and Criminals (where Criminals are a subcategory of People that conform to the property criminality). With that in mind, your original phrase becomes:

Members of category People who assist instances of category Criminal negate their property morality

In effect, we are asserting the property 'morality' on the category People and associating it (inversely) to the act of assisting criminals. We could translate this into a simple syllogism, e.g.:

• All people who assist criminals are immoral
• x is a person who assists criminals
• x is immoral

When we negate things in the contraposition we implicitly switch from talking about categories to talking about instances (e.g. if all members of category X have Y, then no instance of category X does not have Y). So if we start from the statement above, we evolve it like so (where I'm using 'assert' as the logical opposite of 'negate'):

Members of category People who assist instances of category Criminal negate their property morality

No instance of category People who assists instances of category Criminal asserts their property morality

No instance of category People who asserts their property morality assists instances of category Criminal

Members of category People who assert their property morality do not assist instances of category Criminal

What this shows is that the question about the quantification of category Criminal is red herring. It simply doesn't matter whether instances of category Criminal means one, or some, or any, or all; it translates as 'not-none' (a simple 'exists' claim). The focus is on the evolution of the morality property of the category People.