We know: the conjunction rules were boring. Not so the disjunction rules, particularly disjunction elimination.
Disjunction introduction
The rule of disjunction introduction allows you to go from a sentence Pi to any disjunction that has Pi among its disjuncts, say P1∨ . . . ∨ Pi∨ . . . ∨ Pn. In schematic form:
Disjunction Introduction (∨ Intro):
Pi
...
. P1∨ . . . ∨ Pi∨ . . . ∨ Pn
Once again, we stress that Pimay be the first or last disjunct of the conclusion.
Further, as with conjunction introduction, some thought ought to be given to whether parentheses must be added to Pi to prevent ambiguity.
As we explained in Chapter 5, disjunction introduction is a less peculiar rule than it may at first appear. But before we look at a sensible example of how it is used, we need to have at our disposal the second disjunction rule.
Disjunction rules/ 149
Disjunction elimination
We now come to the first rule that corresponds to what we called a method of proof in the last chapter. This is the rule of disjunction elimination, the formal counterpart of proof by cases. Recall that proof by cases allows you to conclude a sentence S from a disjunction P1∨ . . . ∨ Pn if you can prove S from each of P1 through Pn individually. The form of this rule requires us to discuss an important new structural feature of the Fitch-style system of deduction. This is the notion of a subproof.
A subproof, as the name suggests, is a proof that occurs within the context subproofs of a larger proof. As with any proof, a subproof generally begins with an
as-sumption, separated from the rest of the subproof by the Fitch bar. But the
assumption of a subproof, unlike a premise of the main proof, is only temporar- temporary assumptions ily assumed. Throughout the course of the subproof itself, the assumption acts
just like an additional premise. But after the subproof, the assumption is no longer in force.
Before we give the schematic form of disjunction elimination, let’s look at a particular proof that uses the rule. This will serve as a concrete illustration of how subproofs appear in F.
1. (A ∧ B) ∨ (C ∧ D) 2. A ∧ B
3. B ∧ Elim: 2
4. B ∨ D ∨ Intro: 3
5. C ∧ D
6. D ∧ Elim: 5
7. B ∨ D ∨ Intro: 6
8. B ∨ D ∨ Elim: 1, 2–4, 5–7
With appropriate replacements for A, B, C, and D, this is a formalization of the proof given on page 133. It contains two subproofs. One of these runs from line 2 to 4, and shows that B ∨ D follows if we (temporarily) assume A ∧ B. The other runs from line 5 to 7, and shows that the same conclu-sion follows from the assumption C ∧ D. These two proofs, together with the premise (A ∧ B) ∨ (C ∧ D), are just what we need to apply the method of proof by cases—or as we will now call it, the rule of disjunction elimination.
Look closely at this proof and compare it to the informal proof given on page 133 to see if you can understand what is going on. Notice that the
Section 6.2
150 /Formal Proofs and Boolean Logic
assumption steps of our two subproofs do not have to be justified by a rule any more than the premise of the larger “parent” proof requires a justification.
This is because we are not claiming that these assumptions follow from what comes before, but are simply assuming them to show what follows from their supposition. Notice also that we have used the rule ∨ Intro twice in this proof, since that is the only way we can derive the desired sentence in each subproof.
Although it seems like we are throwing away information when we infer B ∨ D from the stronger claim B, when you consider the overall proof, it is clear that B ∨ D is the strongest claim that follows from the original premise.
We can now state the schematic version of disjunction elimination.
Disjunction Elimination (∨ Elim):
P1∨ . . . ∨ Pn ...
P1
... S
⇓ Pn
... S ... . S
What this says is that if you have established a disjunction P1∨. . . ∨Pn, and you have also shown that S follows from each of the disjuncts P1 through Pn, then you can conclude S. Again, it does not matter what order the subproofs appear in, or even that they come after the disjunction. When applying the rule, you will cite the step containing the disjunction, plus each of the required subproofs.
Let’s look at another example of this rule, to emphasize how justifications involving subproofs are given. Here is a proof showing that A follows from the sentence (B ∧ A) ∨ (A ∧ C).
Disjunction rules/ 151
1. (B ∧ A) ∨ (A ∧ C) 2. B ∧ A
3. A ∧ Elim: 2
4. A ∧ C
5. A ∧ Elim: 4
6. A ∨ Elim: 1, 2–3, 4–5
The citation for step 6 shows the form we use when citing subproofs. The citation “n–m” is our way of referring to the subproof that begins on line n and ends on line m.
Sometimes, in using disjunction elimination, you will find it natural to use the reiteration rule introduced in Chapter 3. For example, suppose we modify the above proof to show that A follows from (B ∧ A) ∨ A.
1. (B ∧ A) ∨ A 2. B ∧ A
3. A ∧ Elim: 2
4. A
5. A Reit: 4
6. A ∨ Elim: 1, 2–3, 4–5
Here, the assumption of the second subproof is A, exactly the sentence we want to prove. So all we need to do is repeat that sentence to get the subproof into the desired form. (We could also just give a subproof with one step, but it is more natural to use reiteration in such cases.)
You try it
. . . .
J 1. Open the file Disjunction 1. In this file, you are asked to prove
Medium(c) ∨ Large(c) from the sentence
(Cube(c) ∧ Large(c)) ∨ Medium(c)
Section 6.2
152 /Formal Proofs and Boolean Logic
We are going to step you through the construction of the following proof:
1. (Cube(c) ∧ Large(c)) ∨ Medium(c) 2. Cube(c) ∧ Large(c)
3. Large(c) ∧ Elim: 2
4. Medium(c) ∨ Large(c) ∨ Intro: 3 5. Medium(c)
6. Medium(c) ∨ Large(c) ∨ Intro: 5
7. Medium(c) ∨ Large(c) ∨ Elim: 1, 2–4, 5–6 I 2. To use ∨ Elim in this case, we need to get two subproofs, one for each
of the disjuncts in the premise. It is a good policy to begin by specifying both of the necessary subproofs before doing anything else. To start a subproof, add a new step and choose New Subproof from the Proof menu. Fitch will indent the step and allow you to enter the sentence you want to assume. Enter the first disjunct of the premise, Cube(c) ∧ Large(c), as the assumption of this subproof.
I 3. Rather than work on this subproof now, let’s specify the second case before we forget what we’re trying to do. To do this, we need to end the first subproof and start a second subproof after it. You end the current subproof by choosing End Subproof from the Proof menu. This will give you a new step outside of, but immediately following the subproof.
I 4. Start your second subproof at this new step by choosing New Subproof from the Proof menu. This time type the other disjunct of the premise, Medium(c). We have now specified the assumptions of the two cases we need to consider. Our goal is to prove that the conclusion follows in both of these cases.
I 5. Go back to the first subproof and add a step following the assumption. (Fo-cus on the assumption step of the subproof and choose Add Step After from the Proof menu.) In this step use ∧ Elim to prove Large(c). Then add another step to that subproof and prove the goal sentence, using ∨ Intro. In both steps, you will have to cite the necessary support sentences.
I 6. After you’ve finished the first subproof and all the steps check out, move the focus slider to the assumption step of the second subproof and add a new step. Use ∨ Intro to prove the goal sentence from your assumption.
Disjunction rules/ 153
J 7. We’ve now derived the goal sentence in both of the subproofs, and so are
ready to add the final step of our proof. While focussed on the last step of the second subproof, choose End Subproof from the Proof menu. Enter the goal sentence into this new step.
J 8. Specify the rule in the final step as ∨ Elim. For support, cite the two
subproofs and the premise. Check your completed proof. If it does not check out, compare your proof carefully with the proof displayed above.
Have you accidentally gotten one of your subproofs inside the other one?
If so, delete the misplaced subproof by focusing on the assumption and choosing Delete Step from the Proof menu. Then try again.
J 9. When the entire proof checks out, save it as Proof Disjunction 1.
. . . .
CongratulationsDefault and generous uses of the ∨ rules
There are a couple of ways in which Fitch is more lenient in checking ∨ Elim than the strict form of the rule suggests. First, the sentence S does not have to be the last sentence in the subproof, though usually it will be. S simply has to appear on the “main level” of each subproof, not necessarily as the very last step. Second, if you start with a disjunction containing more than two disjuncts, say P ∨ Q ∨ R, Fitch doesn’t require three subproofs. If you have one subproof starting with P and one starting with Q ∨ R, or one starting with Q and one starting with P ∨ R, then Fitch will still be happy, as long as you’ve proven S in each of these cases.
Both disjunction rules have default applications, though they work rather default uses of disjunction rules differently. If you cite appropriate support for ∨ Elim (i.e., a disjunction
and subproofs for each disjunct) and then check the step without typing a sentence, Fitch will look at the subproofs cited and, if they all end with the same sentence, insert that sentence into the step. If you cite a sentence and apply ∨ Intro without typing a sentence, Fitch will insert the cited sentence followed by ∨, leaving the insertion point after the ∨ so you can type in the rest of the disjunction you had in mind.
You try it
. . . .
J 1. Open the file Disjunction 2. The goal is to prove the sentence
(Cube(b) ∧ Small(b)) ∨ (Cube(b) ∧ Large(b))
Section 6.2
154 /Formal Proofs and Boolean Logic
The required proof is almost complete, though it may not look like it.
I 2. Focus on each empty step in succession, checking the step so that Fitch will fill in the default sentence. On the second empty step you will have to finish the sentence by typing in the second disjunct, (Cube(b) ∧ Large(b)), of the goal sentence. (If the last step does not generate a default, it is because you have not typed the right thing in the ∨ Intro step.)
I 3. When you are finished, see if the proof checks out. Do you understand the proof? Could you have come up with it on your own?
I 4. Save your completed proof as Proof Disjunction 2.
. . . .
CongratulationsExercises
6.1
➶
If you skipped any of the You try it sections, go back and do them now. Submit the files Proof Conjunction 1, Proof Conjunction 2, Proof Conjunction 3, Proof Conjunction 4, Proof Disjunction 1, and Proof Disjunction 2.
6.2
➶
Open the file Exercise 6.2, which contains an incomplete formal proof. As it stands, none of the steps check out, either because no rule has been specified, no support steps cited, or no sentence typed in. Provide the missing pieces and submit the completed proof.
Use Fitch to construct formal proofs for the following arguments. You will find Exercise files for each argument in the usual place. As usual, name your solutions Proof 6.x.
6.3
➶ a = b ∧ b = c ∧ c = d a = c ∧ b = d
6.4
➶ (A ∧ B) ∨ C
C ∨ B
6.5
➶ A ∧ (B ∨ C)
(A ∧ B) ∨ (A ∧ C)
6.6
➶ (A ∧ B) ∨ (A ∧ C) A ∧ (B ∨ C)
Section 6.3