Negating the verb and negating the subject 's property

Question

What is the strict and exact relation (implication, equivalence etc.) between these two sentences?:

I. Alcibiades is not wise. (Negating the subject 's property)

II. Alcibiades is not (=isn 't) wise. (Negating the copula)

At a first glance we have something similar to set theoretic equivalence "x belongs to set not-S (S' complement), iff x doesn 't belong to set S". Though in sentences

I'. Alcibiades is in not-Sparta,

II'. Alcibiades isn' t in Sparta

it seems we have something different: at (I') Alcibiades is surely somewhere but somewhere else than Sparta and at (II') maybe Alcibiades is nowhere but for sure he isn 't in Sparta.

My question is about generalizing such forms -and even wider predicates, i.e. not necessarily sentences with copula but also sentences with subject, verb and object (e.g. I''. Paganini plays not cello and II''. Paganini doesn 't play cello)- and find if there is some dependence from verb 's or other terms' semantics. Every bibliographic reference is welcome.

Answer 1

What is the strict and exact relation (implication, equivalence etc.) between these two sentences?:

I. Alcibiades is not wise. (Negating the subject 's property)

II. Alcibiades is not (=isn 't) wise. (Negating the copula)

my best guess at a constructive reading would be that I. is a assertion of negation, as in

⊢ ¬W(a)

while II. is about lack/absence of proof/evidence for W(a), maybe even, though not necessarily,

⊬ W(a)

I'. Alcibiades is in not-Sparta,

to affirm such it seems to be necessary to pick some especific place P(-) such that

⊢ ∀x( [ P(x) → ¬S(x) ] ∧ [ S(x) → ¬P(x) ] )

and moreover

⊢ P(a)

[which i suppose matches your interpretation,] and in particular this implies

II'. Alcibiades isn' t in Sparta

⊢ ¬S(a)

[again matching your interpretation,] but this by itself may not provide especific P(-)

Answer 2

I’m going to use a different example to elucidate what I think is the main difference between negating the verb and negating the subject.

1. The sky is not red.
2. The sky is not red.

The first is just saying the sky doesn’t have the property ‘red’, while the second is actually affirming that the sky has the property ‘not red’. If “A is not B” is the extensional negation of “A is B” and if ‘not-red’ is the complement of ‘red’, then they’re just saying the same thing.

That is not to say that there are not different ways to understand these concepts, especially since incorporating human notions of collections into formal set theory isn’t usually feasible. The collection of everything that isn’t a turtle contains itself, so in most set theories, no such collection can be modeled. I really don’t know much about NBG and MK, but I think you might have a chance at formalizing it in one of those theories by calling such a collection the class of everything that is not a turtle. In those theories, proper classes cannot be members of classes, so Russell’s Paradox doesn’t apply. You can see more about these class theories here: https://ncatlab.org/nlab/show/von+Neumann%E2%80%93Bernays%E2%80%93G%C3%B6del+set+theory

That is all to say that not every property we can cook up in natural language corresponds to a set, but since there are class theories with the Class Existence Theorem, we can be sure that in those theories we have a class of all and only those classes that have property A, where A is definable in terms of the language of class theory. As it stands, class theories have complements because they have a universal class. So, the examples mentioned would, depending on the way you formalize them, still yield the same result of being equivalent.

Answer 3

Being natural language phrases, there is no one way to capture their semantics using just syntax. Don't be surprised when there's multiple answers.

That being said, a common case where the pattern of "Subject negated-verb Object" is different from "Subject verb negated-object" is in a language of assertions, which is inherently modal. In a modal language, there is some concept of something between true and false. The most obvious of them is one which is "unknown" -- potentially either true or false, but not yet known.

In that sense, "Alcibiades is not-wise" may explicitly declare that Wise(Alcibades) is false, while "Alchebades isn't wise" may declare that Wise(Alcibades) isn't true, but may be unknown.

Answer 4

I will try to give an interpretation.

I consider that affirmation of a negative property inserts the subject into a frame submitted by the property. On the other hand negation of a property does not oblige subject to penetrate any frame. Specifically the affirmation

X is not-P

inserts X into a genus-frame containing P (e.g. "My garden 's rose is not-red" implies that my garden 's rose has a colour, which is other than red). The negation

X is-not P

just says that X has not the property P without any innuendo about X belonging to some frame (e.g. "Euclidean geometry is-not red" is acceptable, because the subject has no colour at all).

If by coincidence X belongs to the frame submitted by property, then the two forms have identical meanings.

Answer 5

I suppose there is a slight confusion here. Let's take the first comparative example:

"Alcibiades is not wise"

"Alcibiades is not wise" (is not) = (isnt)

We predicate the negation of the property in the first sentence, and we do not predicate the property in the second sentence. (So we deny the copula through which the predicate "is wise" holds). Those are rightfully equivalent. Now let's take the second comparative example

"Alcibiades is not in Sparta"

"Alcibiades is in not-Sparta"

The paradoxical element comes from the fact that you didnt follow the form of the first comparative example. The first sentence posits the denial of predicate "in Sparta", but the second sentence doesnt actually follow the form of the first sentence of the first comparative example, since you are still preserving the term "in", which composes the predicate. That is, "Alcibiades is in not-Sparta" is not really positing the denial of the predicate, rather the correct correlate is exactly the same: "alcibiades is not in Sparta", in which "not in Sparta" is the negative predicate which is posited by the copula "is". Ultimately you forgot to make the "not" antecede the predicate since you included the "in". The same problem arises in the Paganini example. "Paganini plays not Cello" is not really a sentence which posits the denial of the predicate "plays Cello". One way to see this is converting it to "Paganini does play not Cello". You still included part of the predicate. In reality, the proper correlate is "Paganini does not play Cello", in which "not play Cello" is the negative predicate posited by "does". The notion of x being in not-S if x is not in S is simply to affirm the negation of "x is in S". Basically, if we deny that x is in S, we affirm that we deny that x is in S, and thus we posit that x is in not-S. We are playing with the fact that anything stands so long as it is posited/affirmed. That's also why most people think that untruth entails falsity, (That is, if p is not true then ~p is true) since falsity is reduced to the truth of untruth, the truth of negation.

Answer 6

The complement of a set can only be defined if a universal set is defined. There is no complement of a set in ZF set theory. Your confusion can be resolved using the logic of naive set theory instead of ZFC set theory.

Alcibiades is non-wise = Alcibiades is an element of the set of things that aren't wise.

Alcibiades isn't wise = not (Alcibiades is wise) = Alcibiades doesn't have the property 'wise'.

Symbolize Alcibiades is wise by W(a).

Thus Alcibiades isn't wise = ~W(a).

Let W denote the set of all things that are wise and only things that are wise.

W = {x| W(x) }

Let W' denote the complement of W.

W' = {x| ~W(x) }

Alcibiades is non-wise means, in naive set theory that, ~W(a).

If Alcibiades isn't wise, then he's an element of W', and conversely.

Thus the two statements are equivalent if you use naive set theory.

Alcibiades isn't in Sparta = not (Alcibiades is in Sparta).

Here, Sparta isn't a set, it's a place. Thus, is-in isn't the elementhood relation of set theory.

Alcibiades is in non-Sparta has the same confusion. Sparta isn't a set.