Lecture 24: Fitch rules for Universal Identification

Now that we know how to eliminate the universal quantifier, let’s find a rule to introduce it.

Two attempts at formulating the 8 I rule

First (naive) attempt:

• Simply reversing the 8 E rule. . . does not work, because that would not always be truth preserving! (The ‘always’ is important here!)

• Eg. consider this bad rule: k A(c) 8x A(x) This surely does not follow from k

• Why doesn’t the conclusion follow?

Well, we had a name c in the formula at line k ,but this name must be a name for a specific object, which means that at line k we have a specific instance.

But clearly you cannot conclude a universal from a single instance:

– If you know that Sylvester is a cat, you cannot conclude that everybody is a cat!

– Or: From ‘2 is even’ it does not follow that ‘all numbers are even’ !

– And: ‘All swans are white’ does not follow from one observation of a white swan!

Second (improved) attempt:

• There is a way of making it work, if we impose certain restrictions.

• The stress on the ‘always’ seems to imply that sometimes the above rule works!

For instance, it worked quite well in our informal arguments where we

carefully considered arbitrary interpretations, with arbitrary variable assignments.

• So what we need here is a name for an arbitrary, unspecified object.

• Thus, if ‘ A(c) ’ is true for whatever ‘ c’ could name,

then ‘ A(x) ’ is satisfied by every object.

• This is not as unusual as it might sound.

In fact, this method of considering arbitrary objects is quite popular in mathematics:

‘Let x be any triangle. Then . . . ’

• To guarantee that the use of the name ‘ c’ is arbitrary enough,

we need to impose two conditions on our rule:

8 I Rule

k A(c)

8x A(x) 8I k

Under the conditions that

1) ‘ c’ must not appear in any premise or any assumption of an unfinished subproof.

2) ‘ A(x) ’ must be obtained from ‘ A(c) ’ by replacing all occurrences of ‘ c’ by ‘ x’.

23.1.2 Some comments on the conditions on 8 I

1) The first condition means that

– if ‘ c’ does appear in an assumption of a subproof,

that assumption must be discharged (i. e., the subproof must be closed)

by the time the 8 I rule involving it is applied.

This tells us that no special knowledge of whatever object ‘ c’ might denote

is available at any important stage of the proof.

2) The second condition means that

– ‘ x’ must not appear at all in the formula ‘ A(c) ’.

– In other words, any connection to ‘ c’ must be fully expunged from the formula.

The name ‘ c’ is just something like a useful and temporary place-holder

on the way to a formula which does not contain it any more.

It’s extremely important that nothing specific be known about the object named:

the object must remain arbitrary — Why?

• On the one hand, in any interpretation, a specific object must be assigned to ‘ c’.

• But, on the other hand, to secure the universal claim,

any object should make the claim true.

• And the only way we can prove this,

is if nothing whatsoever is known about the object

e. g., that it is a fluffy pink cat named Bob which likes to eat zebras.

Another way to put this is this:

• We use the same justification for concluding ‘ 8x A(x) ’ as the one given

in the explanation of the truth-conditions for this universal in an interpretation:

– In order for ‘ 8x A(x) ’ to be true in an interpretation

under a variable assignment ,

each of the variant variable assignments (covering the whole domain of discourse)

must make ‘ A(x) ’ true, no matter which object is assigned to ‘ x’.

• In this way, if we can guarantee that we have an arbitrary object from the domain,

and we see that that makes ‘ A(x) ’ true,

then we can safely conclude that the universal ‘ 8x A(x) ’ must be true as well.

23.1.3 Invalid arguments without the conditions on 8 I

You might ask yourself: Do we really need both conditions for the 8 I rule?

The answer is, yes. Otherwise, the following can happen.

Motivation for condition (1): The reason the first condition becomes clearer

when we consider what happens when we ignore it.

• Consider this silly argument:

Everybody loves Bob Dylan.

) Everyone loves themselves.

• Or, translated into FOL:

8x L(x, d)

) 8x L(x, x)

• A Fitch proof — without condition 1 — would look like this:

1 8x L(x, d)

2 L(d, d) 8E 1

3 8x L(x, x) 8I. — But this does not follow from 2

• Line 3 is clearly wrong:

Just because Bob likes himself, does not imply that everybody does!

• So, what went wrong?

The name we are trying to generalize over here (‘ d ’) is in the premise at line 1,

and we have used (at line 2) the specific knowledge it provides.

In other words, d is not an arbitrary object

(because we already know something about it)!

• If we try to do it correctly — with condition 1 — by introducing a new name at line 2,

i. e., one that is not in ‘ L(x, d) ’,

then we just get back the premise, and we have gotten nowhere:

1 8x L(x, d)

2 L(a, d) 8E 1

3 8x L(x, d) 8I 2

So we have a correct proof, but a pointless one.

• Notice here, that the name a was introduced by 8 E.

This is a correct usage of 8 I, since the name ‘ a ’ itself is arbitrary (or ‘fresh’),

and we have no special knowledge about the object it stands for,

other than they love Bob Dylan, but that is just what was given in the premise (line 1)

and that holds for every object.

• Notice also the following:

By itself, there was nothing wrong with the introduction of ‘ d ’ in line 2 of

the example without condition 1.

Only, we could not use the formula in line 2 to continue with the 8 I.

Thus, when using quantifiers, we have to think ahead,

or, reason from the bottom up as we did before!

Motivation for condition (2): If we violate the second condition,

which demands that we expunge all occurrences of the name we use to generalize over,

we can also derive bad arguments like this:

Everybody is as old as themselves. [Correct]

) Everyone is as old as Judi Dench. [Nonsense!]

• Translated into FOL:

8x O(x, x)

) 8x O(x, d)

• Without condition 2, we could prove this as follows:

1 8x O(x, x)

2 O(d, d) 8E 1

3 8x O(x, d) 8I. — But this does not follow from 2

• Clearly line 3 has to be wrong!

• It violates the second condition, because it does not replace all occurrences of ‘ d ’,

but leaves one instance of the name ‘ d ’ in place.

• If we did what the condition insists on, we would have ‘ 8x O(x, x) ’ at line 3,

which just gives us back our premise.

23.2 Simple proofs with 8 I

23.2.1 Universal conditionals

Let us now look at a correct application of the 8 I rule,

proving something of the form that ‘All A s are B s’.

This follows the pattern:

k A(c)

...

m B(c)

m + 1 A(c) ! B(c) !I k–m

m + 2 8x (A(x) ! B(x)) 8I m + 1

Notice:

• We are using (at line k ) an assumption that involves the name ‘ c ’.

• But this is part of a subproof which is closed by the time that 8 I is employed:

so the assumption is already discharged when use use 8 I.

• Thus, the condition for the application of the rule

1) ‘ c’ must not appear in any premise or any assumption of an unfinished subproof.

is fulfilled!

• Note that ‘ c ’ is still integral to the formula used at line m + 1

on which the generalization (the 8 I) at line m + 2 is based:

‘ c ’ here remains a temporary and arbitrary place-holder.

23.2.2 Example (with strategy)

Let’s look at a more concrete example of this conditional reasoning.

All heroes admire Greta.

Only people who wear capes admire Greta.

) All heroes wear capes.

In FOL, this appears as:

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

8x (A(x, g) ! C (x))

) 8x (H(x) ! C (x))

Notes on the proof:

• Since the conclusion is a 8 -formula, the last step is very likely to be 8 I.

• Recall that quantifiers function in some sense like connectives:

they build up complex formulas from simpler ones.

• In this case, the quantified formula is a conditional statement.

So, the step before the last should be ! I, to introduce ‘ H(x) ! C (x) ’,

for some fresh name instead of ‘ x ’,

i. e., one that does not occur in the premises (so, that we can use 8 I in our last step).

• Basically, we can pick any name besides ‘ g ’, so let’s pick ‘ c ’.

So, our next goal becomes to prove ‘ H(c) ! C (c) ’.

• Recalling your strategy for TFL proofs, it now seems a good move

to assume the antecedent ‘ H(c) ’.

These considerations already give us the overall structure of a possible Fitch proof:

1 8x (H(x) ! A(x, g))

2 8x (A(x, g) ! C (x))

3 H(c)

...

n 2 C (c)

n 1 H(c) ! C (c) !I 3–n 2

n 8x (H(x) ! C (x)) 8I n 1

All we need to do, is to fill out the middle part. . .

Luckily, the premises provide us with the necessary material, so we get:

1 8x (H(x) ! A(x, g))

2 8x (A(x, g) ! C (x))

3 H(c)

4 H(c) ! A(c, g) 8E 1

5 A(c, g) !E 4, 3

6 A(c, g) ! C (c) 8E 2

7 C (c) !E 6, 5

8 H(c) ! C (c) !I 3–7

9 8x (H(x) ! C (x)) 8I 8

Notice:

• There are no restrictions for the 8 E, so we can introduce ‘ c ’ in lines 4 and 6.

• The name ‘ c ’, which we’re generalizing over, appears at line 3 in an assumption,

but the subproof that it begins in line 3 ends at line 7,

so that the assumption is discharged before we apply the 8 I rule in line 9.

Our Fitch proof suggests a way of construing the English-language argument we started from:

All heroes admire Greta.

Only people who wear capes admire Greta.

) All heroes wear capes.

Informal argument (based on the above Fitch proof):

Let Carl be any hero.

Since all heroes admire Greta, Carl admires Greta.

Since only people who wear capes admire Greta, Carl wears a cape.

But ‘Carl’ stands for any hero.

) So all heroes wear capes.

Compare this to the formal proof above: the steps are really very much the same.

And this is like the way that mathematicians argue, for example:

Let k be any prime number.

Then, . . . (without using any specific information about whatever ‘ k ’ denotes)

) All primes are . . .

Warning:

• In the ‘heroes who admire Greta’ example,

there is no information about the named individual (Carl) in the premises.

• So we have to be particularly careful in an informal argument like this

that no specific information about Carl enters into to the argument!

• This is where implicit assumptions like to creep in!

(Recall the Seaworld example, in Section 21.3.)

23.2.3 One more example: 8x A(x) 8x B(x) ` 8x (A(x) B(x))

Notice the difference that the placement of the parentheses makes!

• How to start?

– Look at the conclusion, this is where you want to go.

– Look at the premises, this is what is going to help you to get there.

– Find a way to linking them!

• Proof:

1 8x A(x) _ 8x B(x)

2 8x A(x)

3 A(c) 8E 2

4 A(c) B(c) I 3

5 8x B(x)

6 B(c) 8E 5

7 A(c) B(c) I 6

8 A(c) B(c) E 1, 2–4, 5–7

9 8x (A(x) _ B(x)) 8I 8

Discussion:

• The overall strategy is to use _ E to get

from the disjunctive information in the premise (line 1) to line 8,

which is then generalized at line 9.

• But to do that, we needed to take quantifiers off at lines 3 and 6.

• So, what is instructive in this example is the auxiliary role of the universal quantifier

rules in the subproofs, and in them, the auxiliary role of the name ‘ c ’.

• The name ‘ c ’ that we use here does not appear in the premises,

and neither does it appear in the assumptions of the subproofs.

• We get it just from the universal instantiations at lines 3 and 6,

and it is introduced into the main line of the proof only at line 8.

• Here, you also see why it is important on line 9

to do away with all the c s on line 8 when we do universal generalization ( 8 I).

If we didn’t, the conclusion would not be guaranteed to be true!