58 /The Logic of Atomic Sentences
Section 2.4
Constructing proofs in Fitch/ 59
a new step to the proof by choosingAdd Step After from the Proof menu. In the new step, enter the sentencea=b, either by typing it in or by using the toolbar at the top of the proof window. We will first use this step to get our conclusion and then go back and prove this step.
4. Once you have entered a=b, add another step below this and enter the
goal sentenceSameRow(b,a). Use the mouse to click on the word Rule?
that appears to the right ofSameRow(b,a). In the menu that pops up, go to the Elimination Rules and select=. If you did this right, the rule name should now say = Elim. If not, try again.
5. Next cite the first premise and the intermediate sentence you first entered.
You do this in Fitch by clicking on the two sentences, in either order. If you click on the wrong one, just click again and it will be un-cited. Once you have the right sentences cited, chooseVerify Prooffrom theProof menu. The last step should now check out, as it is a valid instance of= Elim. The step containinga=bwill not check out, since we haven’t yet indicated what it follows from. Nor will the goal check out, since we don’t yet have a complete proof ofSameRow(b,a). All in good time.
6. Now add a step before the first introduced step (the one containinga=b),
and enter the sentence b=b. Do this by moving the focus slider (the triangle in the left margin) to the step containing a=b and choosing Add Step Before from the Proof menu. (If the new step appears in the wrong place, chooseDelete Step from the Proof menu.) Enter the sentenceb=band justify it by using the rule= Intro. Check the step.
7. Finally, justify the step containing a=bby using the= Elimrule. You
will need to move the focus slider to this step, and then cite the second premise and the sentenceb=b. Now the whole proof, including the goal, should check out. To find out if it does, choose Verify Proof from the Proofmenu. The proof should look like the completed proof on page 57, except for the absence of numbers on the steps. (Try out Show Step Numbersfrom theProofmenu now. The highlighting on support steps will go away and numbers will appear, just like in the book.)
8. We mentioned earlier that Fitch lets you take some shortcuts, allowing
you to do things in one step that would take several if we adhered strictly toF. This proof is a case in point. We have constructed a proof that falls underF but Fitch actually has symmetry of identity built into= Elim.
So we could prove the conclusion directly from the two premises, using a single application of the rule= Elim. We’ll do this next.
Section 2.4
60 /The Logic of Atomic Sentences
9. Add another step at the very end of your proof. Here’s a trick you will find handy: Click on the goal sentence at the very bottom of the window. This puts the focus on the goal sentence. ChooseCopy from the Edit menu, and then click back on the empty step at the end of your proof. Choose Pastefrom theEditmenu and the goal sentence will be entered into this step. This time, justify the new step using= Elimand citing just the two premises. You will see that the step checks out.
10. Save your proof asProof Identity 1.
. . . .
Congratulations Since the proof systemF does not have any rules for atomic predicates other than identity, neither does Fitch. However, Fitch does have a mecha-nism that, among other things, lets you check for consequences among atomic sentences that involve many of the predicates in the blocks world language.1 This is a rule we call Analytic ConsequenceorAna Confor short.Ana Analytic ConsequenceCon is not restricted to atomic sentences, but that is the only application of the rule we will discuss at the moment. This rule allows you to cite some sentences in support of a claim if any world that makes the cited sentences true also makes the conclusion true, given the meaning of the predicates as used in Tarski’s World. Let’s get a feeling forAna Conwith some examples.
You try it . . . .
1. Use Fitch to open the fileAna Con 1. In this file you will find nine premises followed by six conclusions that are consequences of these premises. Indeed, each of the conclusions follows from three or fewer of the premises.
2. Position the focus slider (the little triangle) at the first conclusion following the Fitch bar,SameShape(c,b). We have invoked the ruleAna Conbut we have not cited any sentences. This conclusion follows fromCube(b) and Cube(c). Cite these sentences and check the step.
3. Now move the focus slider to the step containing SameRow(b,a). Since the relation of being in the same row is symmetric and transitive, this follows from SameRow(b,c) and SameRow(a,c). Cite these two sentences and check the step.
1This mechanism does not handle the predicatesAdjoinsandBetween, due to the com-plexity of the ways the meanings of these predicates interact with the others.
Constructing proofs in Fitch/ 61
4. The third conclusion,BackOf(e,c), follows from three of the premises. See
if you can find them. Cite them. If you get it wrong, Fitch will give you anXwhen you try to check the step.
5. Now fill in the citations needed to make the fourth and fifth conclusions
check out. For these, you will have to invoke theAna Con rule yourself.
(You will find the rule on theConsubmenu of theRule? popup.)
6. The final conclusion,SameCol(b,b), does not require that any premises be
cited in support. It is simply an analytic truth, that is, true in virtue of its meaning. Specify the rule and check this step.
7. When you are done, choose Verify Proof to see that all the goals check
out. Save your work as Proof Ana Con 1.
. . . .
CongratulationsTheAna Conmechanism is not really a rule, technically speaking, though rules vs.Con mechanisms we will continue to call it that since it appears on the Rule?menu in Fitch.
This mechanism, along with the two others appearing on the Consubmenu, apply complicated procedures to see whether the sentence in question follows from the cited sentences. As we will explain later, these three items try to find proofs of the sentence in question “behind the scenes,” and then give you a checkmark if they succeed. The proof they find may in fact apply many, many different rules in getting from the cited steps to the target sentence.
The main difference you will run into between the genuine rules in Fitch and the mechanisms appearing on the Con menu is that the latter “rules”
will sometimes fail even though your step is actually correct. With the genuine rules, Fitch will always give your step either a checkmark or anX, depending on whether the rule is applied correctly. But with theConmechanisms, Fitch will sometimes try to find a proof of the target sentence but fail. In these cases, Fitch will give the step a question mark rather than a check or anX, since there might be a complicated proof that it just couldn’t find.
To mark the difference between the genuine rules ofF and the three con-sequence mechanisms, Fitch displays the rule names in green and the conse-quence mechanisms in blue. Because the Con mechanisms look for a proof behind the scenes, we will often ask you not to use them in giving solutions to homework problems. After all, the point is not to have Fitch do your home-work for you! In the following problems, you should only use theAna Con rule if we explicitly say you can. To see whether a problem allows you to use any of the Con mechanisms, double click on the goal or choose See Goal Constraintsfrom theGoalmenu.
Section 2.4
62 /The Logic of Atomic Sentences
Remember
The deductive system you will be learning is a Fitch-style deductive sys-tem, namedF. The computer application that assists you in constructing proofs inFis therefore called Fitch. If you write out your proofs on paper, you are using the systemF, but not the program Fitch.
Exercises
2.15
➶
If you skipped the You try it sections, go back and do them now. Submit the files Proof Identity 1andProof Ana Con 1.
2.16
➶
Use Fitch to give a formal version of the informal proof you gave in Exercise 2.5. Remember, you will find the problem setup in the fileExercise 2.16. You should begin your proof from this saved file. Save your completed proof asProof 2.16.
In the following exercises, use Fitch to construct a formal proof that the conclusion is a consequence of the premises. Remember, begin your proof by opening the corresponding file, Exercise 2.x, and save your solution as Proof 2.x. We’re going to stop reminding you.
2.17
➶ SameCol(a,b) b=c
c=d
SameCol(a,d)
2.18
➶ Between(a,d,b) a=c
e=b
Between(c,d,e)
2.19
➶ Smaller(a,b) Smaller(b,c) Smaller(a,c)
You will need to useAna Con in this proof. This proof shows that the pred-icate Smaller in the blocks language is transitive.
2.20
➶ RightOf(b,c) LeftOf(d,e) b=d LeftOf(c,e)
Make your proof parallel the informal proof we gave on page 52, using both an identity rule and Ana Con (where necessary).