4A. First example of a two-column proof

38



4. Two-column proofs

Hypotheses:

1. P ⇒ (Q & R)

2. P

Conclusion: R



You could verify that this deduction is valid by evaluating the conclusion for all possible values

of the variables that make the hypotheses true. That would not be difficult, but let us take

a different approach. Namely, we will prove that the deduction is valid by showing that it is

a combination of deductions that are already known to be valid. Informally, we could try to

convince someone that the deduction is valid by making the following explanation:

Assume the Hypotheses (1) and (2) are true. Then applying ⇒-elimination

(with P in the role of A, and Q & R in the role of B) establishes that Q &

R is true. This is an intermediate conclusion. It follows logically from the

hypotheses, but it is not the conclusion we want. Now, applying &-elimination

(with Q in the role of A, and R in the role of B) establishes that R is true.

This is the conclusion of the deduction. Thus, we see that if the hypotheses

of this deduction are true, then the conclusion is also true. So the deduction

is valid.

For emphasis, let us repeat that this explanation allows us to avoid considering all the possible

values of the variables; instead, we showed that the deduction is merely a combination of

deductions that were already verified.

Remark 4.2. Notice that we are using the fact that the symbolic deductions are true regardless

of the symbolization key we use. This is what allows us to talk about using (for example) “Q&R

in the role of B.” Another way of saying this, is that we are introducing a new symbolization

key in which we let A stand for P , and let B stand for Q & R.

Formally, a proof is a sequence of assertions. The first assertions of the sequence are

assumptions; these are the hypotheses of the deduction. It is required that every assertion later

in the sequence is an immediate consequence of earlier assertions. (There are specific rules

that determine which assertions are allowed to appear at each point in the proof.) The final

assertion of the sequence is the conclusion of the deduction.

In this chapter, we use the format known as “two-column proofs” for writing our proofs.

As indicated in the tableau below:

• Assertions appear in the left column.

• The reason (or “justification”) for including each assertion appears in the right column.

(The allowable justifications will be discussed in the later sections of this chapter.)

assertion



justification



Every assertion in a two-column proof needs to have a justification in the second column.

For clarity, we draw a dark horizontal line to separate the hypotheses from the rest of the

proof. (In addition, we will number each row of the proof, for ease of reference, and it is good

to make the left border of the figure a dark line.) For example, here is a two-column proof that

justifies the deduction above. It starts by listing the hypotheses of the deduction, and ends

with the correct conclusion.



4. Two-column proofs

1



P ⇒ (Q & R)



hypothesis



2



P



hypothesis



3



Q&R



4



R



39



⇒-elim (lines 1 and 2)

&-elim (line 3)



In this example, the assertions were written in the language of Propositional Logic, but

sometimes we will write our proofs in English. For example, here is a symbolization key that

allows us to translate P , Q, and R into English. For convenience, this same symbolization key

will be used in many of the examples in this chapter.

P : The Pope is here.

Q: The Queen is here.

R: The Russian is here.

Now, we can translate the deduction into English:

Hypotheses:

1. If the Pope is here, then the Queen and the Russian are also here.

2. The Pope is here.

Conclusion: The Russian is here.

And we can provide a two-column proof in English:

1



If the Pope is here, then the Queen

and the Russian are also here.



hypothesis



2



The Pope is here.



hypothesis



3



The Queen and the Russian are both here.



4



The Russian is here.



⇒-elim (lines 1 and 2)

&-elim (line 3)



Hypotheses:

(1) P ⇒ (Q & R) If the Pope is here, then the Queen

and the Russian are also here.

(2) P

The Pope is here.

Conclusion: R



Therefore, the Russian is here.



While you are getting accustomed to two-column proofs, it will probably be helpful to see

examples in both English and Propositional Logic. To save space, and make it easier to compare

the two, the text will sometimes combine both proofs into one figure, by adding a third column

at the right that states the English-language versions of the assertions:

assertion in

Propositional Logic



justification



English-language version

of the assertion



For example, here is what we get by combining the two proofs above:



40



4. Two-column proofs



1



P ⇒ (Q & R)



hypothesis



2



P



hypothesis



The Pope is here.



3



Q&R



⇒-elim

(lines 1 and 2)



The Queen and the Russian are both here.



4



R



&-elim (line 3)



The Russian is here.



If the Pope is here, then the Queen

and the Russian are also here.



The next few sections will explain the justifications that are allowed in a two-column proof.

4B. Hypotheses and theorems in two-column proofs

A two-column proof must start by listing all of the hypotheses of the deduction, and each

hypothesis is justified by writing the word hypothesis in the second column. (This is the

only rule that is allowed above the dark horizontal line, and it is not allowed below the dark

horizontal line.) We saw this rule in the above examples of two-column proofs. As a synonym

for “hypothesis,” one sometimes says “given” or “assumption.”

Any deduction that is already known to be valid can be used as a justification if its hypotheses have been verified earlier in the proof. (And the lines where the hypotheses appear are

written in parentheses after the name of the theorem.) For example, the theorems “⇒-elim”

and “∨-elim were used in our first examples of two-column proofs. These and several other very

useful theorems were given in Chapter 3. You will be expected to be familiar with all of them.

EXAMPLE 4.3. Here is a proof of the deduction

P ∨ Q,



Q ⇒ R,



¬P,



.˙. R.



We provide an English translation by using the symbolization key on page 39.

1



P ∨Q



hypothesis



Either Pope is here, or the Queen is here.



2



Q⇒R



hypothesis



If the Queen is here, then

the Russian is also here.



3



¬P



hypothesis



The Pope is not here.



4



Q



∨-elim (lines 1 and 3)



The Queen is here.



5



R



⇒-elim (lines 2 and 4)



The Russian is here.



EXAMPLE 4.4. Here is a short proof of the deduction in Example 3.27.

1



¬L ⇒ (J ∨ L)



hypothesis



2



¬L



hypothesis



3



J ∨L



⇒-elim (lines 1 and 2)



4



J



∨-elim (lines 3 and 2)



This proof is much shorter, and easier to check, than the case-by-case analysis of our original

solution.



4. Two-column proofs



41



EXERCISES 4.5. Write a two-column proof of each of the following deductions:

1) P ∨ Q, Q ∨ R, ¬Q, .˙. P & R.

2) (E ∨ G) ∨ F , ¬G & ¬F , .˙. E.

EXERCISE 4.6. Provide a justification (rule and line numbers) for each line of this proof.

1



W ⇒ ¬B



2



A&W



3



¬B ⇒ (J & K)



4



W



5



¬B



6



J &K



7



K



EXERCISES 4.7. Write a two-column proof of each of the following deductions. (Write the

assertions in English.)



1)



Hypotheses:

The Pope and the Queen are here.

Conclusion: The Queen is here.



Hypotheses:

The Pope is here.

2)



The Russian and the Queen are here.

Conclusion: The Queen and the Pope are here.



Hypotheses:

If the Pope is here, then the Queen is here.

3)



If the Queen is here, then the Russian is here.

The Pope is here.

Conclusion: The Russian is here.



4) Grace is sick.

Frank is sick.

.˙. Either Grace and Frank are both sick, or Ellen is sick.

EXAMPLE 4.8. Many proofs use De Morgan’s Laws (in other words, the rules for negation) or

the fact that any statement is logically equivalent to its contrapositive. Here is an example.