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
Negation rules
Last but not least are the negation rules. It turns out that negation introduc-tion is our most interesting and complex rule.
Negation rules/ 155
Negation elimination
The rule of negation elimination corresponds to a very trivial valid step, from
¬¬P to P. Schematically:
Negation Elimination (¬ Elim):
¬¬P...
. P
Negation elimination gives us one direction of the principle of double nega-tion. You might reasonably expect that our second negation rule, negation introduction, would simply give us the other direction. But if that’s what you guessed, you guessed wrong.
Negation introduction
The rule of negation introduction corresponds to the method of indirect proof or proof by contradiction. Like ∨ Elim, it involves the use of a subproof, as will the formal analogs of all nontrivial methods of proof. The rule says that if you can prove a contradiction ⊥ on the basis of an additional assumption P, then you are entitled to infer ¬P from the original premises. Schematically:
Negation Introduction (¬ Intro):
P ...
⊥ . ¬P
There are different ways of understanding this rule, depending on how we interpret the contradiction symbol ⊥. Some authors interpret it simply as shorthand for any contradiction of the form Q ∧ ¬Q. If we construed the schema that way, we wouldn’t have to say anything more about it. But we will treat ⊥ as a symbol in its own right, to be read “contradiction.” This has several advantages that will become apparent when you use the rule. The one disadvantage is that we need to have rules about this special symbol. We introduce these rules next.
Section 6.3
156 /Formal Proofs and Boolean Logic
⊥ Introduction
The rule of ⊥ Introduction (⊥ Intro) allows us to obtain the contradiction symbol if we have established an explicit contradiction in the form of some sentence P and its negation ¬P.
⊥ Introduction (⊥ Intro):
P...
¬P...
. ⊥
Ordinarily, you will only apply ⊥ Intro in the context of a subproof, to show that the subproof’s assumption leads to a contradiction. The only time you will be able to derive ⊥ in your main proof (as opposed to a subproof) is when the premises of your argument are themselves inconsistent. In fact, this is how we give a formal proof that a set of premises is inconsistent. A formal formal proofs of
inconsistency proof of inconsistency is a proof that derives ⊥ at the main level of the proof.
Let’s try out the rules of ⊥ Intro and ¬ Intro to see how they work.
You try it
. . . .
I 1. To illustrate these rules, we will show you how to prove ¬¬A from A.
This is the other direction of double negation. Use Fitch to open the file Negation 1.
I 2. We will step you through the construction of the following simple proof.
1. A 2. ¬A
3. ⊥ ⊥ Intro: 1, 2
4. ¬¬A ¬ Intro: 2–3
I 3. To construct this proof, add a step immediately after the premise. Turn it into a subproof by choosing New Subproof from the Proof menu. Enter the assumption ¬A.
I 4. Add a new step to the subproof and enter ⊥, changing the rule to ⊥ Intro.
Cite the appropriate steps and check the step.
Negation rules/ 157
5. Now end the subproof and enter the final sentence, ¬¬A, after the sub- J proof. Specify the rule as ¬ Intro, cite the preceding subproof and check the step. Your whole proof should now check out.
J 6. Notice that in the third line of your proof you cited a step outside the
subproof, namely the premise. This is legitimate, but raises an important issue. Just what steps can be cited at a given point in a proof? As a first guess, you might think that you can cite any earlier step. But this turns out to be wrong. We will explain why, and what the correct answer is, in the next section.
J 7. Save your proof as Proof Negation 1.
. . . .
Congratulations The contradiction symbol ⊥ acts just like any other sentence in a proof. In particular, if you are reasoning by cases and derive ⊥ in each of your subproofs, then you can use ∨ Elim to derive ⊥ in your main proof. For example, here is a proof that the premises A ∨ B, ¬A, and ¬B are inconsistent.1. A ∨ B 2. ¬A 3. ¬B 4. A
5. ⊥ ⊥ Intro: 4, 2
6. B
7. ⊥ ⊥ Intro: 6, 3
8. ⊥ ∨ Elim: 1, 4–5, 6–7
The important thing to notice here is step 8, where we have applied ∨ Elim to extract the contradiction symbol from our two subproofs. This is clearly justified, since we have shown that whichever of A or B holds, we immediately arrive at a contradiction. Since the premises tell us that one or the other holds, the premises are inconsistent.
Other ways of introducing ⊥
The rule of ⊥ Intro recognizes only the most blatant contradictions, those where you have established a sentence P and its negation ¬P. What if in the course of a proof you come across an inconsistency of some other form? For
Section 6.3
158 /Formal Proofs and Boolean Logic
example, suppose you manage to derive a single tt-contradictory sentence like ¬(A ∨ ¬A), or the two sentences ¬A ∨ ¬B and A ∧ B, which together form a tt-contradictory set?
It turns out that if you can prove any tt-contradictory sentence or sen-tences, the rules we’ve already given you will allow you to prove ⊥. It may take a fair amount of effort and ingenuity, but it is possible. We’ll eventually prove this, but for now you’ll have to take our word for it.
One way to check whether some sentences are tt-contradictory is to try to derive ⊥ from them using a single application of Taut Con. In other words, introducing ⊥
with Taut Con enter ⊥, cite the sentences, and choose Taut Con from the Rule? menu. If Taut Con tells you that ⊥ follows from the cited sentences, then you can be sure that it is possible to prove this using just the introduction and elimination rules for ∧, ∨, ¬, and ⊥.
Of course, there are other forms of contradiction besides tt-contradictions.
For example, suppose you manage to prove the three sentences Cube(b), b = c, and ¬Cube(c). These sentences are not tt-contradictory, but you can see that a single application of = Elim will give you the tt-contradictory pair Cube(c) and ¬Cube(c). If you suspect that you have derived some sentences whose inconsistency results from the Boolean connectives plus the identity predicate, you can check this using the FO Con mechanism, since FO Con introducing ⊥
with FO Con understands the meaning of =. If FO Con says that ⊥ follows from the cited sentences (and if those sentences do not contain quantifiers), then you should be able to prove ⊥ using just the introduction and elimination rules for =, ∧,
∨, ¬, and ⊥.
The only time you may arrive at a contradiction but not be able to prove
⊥ using the rules of F is if the inconsistency depends on the meanings of predicates other than identity. For example, suppose you derived the contra-diction n < n, or the contradictory pair of sentences Cube(b) and Tet(b). The rules of F give you no way to get from these sentences to a contradiction of the form P and ¬P, at least without some further premises.
What this means is that in Fitch, the Ana Con mechanism will let you introducing ⊥
with Ana Con establish contradictions that can’t be derived in F . Of course, the Ana Con mechanism only understands predicates in the blocks language (and even there, it excludes Adjoins and Between). But it will allow you to derive ⊥ from, for example, the two sentences Cube(b) and Tet(b). You can either do this directly, by entering ⊥ and citing the two sentences, or indirectly, by using Ana Con to prove, say, ¬Cube(b) from Tet(b).
Negation rules/ 159
You try it
. . . .
J 1. Open Negation 2 using Fitch. In this file you will find an incomplete proof.
As premises, we have listed a number of sentences, several groups of which are contradictory.
J 2. Focus on each step that contains the ⊥ symbol. You will see that various
sentences are cited in support of the step. Only one of these steps is an application of the ⊥ Intro rule. Which one? Specify the rule for that step as ⊥ Intro and check it.
J 3. Among the remaining steps, you will find one where the cited sentences
form a tt-contradictory set of sentences. Which one? Change the justifi-cation at that step to Taut Con and check the step. Since it checks out, we assure you that you can derive ⊥ from these same premises using just the Boolean rules.
J 4. Of the remaining steps, the supports of two are contradictory in view of the
meaning of the identity symbol =. Which steps? Change the justification at those step to FO Con and check the steps. To derive ⊥ from these premises, you would need the identity rules (in one case = Elim, in the other = Intro).
J 5. Verify that the remaining steps cannot be justified by any of the rules ⊥
Intro, Taut Con or FO Con. Change the justification at those steps to Ana Con and check the steps.
J 6. Save your proof as Proof Negation 2. (Needless to say, this is a formal proof
of inconsistency with a vengeance!)
. . . .
Congratulations⊥ Elimination
As we remarked earlier, if in a proof, or more importantly in some subproof, you are able to establish a contradiction, then you are entitled to assert any fol sentence P whatsoever. In our formal system, this is modeled by the rule of ⊥ Elimination (⊥ Elim).
⊥ Elimination (⊥ Elim):
⊥...
. P
Section 6.3
160 /Formal Proofs and Boolean Logic
The following You try it section illustrates both of the ⊥ rules. Be sure to go through it, as it presents a proof tactic you will have several occasions to use.
You try it
. . . .
I 1. It often happens in giving proofs using ∨ Elim that one really wants to eliminate one or more of the disjuncts, because they contradict other assumptions. The form of the ∨ Elim rule does not permit this, though.
The proof we will construct here shows how to get around this difficulty.
I 2. Using Fitch, open the file Negation 3. We will use ∨ Elim and the two ⊥ rules to prove P from the premises P ∨ Q and ¬Q.
I 3. Start two subproofs, the first with assumption P, the second with assump-tion Q. Our goal is to establish P in both subproofs.
I 4. In the first subproof, we can simply use reiteration to repeat the assump-tion P.
I 5. In the second subproof, how will we establish P? In an informal proof, we would simply eliminate this case, because the assumption contradicts one of the premises. In a formal proof, though, we must establish our goal sentence P in both subproofs, and this is where ⊥ Elim is useful. First use
⊥ Intro to show that this case is contradictory. You will cite the assumed sentence Q and the second premise ¬Q. Once you have ⊥ as the second step of this subproof, use ⊥ Elim to establish P in this subproof.
I 6. Since you now have P in both subproofs, you can finish the proof using ∨ Elim. Complete the proof.
I 7. Save your proof as Proof Negation 3.
. . . .
Congratulations It turns out that we do not really need the rule of ⊥ Elim. You can prove any sentence from a contradiction without it; it just takes longer. Suppose, for example, that you have established a contradiction at step 17 of some proof.Here is how you can introduce P at step 21 without using ⊥ Elim.
Negation rules/ 161
17. ⊥ 18. ¬P
19. ⊥ Reit: 17
20. ¬¬P ¬ Intro: 18–19
21. P ¬ Elim: 20
Still, we include ⊥ Elim to make our proofs shorter and more natural.
Default and generous uses of the ¬ rules
The rule of ¬ Elim allows you to take off two negation signs from the front of a sentence. Repeated uses of this rule would allow you to remove four, six, or indeed any even number of negation signs. For this reason, the implementation of ¬ Elim in Fitch allows you to remove any even number of negation signs in one step.
Both of the negation rules have default applications. In a default application default uses of negation rules of ¬ Elim, Fitch will remove as many negation signs as possible from the front
of the cited sentences (the number must be even, of course) and insert the resulting sentence at the ¬ Elim step. In a default application of ¬ Intro, the inserted sentence will be the negation of the assumption step of the cited subproof.
You try it
. . . .
J 1. Open the file Negation 4. First look at the goal to see what sentence we
are trying to prove. Then focus on each step in succession and check the step. Before moving to the next step, make sure you understand why the step checks out and, more important, why we are doing what we are doing at that step. At the empty steps, try to predict which sentence Fitch will provide as a default before you check the step.
J 2. When you are done, make sure you understand the completed proof. Save
your file as Proof Negation 4.
. . . .
CongratulationsExercises
6.7
➶
If you skipped any of the You try it sections, go back and do them now. Submit the files Proof Negation 1, Proof Negation 2, Proof Negation 3, and Proof Negation 4.
Section 6.3
162 /Formal Proofs and Boolean Logic
6.8
➶
(Substitution) In informal proofs, we allow you to substitute logically equivalent sentences for one another, even when they occur in the context of a larger sentence. For example, the following inference results from two uses of double negation, each applied to a part of the whole sentence:
P ∧ (Q ∨ ¬¬R)
¬¬P ∧ (Q ∨ R)
How would we prove this using F, which has no substitution rule? Open the file Exercise 6.8, which contains an incomplete formal proof of this argument. As it stands, none of the proof’s steps check out, because no rules or support steps have been cited. Provide the missing justi-fications and submit the completed proof.
Evaluate each of the following arguments. If the argument is valid, use Fitch to give a formal proof using the rules you have learned. If it not valid, use Tarski’s World to construct a counterexample world. In the last two proofs you will need to use Ana Con to show that certain atomic sentences contradict one another to introduce ⊥. Use Ana Con only in this way. That is, your use of Ana Con should cite exactly two atomic sentences in support of an introduction of ⊥. If you have difficulty with any of these exercises, you may want to skip ahead and read Section 6.5.
6.9
➶ SameRow(b, f) ∨ SameRow(c, f)
∨ SameRow(d, f)
¬SameRow(c, f) FrontOf(b, f)
¬(SameRow(d, f) ∧ Cube(f))
¬Cube(f)