144 /Formal Proofs and Boolean Logic
3. Next, go to the popup Rule? menu and under the Elimination Rules, choose∧.
4. If you try to check this step, you will see that it fails, because you have have not yet cited any sentences in support of the step. In this example, you need to cite the single premise in support. Do this and then check the step.
5. You should be able to prove each of the other sentences similarly, by means of a single application of∧Elim. When you have proven these sentences, check your goals and save the proof asProof Conjunction 1.
. . . .
CongratulationsConjunction introduction
The corresponding introduction rule, conjunction introduction, allows you to assert a conjunctionP1∧. . .∧Pn provided you have already established each of its constituent conjuncts P1throughPn. We will symbolize this rule in the following way:
Conjunction Introduction (∧ Intro):
P1
⇓ Pn
...
P1∧. . .∧Pn
In this rule, we have used the notation:
P1
⇓ Pn
to indicate that each of P1 through Pn must appear in the proof before you can assert their conjunction. The order in which they appear does not matter, and they do not have to appear one right after another. They just need to appear somewhere earlier in the proof.
Here is a simple example of our two conjunction rules at work together. It is a proof ofC∧BfromA∧B∧C.
Conjunction rules/ 145
1.A∧B∧C
2.B ∧ Elim: 1
3.C ∧ Elim: 1
4.C∧B ∧ Intro: 3, 2
Let’s try our hand using both conjunction rules in Fitch.
You try it . . . .
1. Open the file Conjunction 2. We will help you prove the two sentences
requested in the goals. You will need to use both of the conjunction rules in each case.
2. The first goal is Medium(d)∧ ¬Large(c). Add a new step and enter this
sentence. (Remember that you can copy the sentence from the goal pane and paste it into the new step. It’s faster than typing it in.)
3. Above the step you just created, add two more steps, typing one of the
conjuncts in each. If you can prove these, then the conclusion will follow by ∧ Intro. Show this by choosing this rule at the conjunction step and citing the two conjuncts in support.
4. Now all you need to do is prove each of the conjuncts. This is easily done
using the rule∧Elimat each of these steps. Do this, cite the appropriate support sentences, and check the proof. The first goal should check out.
5. Prove the second goal sentence similarly. Once both goals check out, save
your proof asProof Conjunction 2.
. . . .
CongratulationsDefault and generous uses of the∧ rules
As we said, Fitch is generous in its interpretation of the inference rules ofF. For example, Fitch considers the following to be an acceptable use of∧Elim:
17.Tet(a)∧Tet(b)∧Tet(c)∧Tet(d) ...
26.Tet(d)∧Tet(b) ∧ Elim: 17
Section 6.1
146 /Formal Proofs and Boolean Logic
What we have done here is pick two of the conjuncts from step 17 and assert the conjunction of these in step 26. Technically, F would require us to de-rive the two conjuncts separately and, like Humpty Dumpty, put them back together again. Fitch does this for us.
Since Fitch lets you take any collection of conjuncts in the cited sentence and assert their conjunction in any order, Fitch’s interpretation of ∧ Elim allows you to prove that conjunction is “commutative.” In other words, you can use it to take a conjunction and reorder its conjuncts however you please:
13.Tet(a)∧Tet(b) ...
21.Tet(b)∧Tet(a) ∧ Elim: 13
You try it . . . .
1. Open the fileConjunction 3. Notice that there are two goals. The first goal asks you to proveTet(c)∧Tet(a) from the premise. Strictly speaking, this would take two uses of∧ Elimfollowed by one use of∧ Intro. However, Fitch lets you do this with a single use of∧Elim. Try this and then check the step.
2. Verify that the second goal sentence also follows by a single application of Fitch’s rule of∧Elim. When you have proven these sentences, check your goals and save the proof asProof Conjunction 3.
3. Next try out other sentences to see whether they follow from the given sentence by ∧ Elim. For example, does Tet(c)∧Small(a) follow? Should it?
4. When you are satisfied you understand conjunction elimination, close the file, but don’t save the changes you made in step 3.
. . . .
Congratulations The∧Introrule implemented in Fitch is also less restrictive than our dis-cussion of the formal rule might suggest. First of all, Fitch does not care about the order in which you cite the supporting sentences. Second, if you cite a sen-tence, that sentence can appear more than once as a conjunct in the concluding sentence. For example, you can use this rule to conclude Cube(a)∧Cube(a) from the sentenceCube(a), if you want to for some reason.Conjunction rules/ 147
Both of the conjunction rules have default uses. If at a new step you cite
a conjunction and specify the rule as∧ Elim, then when you check the step default uses of conjunction rules (or chooseCheck Proof), Fitch will fill in the blank step with the leftmost
conjunct in the cited sentence. If you cite several sentences and apply∧Intro, Fitch will fill in the conjunction of those steps, ordering conjuncts in the same order they were cited.
You try it . . . .
1. Open the file Conjunction 4.
2. Move the focus to the first blank step, the one immediately following the
premises. Notice that this step has a rule specified, as well as a support sentence cited. Check the step to see what default Fitch generates.
3. Then, focus on each successive step, try to predict what the default will
be, and check the step. (The last two steps give different results because we entered the support steps in different orders.)
4. When you have checked all the steps, save your proof as Proof
Conjunc-tion 4.
5. Feel free to experiment with the rule defaults some more, to see when they
are useful.
. . . .
Congratulations One final point: In applying conjunction introduction, you will sometimeshave to be careful about parentheses, due to our conventions about dropping parentheses and conjunction rules outermost parentheses. If one of the conjuncts is itself a conjunction, then
of course there is no need to add any parentheses before forming the larger conjunction, unless you want to. For example, the following are both correct applications of the rule. (The first is what Fitch’s default mechanism would give you.)
Correct: 1.A∧B 2.C
3. (A∧B)∧C ∧ Intro: 1, 2
Correct: 1.A∧B 2.C
3.A∧B∧C ∧ Intro: 1, 2
Section 6.1
148 /Formal Proofs and Boolean Logic
However, if one of the conjuncts is a disjunction (or some other complex sentence), to prevent ambiguity you may need to reintroduce the parentheses that you omitted before. Thus the first of the following is a correct proof, but the second contains a faulty application of conjunction introduction, since it concludes with an ambiguous sentence.
Correct: 1.A∨B 2.C
3. (A∨B)∧C ∧ Intro: 1, 2
Wrong: 1.A∨B 2.C
3.A∨B∧C ∧ Intro: 1, 2
Section 6.2