Lecture 18: First order logic: The Identity

languages of ﬁrst-order logic, namely

• identity, which is a 2-place relation that we always symbolize with ‘ = ’.

17.3 Identity

17.3.1 Trivial or tricky?

Some ideas seem almost trivial until one starts to think about them very carefully:

identity is an example for this!

After all, identity seem to be saying that this thing is the same as this thing, and how can

that not be the case? (Aristotle named the Law of Identity his ‘First Law of Thought’.)

However, we have to keep in mind the distinction between

(1) what the world is like, and (2) what we are able to assert about the world.

In particular,

• one thing might possess two diﬀerent names,

e. g., the names, ‘George Eliot’ and ‘Mary Ann Evans’, and

• a person can learn the truth of the identity ‘George Eliot = Mary Ann Evans’

and be genuinely enlightened by it.

In other words,

• identity statements are not always trivial, and

• information about identity can be cognitively signiﬁcant.

Another example:

• The Babylonians famously discovered that Hesperus = Phosphorus,

i. e., that two heavenly bodies,

– one appearing in the evening (the Evening Star) and

– one appearing in the morning (the Morning Star)

were one and the same. In fact, we now know this body today as

– the planet Venus.

• This was an extraordinarily signiﬁcant discovery, and one which shows

that what were thought to be two distinct things were actually one thing.

• Again: Identities can be tricky!

17.3.2 Two principles about identity

There are two important facts about the way we understand identity:

1. The Principle that everything is identical to itself

(i. e., Aristotle’s Law of Identity), which we might formalize as:

8x (x = x).

2. The Principle of the indiscernibility of identicals

(one of Leibniz’s fundamental laws of thought), which might be formulated as:

8x 8y ((x = y ^ P(x)) ! P(y ),

which holds for whatever predicate you put in for the placeholder.

This principle says (roughly speaking) that if two things are identical, then

whatever predicate you can assert of one of them,

must also hold when you refer to the other.

Even more roughly speaking:

no predicate can mark a diﬀerence (‘discern’) between identical things.

Or: If two things are identical, you cannot tell them apart!

It might help to look at this negatively: if two things are genuinely diﬀerent,

then there must be some property which marks the diﬀerence.

8x 8y (P(x) ^ ¬P(y )) ! ¬(x = y ).

These two principles are crucial in reasoning about identity,

and will play a signiﬁcant role when we come to adopt Fitch rules.

Note:

• We adopt identity in all of our FOLs, and indeed we consider it to be what is sometimes

called a logical particle, alongside the connectives and the quantiﬁers.

• This means that we always interpret it in the same way, as expressing the identity

or (if negated) the non-identity of objects in whatever domain is speciﬁed.

17.4 ‘Else’

Let us see how we can use identity to express English phrases.

In a domain consisting only of people,

8x 8y O(x, y )

means ‘everyone observes everyone’, not that everyone observes everyone else.

Similarly,

9x 9y A(x, y )

does not mean that someone admires someone else.

Instead, it is also true, if there is someone who admires themself!

In fact, both formulas can be true even in a domain that has only one single element!

If we want to symbolize that everybody observes everybody else,

we can make this explicit by using the identity predicate:

8x 8y (¬(x = y ) ! O(x, y ))

It says ‘For all x, for all y , if x is diﬀerent from y , then O(x, y ) ’.

So, it is true if everyone satisﬁes ‘ x observes everyone who is not x ’.

Now, we can also use this technique to symbolize

‘Someone observes everyone else’,

and (for practice) we also use ‘ P(x) ’ to restrict ‘everyone’ and ‘someone’ to people,

i. e., to say explicitly that there is someone who observes every person other than them:

9x P(x) ^ 8y ((P(y ) ^ ¬(x = y )) ! O(x, y ))

Note:

• In the previous formulas, I have written ‘ ¬(x = y ) ’, with parentheses around ‘ x = y ’

to be very explicit about the formula that is being negated.

• However, to reduce the clutter of using many parentheses, our textbook adopts the

convention of writing the negation of the identity simply as ‘ ¬x = y ’.

After all, negation is a unary connective that applies only to formulas,

not to individual variables, so no ambiguity arises.

From now on, we shall also adopt this usage.

• (Another convention that you can also see in the literature is ‘ x 6 = y ’.)

17.5 Uniqueness

To say that someone is a hero is to say that at least one person satisﬁes H(x) ,

i. e., ‘ 9x H(x) ’.

To say that there is exactly one hero is to say that

(1) there is a person who satisﬁes H(x) and (2) no-one else does.

How can we symbolize ‘no one else’ ?

To say ‘no-one other than x is a hero’ is to say that

no hero exists who is diﬀerent from x .

We can do this in any of the following ways:

1. 9x (H(x) ^ ¬9y (H(y ) ^ ¬x = y )),

2. 9x 8y (H(y ) $ x = y ),

3. 9x (H(x) ^ 8y (H(y ) $ x = y )),

4. 9x (H(x) ^ 8y (H(y ) ! x = y )).

As it turns out, these are all equivalent!

This is a lot to take in. . .

For doing translations, it is a good idea

to pick the one which you ﬁnd the most natural, usually the ﬁrst or the fourth.

Then, you can use what we have already said about quantiﬁer equivalence,

together with the familiar rules for TFL connectives, to see that these are equivalent.

17.6 ‘Only’, ‘at least’, ‘at most’, ‘exactly’

Let us see how we can use identity to express other notions about quantities.

17.6.1 ‘Only’ — ‘Exactly one’

We can use the same procedures to capture something like:

‘Greta is the only hero’, or ‘Potter is the only wizard’.

These say that Greta is a hero — and no-one other than Greta is also a hero —

and the same respectively for Potter and the wizards. So we have, for example:

(W (p) ^ ¬9y (W (y ) ^ ¬p = y )).

You can practice using the other formulations, for instance:

(W (p) ^ 8y (W (y ) ! p = y )).

As we can see, ‘only’ is really ‘exactly one’.

And the ordinary ‘ 9y ’ quantiﬁer by itself is really ‘at least one’.

17.6.2 ‘At least two’

Remember that

9x 9y (H(x) ^ H(y ))

does not say that there are at least two heroes:

it is also true if someone is a hero, and there is just one.

Because the object variables x and y can be instantiated by the same element in the domain.

So Greta being a hero would make it true, even if there were no other heroes.

If you want to insist that there are distinct heroes, i. e., say that there are at least two

heroes, you have to insist that they are diﬀerent by using the identity predicate.

So here you would include ‘ ¬x = y ’, as follows:

9x 9y ((H(x) ^ H(y )) ^ ¬x = y )

9x H(x) ^ 9y (H(y ) ^ ¬x = y ).

17.6.3 ‘At most one’

The previous formula also lets us express that there is at most one hero,

for we can simply deny that there are at least two. So:

¬9x (H(x) ^ 9y (H(y ) ^ ¬x = y ))

This will be true in just the cases where there are 0 or 1 heroes,

and false when there are 2 or more.

(Exercise: Convince yourself that these are good translations!)

• There is another way of expressing ‘at most one’, namely

to say that if x and y are heroes, then x and y are the same, so:

8x 8y ((H(x) ^ H(y )) ! x = y ).

This has the advantage of making it clear that

• there might be no heroes at all

(the universal conditional is true if both antecedent conditions fail), or

• when there is just one: if there appear to be two things

that satisfy the antecedent conditions, then they have to be the same!

The formula is false in all other possibilities, which is just what we want.

For example,

• if there are indeed just two heroes, Greta and Autumn, then they will make sure the

antecedent conditions are satisﬁed; but then, since they are distinct, the consequent

must be false — they are not identical!

• The same will follow if there are three; just take any two of them, and the same

argument works. In fact, the same reasoning will apply for any number.

17.6.4 ‘At most two’ vs. ‘at least three’

What about ‘There are at most two heroes’ ?

For this, we have to deny that there are at least three heroes.

Now this requires three existential assertions, so will

involve ‘ 9x ’, ‘ 9y ’, ‘ 9z ’, and we then have to say that any two of x , y , z are diﬀerent.

For this, ‘ ¬x = y ^ ¬y = z ’ is not enough. (Because it could still be that x = z .)

Thus, we express ‘there are at least three heroes’ by:

9x (H(x) ^ 9y (((H(y ) ^ ¬x = y )) ^ 9z (H(z) ^ (¬x = z ^ ¬y = z))))

By denying this, we express that there are at most two heroes:

¬(9x (H(x) ^ 9y (((H(y ) ^ ¬x = y )) ^ 9z (H(z) ^ (¬x = z ^ ¬y = z)))))

Alternatively, we can think of it this way: Of any three heroes x , y , z ,

at least two have to be the same (i.e., x = y or x = z or y = z ).

8x 8y 8z ((H(x) ^ (H(y ) ^ H(z))) ! (x = y (x = z y = z))).

By themselves, these formulas look rather daunting, but once

you understand their structure and the logic behind it, they should become clear.

17.6.5 ‘Exactly’

We already know how to say that there is exactly one hero:

some x is a hero, and there is no y that is diﬀerent from x and also a hero.

To say there are exactly two heroes is to say both,

(1) that there are at least two and (2) that there are at most two.

Alternatively, you can think of it this way:

there are two diﬀerent heroes ( x and y ), and any hero ( z ) is one of these two.

9x 9y (H(x) ^ (H(y ) ^ ¬x = y )) ^ 8z (H(z) ! (z = x _ z = y ))

We can also express this by the logically equivalent statement:

9x 9y ¬x = y ^ 8z (H(z) $ (z = x _ z = y ))

(But, this equivalence has to be proved!)

Following this pattern, we could symbolize ‘There are exactly 27 heroes’ or

‘There are at most 15 heroes’, etc., but, the sentences would become extremely long. . .

Nevertheless, all of these diﬀerent notions can be expressed by our two quantiﬁers,

the TFL connectives, and identity!

17.7 Deﬁnite descriptions

17.7.1 ‘The’

In English, the deﬁnite article ‘the’ is used to form phrases called ‘deﬁnite descriptions’

(i. e., expressions that are not proper names, but nevertheless identify an individual object),

such as

• ‘the hero’ or

• ‘the youngest villain’.

Grammatically, deﬁnite descriptions are used in the same way as

• indeﬁnite descriptions (‘a hero’) or

• quantiﬁers (‘all heroes’, ‘no villains’),

i. e., they seem to go inside predicates, as in

• ‘All heroes are young’, or

• ‘The hero of climate marches is young’.

However, we have already seen with ‘a’ and ‘all’ that this is misleading!

So, how should we treat deﬁnite descriptions?

According to the philosopher Bertrand Russell’s (1872–1970) analysis of deﬁnite descriptions,

a sentence containing a deﬁnite description, for example

‘The A is B ’,

should be represented in such a way that it is true iﬀ

(1) exactly one thing is A , and (2) that thing is also B .

Using what we have learned so far, one way of expressing this is

9x (A(x) ^ 8y (A(y ) ! x = y )) ^ B(x).

17.7.2 Singular possessives

Deﬁnite descriptions can also be used also to symbolize singular possessives, as in

‘Autumn wears Greta’s cape’.

We can paraphrase this as: ‘Autumn wears the cape that belongs to Greta’.

Symbolizing this with a deﬁnite description yields (using ‘ E (x) ’ for ‘ x is a cape’):

9x ((E (x) ^ B(x, g)) ^ 8z ((E (z) ^ B(z, g)) ! z = x)) ^ R(a, x),

or, shorter, but logically equivalent:

9x 8z ((E (z) ^ B(z, g)) $ z = x) ^ R(a, x).

Here, of course we need the right predicates for

• ‘ x is a cape’, E (x) ,

• ‘ x belongs to y ’, B(x, y ) ,

• ‘ x wears y ’, R(x, y ) .

17.7.3 ‘Both’ and ‘neither’

Russell’s idea can also be used to deal with ‘both’ and ‘neither’, when they

• are not used as sentence connectives (as in ‘Neither Autumn nor Greta are villains’),

• but as determiners, as in ‘Neither villain is younger than Greta’.

The latter sentence is true iﬀ

(1) there are exactly two villains, and (2) each of these is not younger than Greta.

Using Russell’s method, and V (x) for ‘ x is a villain’

and Y (x, y ) for ‘ x is younger than y ’, we get:

9x 9y (((V (x)^V (y ))^¬x = y )^8z (V (z) ! (z = x _z = y )))^(¬Y (x, g)^¬Y (y , g)),

Again, this looks formidable, but a closer look reveals it’s structure:

it just spells out what we’ve seen before,

but for the case of two objects and not just one, which is what makes it much longer.

What do the components of the formula tell us?

• First, it is a conjunction of things under 2 existential quantiﬁers.

This means that we have a collection of conditions that have to be fulﬁlled.

• Second, the ﬁrst part says that there are 2 diﬀerent things ( x and y ) that are villains;

this is followed by a clause which says anything which is a villain ( z ) must be one or

the other of these two things ( x or y );

• Finally, the last clause says that the ﬁrst ( x ) and the second ( y ) of these two villains

is not younger than Greta.

Alternative formulations are (Look at them carefully to individuate the components!):

1. 9x 9y 8z (V (z) $ (z = x _ z = y )) ^ (¬x = y ^ (¬Y (x, g) ^ ¬Y (y , g)))

2. 9x 9y ((V (x) ^ V (y )) ^ 8z (V (z) ! (z = x _ z = y ))) ^ (¬x = y ^ 8z (V (z) ! ¬Y (z, g)))

3. 9x 9y 8z (V (z) $ (z = x _ z = y )) ^ (¬x = y ^ 8z (V (z) ! ¬Y (z, g)))

If we wanted to symbolize ‘Both villains are younger than Greta’ (instead of ‘neither’),

we would do exactly the same, but would without negating the ‘ Y (x, y ) ’ predicate.

Today we’ve seen how powerful, and certainly not trivial, the notion of identity is!