Why is it that the statement "All goblins are yellow" does not contradict the statement "All goblins are pink?"

Question

From what I know, I think it has something to do with vacuous truths, but my understanding is not quite there yet.

Answer 1

Edit in response to your comment: Okay, long answer:

What is the meaning of "the"?

(A previous version of the question had the statement "The goblins are pink"; this is an elaboration on that formulation.)

First of all, as noted in the comments, the "the" makes things a bit problematic; it is not obvious that "the" is to mean the same as logical "all". There are several viewpoints one could take:

• A meaningful usage of "the" presupposes the existence of goblins; if there exist no goblins, the statements are nonsensical and fail to be assigned any truth value whatsoever.
• The usage of "the" asserts that goblins exist and that the predication applies to all of them. In this case, the two statements are false in the real world and contradictory -- see the last paragraph of the second section.
• "the" means the same as "all" in the standard logical sense: The two universal claims are vacuously true in the real world and not contradictory -- see the second section on that.
• Something more complicated.

Note in particular that the mathematical use of "all" is not identical to the natural language one (and logicians are well aware of that); in informal speech, "all" (and even more so "the") usually does implicate the existence of objects in the restriction. But you seem to be explicitly concerned with the issue of vacuous truth arising from the standard logical usage of "all", so this is what I will elaborate on.

As pointed out in the comments, there is also a difference between modern and classical Greek logic: While in modern standard logic, universal statements become vacuously true when there are no elements to satisfy the restriction, the universal quantifier in Aristotelian has existential import, that is, "all goblins" would entail the existence of goblins. I will be presupposing modern logic here.

Note also that the evaluation of quantifiers such as "all", "some" or "the" is always relative to particular utterance situations. If I claim that "I did all the dishes", you won't call me a liar because I didn't wash all the dishes in the world; what is meant, unless specified otherwise, is that the statement is true in the currently relevant situation, e.g. in my kitchen, with a restricted domain of objects. And of course, if Frodo says that "The goblins are pink", then in that utterance context, namely in the fictional universe of Middle-earth, the statement that there exist goblins certainly is true.

And finally, there is the issue of if and why "x is pink" and "x is yellow" would be contradictory -- more on that in the last section; for simplicity, I will go with "pink" and "not pink" for the time being.

Why is "All goblins are pink" and "All goblins are not pink" not contradictory without further assumptions?

Here about the issue with vacuous truth:

The statement

All goblins are pink

translates to

For all x: If x is a goblin, then x is pink

which, in classical logic, is equivalent to

There is no x such that not: If x is a goblin, then x is pink

which is equivalent to

There is no x such that: x is a goblin and x is not pink

Analogous for the other statement with "are not pink", which ends up as

There is no x such that: x is a goblin and x is not not pink

which in classical logic can be turned into

There is no x such that: x is a goblin and x is pink

that is, the two universal sentences can alternatively be phrased as

All goblins are pink
= There is no object which is a goblin but not pink
All goblins are not pink
= There is no object which is a goblin but pink

In classical logic, the only way for a universal statement to become false is if there is a concrete object of which the quantified formula is false. Thus in our case, "All goblins are pink" can only be false if there exists a goblin which is not pink, and "All goblins are not pink" can only be false if there exists a goblin which is pink.
But if there are no goblins to begin with, then in particular there can be no such counter example, and the statements can not be false. If a universal statement is true because there exist, like here, no objects that satisfy the restriction, it is said to be vacuously true.

The two universal claims are not immediately contradictory, because they are not negations of each other: The two sentences are not of the form "A" and "not A"; the negation is embedded deeper inside, and does not cause the two claims to have opposite meaning.

A contradiction could also arise if the statements are not directly negations of each other, but if one could derive from them a pair of statements of the form "a has property P and a does not have property P", for some term a. That is, such a derived contradiction would require that there is a concrete object which is both pink and not pink. If we additionally assume that there is least one goblin, or if we take this as implied by the usage of "the", then by the two universal claims, it would be pink and not pink, which is a contradiction, and hence the conjunction of the two universal and the existential claim is a contradictory.
But if there are no goblins, then there is precisely nothing that entails the existence of any such object with contradictory properties, and thus without additional assumptions, the two universal claims are consistent.

Is "There exist goblins", "All goblins are pink" and "All goblins are yellow" contradictory?

It depends.

Logic is only concerned with the structure of arguments involving logical expressions such as "if ... then", "not", for all". Logic itself doesn't know the meaning of so-called non-logical expressions such as "goblin", "pink" and "yellow".
As stated above, a contradiction arises whenever there is a pair of statements of the form "A and not A". But in the sentences given, with "yellow" instead of "not pink", even under the additional assumption that goblins exist there is just no such pair of explicitly contradictory claims. This has nothing any longer to do with vacuous truth; there is just not enough information in terms of the logical structure of the sentence to derive a contradiction.

However, one could add axioms with world knowledge about the meaning of these non-logical expressions, and e.g. explicitly specify that

For all x: If x is pink, then x is not yellow
For all x: If x is yellow, then x is not pink

Now in addition assume that there exists at least one goblin:

Peter is a goblin.

Then, with the assumptions

For all x: If x is a goblin, then x is pink
For all x: If x is a goblin, then x is yellow

we have by the rules of universal instantiation and modus ponens that

Peter is pink.
Peter is yellow.

Now with the axiom

For all x: If x is pink, then x is not yellow

we can, again with universal instantiation and modus ponens, derive

Peter is not yellow

which is a contradiction to

Peter is yellow.

(and similarly for the other direction from yellow to not pink).

Note that the non-contradictoriness of the yellow/pink sentences is independent of vacuous truth; the reason is that additional axioms are needed. With the axioms added, the existence of goblins is needed to derive a contradiction, similar to above.

Then again, the question is what it means for an object to "be pink". The above axioms are justified if we take "x is pink" to mean that the object is entirely covered in pink and hence can not simultaneously be of any other color. But if goblins are striped, they could very well be both pink and yellow; if "is pink" just means "is at least partially covered in pink", we would not want to accept the above axioms and hence again have nothing to derive a contradiction from.

In sum, it depends on the meaning of the natural language expressions "is pink" and "is yellow", and this is not something logic is concerned with.

TL;DR

•  

  • The sentence "There exist goblins" is false in the real world.
  • The sentences "All goblins are pink" and "All goblins are not pink" are both vacuously true in the real world.
  • The sentences "The goblins are pink" and "The goblins are not pink" may be true, false or nonsensical in the real world depending on the meaning of "the".
• • The set of sentences {All goblins are pink; All goblins are not pink} is not contradictory.
  • The set of sentences {All goblins are pink; All goblins are not pink; There exist goblins} is contradictory.
  • The set of sentences {All goblins are pink; All goblins are yellow} is not contradictory.
  • The set of sentences {All goblins are pink; All goblins are yellow; There exist goblins; All pink things are not yellow} is contradictory.

Answer 2

There are two ways in which these statements can be non-contradictory:

Option A: Non-mutually exclusive

It is possible for a goblin to be both pink and yellow, therefore it is possible for a goblin to be both pink AND yellow simultaneously.

Option B: Vacuous truth (which is what it seems you are angling for)

From wikipedia: In mathematics and logic, a vacuous truth is a conditional or universal statement that is only true because the antecedent cannot be satisfied. For example, the statement "all cell phones in the room are turned off" will be true even if there are no cell phones in the room.

In your specific scenario, due to the fictional nature of goblins, their non-existence implies that any statement applying a universal property to them is automatically true. All goblins wear hats, all goblins do not wear hats, are both true, because the group you are applying a statement to has no members.

Answer 3

This has to do with how we translate statements from natural language into formal logic. There are many different possible ways to do so, and some of those yield different results. Statements like these are typically translated into a Tarskian second-order-logic where "All Goblins are yellow" would first be converted to "For all things, if something is a goblin, that thing is yellow."

We don't even need to continue onto symbols to see that the above statement does not contradict the statement "For all things, if something is a goblin, that thing is pink," just in the case that there are no things that are goblins.

This does not necessarily match our natural language intuitions, which is a symptom of the fact that natural language statements are never exactly equivalent to formal logic statements. It's further worth noting that some logics --possible-world logics, for example --might yield a different conclusion, because they, unlike Tarskian logics, allow for non-existent entities to have defined properties.

Answer 4

From a riddle perspective I imagine both statements are simultaneously possible if you consider the definition of yellow to be cowardly.

All goblins are cowardly and pink.

Answer 5

This is the kind of pseudo-paradox or counterintuitive result that people use to show how formal logic works, in this case classical first order logic.

Also, for the purposes of this example, I think we are supposed to assume that yellow and pink are mutually exclusive predicates. We don't actually need yellow and pink to be mutually exclusive; the person who gave you this example just wanted to show that even mutually exclusive predicates are not a problem. The person who gave you this example likely didn't want to use negation, because negation in natural languages is slippery and probably distracting here.

It's also possible that whoever came up with this example is trying to demonstrate some facts about invalid syllogisms? I'm not sure. Some more context about where you saw this example would be helpful.

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All goblins are yellow (A) does not contradict the statement All goblins are pink (B), could mean one of two things.

• A and B are both true in the real world.
• It is possible for both A and B to simultaneously be true.

The first statement implies the second, but I'll try to answer both in a useful way.

Both true in real world

So, there are no goblins. Let's just accept this.

Since there are no goblins, it doesn't matter what predicate we apply to all zero of them. The statement for every goblin g, P(g) is true is true regardless of what the predicate P is.

Universal quantification for all... / foreach ... / every ... / all ... &c, in classical logic, is equivalent to saying that there are no counterexamples.

Every goblin is yellow is equivalent to It is not the case that there exists a non-yellow goblin.

When phrased this way It is not the case that there exists a non-yellow goblin is true because there are no goblins. Similarly, It is not the case that there exists a non-pink goblin is also true.

Since these statements are both true, they don't contradict each other. If we're referring to the real world, we don't need to consult their structure or contents, we can just consider the truth values of each sentence in isolation.

It is possible for both to simultaneously be true

If we take the hint from the person giving this example and just accept that yellow and pink are mutually exclusive predicates, then both A and B are true exactly when there are no goblins.

If there is at least one goblin, then both sentences are not simultaneously true. After all, if we select one goblin at random and look at it, it will be pink, yellow, or neither pink nor yellow. In all cases, at least one of our statements is false.

Answer 6

Technically, the goblins can be both yellow and pink. When we think about this logic statement, you can think about it from the principle of set theory: a branch of mathematical logic that studies sets, which informally are collections of objects. Philosophers like W. V. Quine utilized and taught set theory to compare items and objects. If goblins can only be either yellow or pink, you may write the statement goblins all yellow OR goblins all pink and the statement you have will be contradictions. However, if a goblin can have both yellow and pink coloring on their bodies, then the statement about goblins becomes goblins all yellow AND goblins all pink, thus the statement you have is not a contradiction and goblins can both be yellow and pink in coloration. Also, it is scientifically proven that animals and different humans have different forms of photo-receptivity in their eyes, leading to different color perceptions. So, maybe goblins are yellow from the perception of certain beings or have a yellowish hue to people viewing them on one end of the electromagnetic spectrum, but appear pink from a different color perception.

Answer 7

Imagine we are trying to find some goats. We can see the whole world, except for the contents of two boxes A and B- and alas, we see no goats. We send out two lackeys, one to each box. The lackey sent to box B reports back "All the goats are in Box A" and the lackey sent to box A reports back "All the goats are in Box B". They aren't contradicting each other, and we have learned that there are no goats.

Answer 8

This crucially depends on how you formalize the statement.

Lets use predicate logic. Let Y(x) be the predicate "x is yellow" and P(x) the predicate "x is Pink".

Both statements can be simultaneously true. First of all, the truth depends on the universe. Lets create a universe with three goblins a,b and c.

Let universe A be the universe in which P_A = {a,b,c} and Y_A = {a,b,c}. Obviously, a,b and c are both yellow and pink in this universe, and the statement is true.

Lets see another universe B, in which P_B = {a} and Y_B = {b,c}. In this universe, both statements aren't simultaneously true.

Lets formalize this in a different way using a function. Let color(x) be the function that assigns each x a color. Lets also use the relation Goblin(x) to be the predicate "x is a Goblin"

Forall x. (Goblin(x) => (color(x) == Pink && color(x) == Yellow)).

Obviously, this is true in every universe where no goblins exist. But in every universe in which goblins do exists, its false, since functions have a unique mapping.

We could also use

[Forall x. (Goblin(x) => (color(x) == Pink))] and [Forall x. (Goblin(x) => (color(x) == Yellow))], it should be easy to see how they can be transformed into each other.

So really, it all just depends on what universe you are using. Too many people assume the universe in which their statements are interpreted is "the real world" or some notion of "reality", failing to understand that reality isn't a formal concept you can use here.

Its absolutely ok to talk about universes in which goblins do exist. And in those, the two statements can't be both true, depending on how you model them.

I use the term universe here, in literature you'll also often find the term model.

Answer 9

Why is it that the statement “All goblins are yellow” does not contradict the statement “All goblins are pink?”

The simplest way to see it is to disprove it.

1. Let's posit: There's no goblins.
2. "All goblins are yellow" then becomes (vacuously) true.
3. "All goblins are pink" then becomes (vacuously) true.

So you have a situation where both statements hold. As such, you cannot say the two statements contradict each other.

(yeah, it's very misleading to use "all of" for sets of size 0. But it's done all the time in some fields, like maths)

Answer 10

We have two basic kinds of quantifiers in quantificational logic: "Existential", and "Universal"

an example of a universal quantifier is like you use: "All goblins are pink", which in other words, means if I take any goblin, it's a pink one.

an example of an existential quantifier is: "A goblin is pink", which in other words means, "at least one goblin is pink", maybe more, or maybe exactly one. That's all the statement says.

Aristotle, when formulating the Aristotelian logic system, took his universal quantifiers to imply the existential. In other words, "All goblins are pink" implies "A goblin is pink". There's nothing inherently contradictory with this formulation, but it turns out it's less convenient to use universal quantifiers in this way.

In modern logic, the formulation has changed such that you cannot infer existential from universal. In other words, the statement All goblins are pink means that all things in this set have this property, but it doesn't actually state that the set has any members.

This is something that might seem rather pedantic, to now include these "vacuous truths" in our logical system, but as it turns out, it is quite important in science and mathematics. In proofs by contradiction, you may need to make use of certain universal properties of the elements in a set, while at the same time ultimately proving that that set is empty by contradicting the assertion that at least one thing is in that set. In other words, there are cases in which you might use vacuous truths about a set of numbers to prove that no such numbers can possibly exist.

For example, if we wanted to prove that there are no even prime numbers greater than 2, we would use the properties of even numbers greater than 2 (divisibility by 2) and the properties of prime numbers (only has factors of 1 and itself) to prove that no such numbers exist (even prime numbers greater than 2). These are both vacuous truths of the set of prime numbers greater than 2, but as we find, that set is empty.

Answer 11

The premises are constructed in such a way that a person will tend to bring in outside information that causes the two predicates to be contradictory.

When interpreted in formal logic, "yellow" and "pink" are just meaningless properties that a thing can have until a rule is introduced to give them meaning. Unless you say otherwise, there is no rule that makes them mutually exclusive, therefore there is no contradiction.

Using a color in both statements fools the reader into implicitly inserting additional rules into the system:

• Pink is a color
• Yellow is a color
• Pink is not yellow
• Everything is, at most, one color

These were never formally stated as premises, so to assume them is incorrect in formal logic.

Answer 12

I will go with a simpler explanation, "yello" is a constituent of "pink" so the statement "all goblins are pink" is simply a more specific statement without any contradiction.

All pink things are yellow but not all yellow things are pink.