In the following two exercises, determine whether the sentences are consistent. If they are, use Tarski’s World to build a world where the sentences are both true. If they are inconsistent, use Fitch to give a proof that they are inconsistent (that is, derive ⊥ from them). You may use Ana Con in your proof, but only applied to literals (that is, atomic sentences or negations of atomic sentences).
6.15
➶
¬(Larger(a, b) ∧ Larger(b, a))
¬SameSize(a, b)
6.16
➶
Smaller(a, b) ∨ Smaller(b, a) SameSize(a, b)
Section 6.4
The proper use of subproofs
Subproofs are the characteristic feature of Fitch-style deductive systems. It is important that you understand how to use them properly, since if you are not careful, you may “prove” things that don’t follow from your premises. For example, the following formal proof looks like it is constructed according to our rules, but it purports to prove that A ∧ B follows from (B ∧ A) ∨ (A ∧ C), which is clearly not right.
1. (B ∧ A) ∨ (A ∧ C) 2. B ∧ A
3. B ∧ Elim: 2
4. A ∧ Elim: 2
5. A ∧ C
6. A ∧ Elim: 5
7. A ∨ Elim: 1, 2–4, 5–6
8. A ∧ B ∧ Intro: 7, 3
The problem with this proof is step 8. In this step we have used step 3, a step that occurs within an earlier subproof. But it turns out that this sort of justification—one that reaches back inside a subproof that has already ended—is not legitimate. To understand why it’s not legitimate, we need to think about what function subproofs play in a piece of reasoning.
A subproof typically looks something like this:
Section 6.4
164 /Formal Proofs and Boolean Logic
P ... Q
R ... S T...
Subproofs begin with the introduction of a new assumption, in this exam-ple R. The reasoning within the subproof depends on this new assumption, discharging
assumptions by ending subproofs
together with any other premises or assumptions of the parent proof. So in our example, the derivation of S may depend on both P and R. When the subproof ends, indicated by the end of the vertical line that ties the subproof together, the subsequent reasoning can no longer use the subproof’s assump-tion, or anything that depends on it. We say that the assumption has been discharged or that the subproof has been ended.
When an assumption has been discharged, the individual steps of its sub-proof are no longer accessible. It is only the subsub-proof as a whole that can be cited as justification for some later step. What this means is that in justifying the assertion of T in our example, we could cite P, Q, and the subproof as a whole, but we could not cite individual items in the subproof like R or S. For these steps rely on assumptions we no longer have at our disposal. Once the subproof has been ended, they are no longer accessible.
This, of course, is where we went wrong in step 8 of the fallacious proof given earlier. We cited a step in a subproof that had been ended, namely, step 3. But the sentence at that step, B, had been proven on the basis of the assumption B ∧ A, an assumption we only made temporarily. The assumption is no longer in force at step 8, and so cannot be used at that point.
This injunction does not prevent you from citing, from within a subproof, items that occur earlier outside the subproof, as long as they do not occur in subproofs that ended before that step. For example, in the schematic proof given above, the justification for S could well include the step that contains Q.
This observation becomes more pointed when you are working in a sub-proof of a subsub-proof. We have not yet seen any examples where we needed to have subproofs within subproofs, but such examples are easy to come by. Here is one, which is a proof of one direction of the first DeMorgan law.
The proper use of subproofs/ 165
1. ¬(P ∧ Q) 2. ¬(¬P ∨ ¬Q)
3. ¬P
4. ¬P ∨ ¬Q ∨ Intro: 3
5. ⊥ ⊥ Intro: 4, 2
6. ¬¬P ¬ Intro: 3–5
7. P ¬ Elim: 6
8. ¬Q
9. ¬P ∨ ¬Q ∨ Intro: 8
10. ⊥ ⊥ Intro: 9, 2
11. ¬¬Q ¬ Intro: 8–10
12. Q ¬ Elim: 11
13. P ∧ Q ∧ Intro: 7, 12
14. ¬(P ∧ Q) Reit: 1
15. ⊥ ⊥ Intro: 13, 14
16. ¬¬(¬P ∨ ¬Q) ¬ Intro: 2–15
17. ¬P ∨ ¬Q ¬ Elim: 16
Notice that the subproof 2–15 contains two subproofs, 3–5 and 8–10. In step 5 of subproof 3–5, we cite step 2 from the parent subproof 2–15. Similarly, in step 10 of the subproof 8–10, we cite step 2. This is legitimate since the subproof 2–15 has not been ended by step 10. While we did not need to in this proof, we could in fact have cited step 1 in either of the sub-subproofs.
Another thing to note about this proof is the use of the Reiteration rule at step 14. We did not need to use Reiteration here, but did so just to illustrate a point. When it comes to subproofs, Reiteration is like any other rule: when you use it, you can cite steps outside of the immediate subproof, if the proofs that contain the cited steps have not yet ended. But you cannot cite a step inside a subproof that has already ended. For example, if we replaced the justification for step 15 with “Reit: 10,” then our proof would no longer be correct.
As you’ll see, most proofs in F require subproofs inside subproofs—what
we call nested subproofs. To create such a subproof in Fitch, you just choose nested subproofs New Subproof from the Proof menu while you’re inside the first subproof.
You may already have done this by accident in constructing earlier proofs. In the exercises that follow, you’ll have to do it on purpose.
Section 6.4
166 /Formal Proofs and Boolean Logic
Remember
◦ In justifying a step of a subproof, you may cite any earlier step con-tained in the main proof, or in any subproof whose assumption is still in force. You may never cite individual steps inside a subproof that has already ended.
◦ Fitch enforces this automatically by not permitting the citation of individual steps inside subproofs that have ended.
Exercises
6.17
✎
Try to recreate the following “proof” using Fitch.
1. (Tet(a) ∧ Large(c)) ∨ (Tet(a) ∧ Dodec(b)) 2. Tet(a) ∧ Large(c)
3. Tet(a) ∧ Elim: 2
4. Tet(a) ∧ Dodec(b)
5. Dodec(b) ∧ Elim: 4
6. Tet(a) ∧ Elim: 4
7. Tet(a) ∨ Elim: 1, 2–3, 4–6
8. Tet(a) ∧ Dodec(b) ∧ Intro: 7, 5
What step won’t Fitch let you perform? Why? Is the conclusion a consequence of the premise?
Discuss this example in the form of a clear English paragraph, and turn your paragraph in to your instructor.
Use Fitch to give formal proofs for the following arguments. You will need to use subproofs within subproofs to prove these.
6.18
➶ A ∨ B
A ∨ ¬¬B
6.19
➶ A ∨ B
¬B ∨ C A ∨ C
6.20
➶ A ∨ B
A ∨ C A ∨ (B ∧ C)