Lecture 17: First-order logic: The universal quantifier

6.2 The universal quantifier

In the last lecture we have learned about the 9 xistential quantifier: ‘there is. . . ’

The second central notion for the quantification of predicates is expressed in these examples:

• ‘Everybody is a hero.’

• ‘Everybody admires Greta.’

• ‘Everything is related to everything.’

Like in the case of ‘something’ or ‘somebody’, these expressions

• do not designate any particular objects that satisfy the predicate,

• instead, they make a claim about all objects in the domain.

To express ‘every’ we introduce a new symbol: ‘ 8 ’ (‘for 8 ll’)

Everybody is a hero

| {z } .

Everybody

| {z } H(x).

8x H(x).

Note:

• ‘ 8 ’ symbolizes the universal quantifier, which forms statements

about everything (in the domain).

• The upside down ‘A’ is to be seen as a mnemonic for ‘all’.

Just as the reversed ‘E’ should remind you of ‘exists’.

• Again we write the variable it refers to after it, e. g., ‘ 8x ’ or ‘ 8y ’.

• The same general considerations that we noted for the existential quantifier apply here:

– The quantifier qualifies the predicate that the variable ‘ x ’ is part of.

– In other words, the quantifier tells us something about that predicate,

which refers to a property/relation.

– Whether it is true or not, depends on the domain in question.

‘Everybody is nice’ is true in the domain of cats, but false in the domain of people.

– Again, the term ‘quantifier’ is justified:

it provides an answer to a ‘How many?’ question:

∗ How many heroes? All in the domain.

∗ How many things are related? All in the domain.

– A universally quantified statement tell us something about the extent

of the property, i. e., the number of objects that have it, the ‘quantity’.

16.2.1 Quantifier restriction

To express universality, i. e., to make a claim about all objects in a domain, English

has many determiners (as these words are called by linguists): e. g., ‘all’, ‘every’, ‘any’.

In fact, English has another way to express universality,

although the grammar suggests otherwise:

• ‘The cat is a mammal.’

But, we leave that for those who study English, not logic. . .

As we saw in the case of ‘some’, universal determiners also usually take modifiers:

• Every hero wears a cape.

• All heroes wear capes.

• Any hero wears a cape.

These statements all mean the same, they are all true in the same case:

‘Every F is G ’ is true iff everything which is F is G .

(Compare this with the existentially quantified statement in Section 15.4.3.)

• Similarly, for ‘All F s . . . ’ and ‘Any F . . . ’.

Question: How do we symbolize such a restricted quantification of the form

‘Every F is G ’ ?

Let us consider three different options and see under which conditions they are true:

a) 8x (F (x) ^ G (x))

– If this formula is true, everything must be F .

– But, this does not have to be the case even if ‘Every F is G ’ is true.

– After all, there might not be any F s, as in ‘Every monster is green’.

– So, this symbolization does not capture the meaning of ‘Every F is G ’ adequately!

b) 8x (F (x) _ G (x))

– One way in which this formula can be true is if everything is F and nothing is G .

– So, it can be true when ‘Every F is G ’ is false.

– Thus, this symbolization does not capture the meaning of ‘Every F is G ’ adequately!

c) 8x (F (x) ! G (x))

– Here, if x is F , then x must also be G .

– If x is not F , it doesn’t matter if it is G or not (see the truth table for implication).

– Here, the can be things that are not F , and things that are G , but not F .

– This fits our understanding of ‘Every F is G ’, e. g., ‘Every cat is a mammal’.

– So, this is the symbolization we were looking for!

16.2.2 Difference between existential and universal restriction

There is a green onion. Every green thing is an onion. Every onion is green.

9x (Gr (x) ^ On(x)) 8x (Gr (x) ! On(x)) 8x (On(x) ! Gr (x))

16.2.3 Examples

Informally, ‘ 8x (F (x) ! G (x)) ’ means: Look at the F s, then check that they are all G .

We use it to represent any of

• All F s are G s.

• Every F is G .

• Any F is G .

For example:

• Every hero wears a cape.

All heroes wear capes. 8x (H(x) ! C (x))

• Everyone who wears a cape observed Greta. 8x (C (x) ! O(x, g))

• Greta admires anyone who wears a cape. 8x (C (x) ! A(g, x))

What is the meaning of 8x (A(g, x) ! C (x)) ?

• Autumn observed everyone who observed Greta. 8x (O(x, g) ! O(a, x))

16.3 ‘No. . . ’ and ‘Only. . . ’

16.3.1 No F is G

Let us now add ‘no’ to our statements.

In English, ‘no’ acts as another so-called determiner, as in ‘No F s are G s’.

However, we will not regard it as primitive,

but rather treat it by combining ‘all’ or ‘exists’ with the negation which we already have.

There are two ways to read ‘No F s are G s’:

1. The first is simply to say ‘There does not exist an F which is also G ’. So:

¬9x (F (x) ^ G (x)).

2. The other is to say: ‘Whatever F you take, that will not be G ’. This gives us:

8x (F (x) ! ¬G (x)).

At first sight, these look very different. But they are in fact equivalent! (See Venn diagram.)

So either one of these ways of reading the assertion would be correct.

Why? We’ll come to this soon!

16.3.1.1 Examples

• No hero wears a cape.

No heroes wear capes. 8x (H(x) ! ¬C (x))

• No one who wears a cape observed Greta. 8x (C (x) ! ¬O(x, g))

• Greta admires no one who wears a cape. ¬9x (C (x) ^ A(g, x))

• Autumn observed no one who observed Greta. 8x (O(x, g) ! ¬O(a, x))

16.3.2 Only F s are G

We symbolize ‘Only F s are G s’ as:

8x (G (x) ! F (x))

16.3.2.1 Examples

• Only heroes wear capes. 8x (C (x) ! H(x))

• Only people who wear capes observed Greta. 8x (O(x, g) ! C (x))

• Greta admires only people who wear capes. 8x (A(g, x) ! C (x))

• Autumn observed only people who observed Greta. 8x (O(a, x) ! O(x, g))

16.4 Quantifier equivalences

The reason why in our earlier example the formulations

¬9x (F (x) ^ G (x)) and 8x (F (x) ! ¬G (x))

are equivalent is because of a general equivalence between the forms

9x P(x) and ¬8x ¬P(x) .

This assumes, as we always do, that properties are determinate (not vague or ambiguous).

The same holds for:

8x P(x) and ¬9x ¬P(x)

¬8x P(x) and 9x ¬P(x)

8x ¬P(x) and ¬9x P(x)

Thus, using negation, we have here a general relationship between the quantifiers

(in classical logic, not necessarily in other logics. . . ).

In other words, to express quantifications, there are two ways of formulating things,

one with ’all’ and one with ’exists’ – and having the negation is put in the right place.

Exercise: Try to explain to yourself why these equivalences hold and how they justify the

equivalence between ¬9x (F (x) ^ G (x)) and 8x (F (x) ! ¬G (x)) . (See 16.3.1.)

Here is a hint and a useful trick:

• Whenever you have a negated quantifier, you are licensed to push in the negation

until it meets either a bracket or a formula (whichever comes first)

iff you flip any quantifier that stands in the way!

• Notice you have the negation of either a conjunction, or an implication statement.

• What equivalences do you know from TFL that will justify our claim

that the two sentences are equivalent?

• Now notice that if the negation is in front of the bracket or the formula,

this is just the reverse procedure, push the negation out and

flip any quantifier that stands in the way!

16.5 Some difficulties translating English into FOL

16.5.1 The indefinite article

• We standardly use ‘is a’ to indicate predication, e. g., ‘Greta is a hero’, H(g) .

• But ‘a’ is often used to claim existence, e. g.,

‘Greta admires a hero’.

so:

9x (H(x) ^ A(g, x))

• However, a generic indefinite article is really closer to a universal quantifier:

‘A hero is someone who inspires’

which is:

8x (H(x) ! I (x))

16.5.2 Universal ‘some’ and existential ‘any’

Normally ‘some’ signifies that one must use the existential quantifier,

and ‘any’ signals the universal.

But it can happen that we have the opposite:

• ‘Someone’, ‘something’ can require a universal quantifier:

If these occur in the antecedent, with a pronoun referring back to it in the consequent,

e. g.,

If someone is a hero, Autum admires them.

Roughly: Autumn admires all heroes.

8x (H(x) ! A(a, x))

• On the other hand, ‘any’ occurring in the antecedent but without pronouns referring

back to them is existential:

If anyone is a hero, Greta is.

Roughly: if there are heroes (at all), Greta is a hero.

9x H(x) ! H(g)

These examples show that

• Translating English into FOL is not mechanical,

but requires a good understanding of what the English sentence actually means.

• Learning to symbolize into FOL forces us to reflect on

what we actually mean with our (informal) language.

• Think about how to represent the difference between:

– Everybody loves somebody.

– There is somebody who is loved by everybody.

16.6 Mixed domains

Sometimes we want to speak of different kinds of things, but we only have one domain.

So, this requires some extra care when using quantifiers.

• A domain can include any mix of things (e. g., people, animals, items of clothing).

• But, to speak of them separately, requires predicates for these different kinds.

Let us look at an example:

Symbolization key

Domain : people currently alive and items of clothing

P(x) : x is a person

I (x) : x inspires

L(x) : x is an item of clothing.

E (x) : x is a cape

C (x) : x wears a cape

R(x, y ) : x wears y

• Some people inspire.

In domain of people only: 9x I (x)

In mixed domain: 9x (P(x) ^ I (x))

• Greta wears something.

In domain of people only: not possible

In mixed domain: 9x ((L(x) ^ P(g)) ^ R(g, x))

• Not everyone is wearing a cape.

In domain of people only: ¬8x C (x)

In mixed domain: ¬8x (P(x) ! C (x))

(alternatively:) ¬8x (P(x) ! 9y (E (y ) ^ R(x, y )))

16.7 Using predicates to express properties, revisited

• One-place predicates express properties

For example:

– ‘ H(x) ’ expresses the property ‘being a hero’

– ‘ A(x, g) ’ expresses ‘ x admires Greta’.

• Combinations of predicates (with connectives, names) express derived properties.

For example:

– ‘ H(x) ^ C (x) ’ expresses ‘ x is a hero who wears a cape’

• But using quantifiers, we can express even more complex properties

For example:

– ‘ 9y (P(y ) ^ A(x, y )) ’ expresses ‘ x admires someone’

16.7.1 From sentences to predicates, and vice versa

• Remember this: if you can say it for Greta, you can say it for x !

– Greta admires a hero. 9y (H(y ) ^ A(g, y )) , which is a sentence

– x admires a hero. 9y (H(y ) ^ A(x, y )) , which is a formula, not a sentence

• This goes the other way: if you can say it for x , you can say it for Greta.

– x wears a cape. 9y (E (y ) ^ R(x, y )) , which is a formula, not a sentence

– Greta wears a cape. 9y (E (y ) ^ R(g, y )) , which is a sentence

16.7.2 Further examples

• x is a person and wears a cape. 9y ((L(y ) ^ E (y )) ^ (P(x) ^ R(x, y ))

• x is a person admired by every person. 8y ((P(y ) ^ P(x)) ! A(y , x))

• x is a person and admires a hero-person. (P(x) ^ 9y ((P(y ) ^ H(y )) ^ A(x, y )))

• x admires only heroes. 8y (((P(x) ^ P(y )) ^ A(x, y )) ! H(y ))

• x is a person who wears no cape. P(x) ^ ¬9y ((L(y ) ^ E (y )) ^ R(x, y ))