What is the truth value of the proposition 'All unicorns are beautiful'?

Question

If we let Fx denotes that which has the property of being a unicorn, and Gx denotes that which has the property of being beautiful, then this proposition would be signified by the following:

∀x(Fx→Gx)

Obviously, we know unicorns don't exist, so this sentence should, at least intuitively, be false. But in terms of logic, Fx is false (because nothing bearing the property of unicorn exists); so in terms of a material condition, since the antecedent is false, this proposition would be true!

I thought of this when I was pondering the idea of a vacuous truth, because this seems like an instance of that. But is this proposition true? That's what I am inclined to believe, but I am not certain.

Answer 1

This specific case is indeed a vacuous truth. A vacuous truth is "a statement that asserts that all members of the empty set have a certain property".

It takes three forms:

• ∀ x : P(x) → Q(x)     where ∀ x : ¬P(x)
• ∀ x ∈ P : Q(x)          where P = ∅
• ∀ ξ : Q(ξ)                  where ξ is a type with no representatives

Your case is the first one. Note that we can define the set P as {x : P(x)} to get to the second form, or define the type ξ : Unicorn to arrive at the third form, and that they are thus intuitively all equivalent.

And yes, since your proposition is a vacuous truth, it is, well, true.

Answer 2

A good way to look at this is through the concepts that Frege introduced - sense (sinn) and reference (bedeutung).

The question becomes whether the proposition

All unicorns are beautiful

has sense and reference: one can ask whether the proper names - unicorn and beautiful refer; one can argue that these names occur in the corpus of written works, that they also occur in speech, that they are not arbitrary strings of letters; thus they refer, but to what? and how?

A unicorn does not occur in the world; but in a fictional world; and in these fictional worlds things are described as beautiful or ugly ie they are the properties of fictional objects.

This is their reference; but what then is their sense?

For a proposition to gain meaning it is not sufficient to focus solely on its logical form; and nor is it enough to gain an understanding of its truth by way of what this proposition refers to in the world - real or fictive; but also by what these words - unicorn and beautiful mean - this is their sense.

Note: a vacuous truth is a proposition that adds nothing to our understanding; that unicorns exist in the fictional world of Narnia, and that there they are considered both wild and beautiful adds to our knowledge of this fictional world.

Thus, it is not a vacuous truth.

A vacuous truth is generally context dependent; it generally means something that is true by reason of its logical form; an example of which is the proposition 'a unicorn is a unicorn'; this is true, but adds nothing to what we didn't know before - thus vacuously true.

Answer 3

Your concern is sound ...

In Aristotle's Logic the inference from :

∀x (Fx → Gx)

to :

∃x (Fx & Gx)

is legitimate. In modern logic, this is not; we say that general terms have existential import.

See the discussion of The Traditional Square of Opposition :

This representation of the four forms is now generally accepted, except for qualms about the loss of subalternation [the above inference]. Most English speakers tend to understand ‘Every S is P’ as requiring for its truth that there be some Ss, and if that requirement is imposed, then subalternation holds for affirmative propositions. Every modern logic text must address the apparent implausibility of letting ‘Every S is P’ be true when there are no Ss. The common defense of this is usually that this is a logical notation devised for purposes of logic, and it does not claim to capture every nuance of the natural language forms that the symbols resemble. So perhaps ‘∀x(Sx → Px)’ does fail to do complete justice to ordinary usage of ‘Every S is P’, but this is not a problem with the logic. If you think that ‘Every S is P’ requires for its truth that there be Ss, then you can have that result simply and easily: just represent the recalcitrant uses of ‘Every S is P’ in symbolic notation by adding an extra conjunct to the symbolization, like this: ∀x(Sx → Px) & ∃xSx.

You can see also Free Logic.

Answer 4

I did my undergrad thesis on fictional characters/objects and truth value so I might be able to help out. It depends on your view of fictional objects.

If you just take a classical logic view of fictional objects (none exist), then the sentence is vacuously true simply because there are no fictional objects. The "x" in "every x" quantifies only over existent objects, since according to this view of logic there are only existent objects in the domain of quantification that "x" represents. Looking at the truth value of the material conditional, then whenever the antecedent is false the conditional is true. So the statement "x is a unicorn" is always false since there is no existent object that is a unicorn, and the statement is always true.

On the Meinongian view, in which there are nonexistent objects for every single set of properties (for example, an object corresponding to the set {unicorn, ugly} exists simply by virtue of the properties existing, so does the set {square, circle} and {square, circle, unicorn, ugly} and so on), the sentence would be false.

On the possibilist view in which fictional statements are true according to a set of possible worlds in which the stories take place, this sentence would be dealt with in the same way as the classical logic view. They assume that an intensional operator is put in front of the sentence "all unicorns are beautiful" and this intensional operator rates the truth value of the sentence according to the world in which the fictional story takes place. But there is no such story in this context, we're merely analyzing the truth value of "all unicorns are beautiful." So it would be vacuously true.

Fictional characters are a huge problem for classical formal semantics, because they just lead to unintuitive results. According to formal semantics, all unicorns are beautiful is vacuously true. But intuitively this is false.

A previous answer stated the following:

A unicorn does not occur in the world; but in a fictional world; and in these fictional worlds things are described as beautiful or ugly ie they are the properties of fictional objects.

According to these views in which there is an intensional operator in front of this sentence, the intensional operator is determined by the context. In this context, there is no intensional operator because we're not talking about any particular story! So this sentence turns out to be vacuously true even if we take the possible world semantics view.

Answer 5

Yes, the proposition is true according to the rules of our normal logic. As you already write: For all entities x holds F(x) is false. And according to the rule ex falso quodlibet the implication

F(x) => G(x) is true.

Of course, by the same means one can prove also: All unicorns are ugly.

Note. There exist non-standard logics like paraconsistent logic which abolish the principle ex falso quodlibet.

Answer 6

If you consider unicorns to be mythical, non-existing creatures, then the proposition is true.

If you consider unicorns to be rumoured creatures for whose existence no evidence has been found yet, then we can say that no observations have been made yet that contradict the proposition, but it is not proven.

Consider the proposition "all yetis are beautiful". There will be many people who seriously claim that the proposition is false. And some will say that yetis are beautiful in their own unique way :-)

Answer 7

The context of the statement is critical. What viewpoint are we looking at? To some people, unicorns are literally metaphors for something unattainable. To others, they're literally a horse-like being that probably doesn't exist. And there are probably thousands of other equally-valid definitions. Beauty is in the eye of the beholder, so it's almost guaranteed that for any unicorn considered beautiful, another person considers it homely, if not outright hideous.

If we're taking the entire set of all things considered unicorns by any person, then asking each person who has considered one of those unicorns whether said unicorn is beautiful, it is highly likely there is at least one instance of a non-beautiful unicorn. Of course, "highly likely" is undefined in strict Boolean-style logic, so your proposition breaks unless it allows for fuzzy truths. (I'll help you out here though: I did not consider the unicorn in Oblivion to be beautiful, which means the above set definitely contains at least one counter-example, so the statement must evaluate to false.)

On the other hand, we can apply any combination of definitions of both unicorns and beauty, which means we can come up with sets for which the proposition definitely evaluates true, and other sets for which it is definitely false.

At the end of the day, this is one of those many "yes or no" questions for which neither "yes" nor "no" is a valid answer.

Answer 8

I'm a bit dismayed the truth table hasn't made it somewhere in the thread already, so here it is:

Fx  |  Gx  | Fx→Gx
-------------------
0   |  0   |   1
0   |  1   |   1
1   |  0   |   0
1   |  1   |   1

I think the other answers have focused on the bottom line of the truth table. All unicorns are beautiful(but, mind you, there are no unicorns). So I won't go further on that.

The more interesting part, to me, is the top two lines. When Fx is false(meaning we're dealing with something that's not a unicorn), Gx can be anything. x could be beautiful, or it could be ugly. ∀x(Fx→Gx) simply says "For all x where x is a unicorn, x is also beautiful". It says nothing about things that aren't unicorns. Assuming unicorns don't exist(there are several definitions in which they do exist), then it turns into something along the lines of "For all x, x can be ugly or beautiful".

This has the bonus of suggesting that everything in existence is either beautiful or ugly.