Section 2.5
Demonstrating nonconsequence
Proofs come in a variety of different forms. When a mathematician proves a theorem, or when a prosecutor proves a defendant’s guilt, they are show-ing that a particular claim follows from certain accepted information, the
information they take as given. This kind of proof is what we call a proof of proofs of consequence consequence, a proof that a particular piece of information must be true if the
given information, the premises of the argument, are correct.
A very different, but equally important kind of proof is a proof of nonconse- proofs of nonconsequence quence. When a defense attorney shows that the crime might have been
com-mitted by someone other than the client, say by the butler, the attorney is trying to prove that the client’s guilt does not follow from the evidence in the case. When mathematicians show that the parallel postulate is not a conse-quence of the other axioms of Euclidean geometry, they are doing the same thing: they are showing that it would be possible for the claim in question (the parallel postulate) to be false, even if the other information (the remaining axioms) is true.
We have introduced a few methods for demonstrating the validity of an argument, for showing that its conclusion is a consequence of its premises. We will be returning to this topic repeatedly in the chapters that follow, adding new tools for demonstrating consequence as we add new expressions to our language. In this section, we discuss the most important method for demon-strating nonconsequence, that is, for showing that some purported conclusion is not a consequence of the premises provided in the argument.
Recall that logical consequence was defined in terms of the validity of arguments. An argument is valid if every possible circumstance that makes the premises of the argument true also makes the conclusion true. Put the other way around, the argument is invalid if there is some circumstance that makes the premises true but the conclusion false. Finding such a circumstance is the key to demonstrating nonconsequence.
To show that a sentence Q is not a consequence of premises P1, . . . , Pn, we must show that the argument with premises P1, . . . , Pn and conclusion Q is invalid. This requires us to demonstrate that it is possible for P1, . . . , Pn to be true while Q is simultaneously false. That is, we must show that there is a possible situation or circumstance in which the premises are all true while
the conclusion is false. Such a circumstance is said to be a counterexample to counterexamples the argument.
Informal proofs of nonconsequence can resort to many ingenious ways for
Section 2.5
64 / The Logic of Atomic Sentences
showing the existence of a counterexample. We might simply describe what is informal proofs of
nonconsequence clearly a possible situation, one that makes the premises true and the conclu-sion false. This is the technique used by defense attorneys, who hope to create a reasonable doubt that their client is guilty (the prosecutor’s conclusion) in spite of the evidence in the case (the prosecution’s premises). We might draw a picture of such a situation or build a model out of Lego blocks or clay.
We might act out a situation. Anything that clearly shows the existence of a counterexample is fair game.
Recall the following argument from an earlier exercise.
Al Gore is a politician.
Hardly any politicians are honest.
Al Gore is dishonest.
If the premises of this argument are true, then the conclusion is likely. But still the argument is not valid: the conclusion is not a logical consequence of the premises. How can we see this? Well, imagine a situation where there are 10,000 politicians, and that Al Gore is the only honest one of the lot. In such circumstances both premises would be true but the conclusion would be false.
Such a situation is a counterexample to the argument; it demonstrates that the argument is invalid.
What we have just given is an informal proof of nonconsequence. Are there such things as formal proofs of nonconsequence, similar to the formal proofs of validity constructed in F? In general, no. But we will define the notion of a formal proof of nonconsequence for the blocks language used in Tarski’s World. These formal proofs of nonconsequence are simply stylized counterparts of informal counterexamples.
For the blocks language, we will say that a formal proof that Q is not a formal proofs of
nonconsequence consequence of P1, . . . , Pn consists of a sentence file with P1, . . . , Pn labeled as premises, Q labeled as conclusion, and a world file that makes each of P1, . . . , Pntrue and Q false. The world depicted in the world file will be called the counterexample to the argument in the sentence file.
You try it
. . . .
I 1. Launch Tarski’s World and open the sentence file Bill’s Argument. This argument claims that Between(b, a, d) follows from these three premises:
Between(b, c, d), Between(a, b, d), and Left(a, c). Do you think it does?
I 2. Start a new world and put four blocks, labeled a, b, c, and d on one row of the grid.
Demonstrating nonconsequence/ 65
J 3. Arrange the blocks so that the conclusion is false. Check the premises. If
any of them are false, rearrange the blocks until they are all true. Is the conclusion still false? If not, keep trying.
J 4. If you have trouble, try putting them in the order d, a, b, c. Now you will
find that all the premises are true but the conclusion is false. This world is a counterexample to the argument. Thus we have demonstrated that the conclusion does not follow from the premises.
J 5. Save your counterexample as World Counterexample 1.
. . . .
CongratulationsRemember
To demonstrate the invalidity of an argument with premises P1, . . . , Pn
and conclusion Q, find a counterexample: a possible circumstance that makes P1, . . . , Pnall true but Q false. Such a counterexample shows that Q is not a consequence of P1, . . . , Pn.
Exercises
2.21
➶
If you have skipped the You try it section, go back and do it now. Submit the world file World Counterexample 1.
2.22
✎
Is the following argument valid? Sound? If it is valid, give an informal proof of it. If it is not valid, give an informal counterexample to it.
All computer scientists are rich. Anyone who knows how to program a computer is a computer scientist. Bill Gates is rich. Therefore, Bill Gates knows how to program a computer.
2.23
✎
Is the following argument valid? Sound? If it is valid, give an informal proof of it. If it is not valid, give an informal counterexample to it.
Philosophers have the intelligence needed to be computer scientists. Anyone who be-comes a computer scientist will eventually become wealthy. Anyone with the intelli-gence needed to be a computer scientist will become one. Therefore, every philosopher will become wealthy.
Section 2.5
66 / The Logic of Atomic Sentences
Each of the following problems presents a formal argument in the blocks language. If the argument is valid, submit a proof of it using Fitch. (You will find Exercise files for each of these in the usual place.) Important: if you use Ana Con in your proof, cite at most two sentences in each application. If the argument is not valid, submit a counterexample world using Tarski’s World.
2.24
➶
Larger(b, c) Smaller(b, d) SameSize(d, e) Larger(e, c)
2.25
➶
FrontOf(a, b) LeftOf(a, c) SameCol(a, b) FrontOf(c, b)
2.26
➶ SameRow(b, c)
SameRow(a, d) SameRow(d, f) LeftOf(a, b) LeftOf(f, c)
2.27
➶ SameRow(b, c)
SameRow(a, d) SameRow(d, f) FrontOf(a, b) FrontOf(f, c)
Section 2.6
Alternative notation
You will often see arguments presented in the following way, rather than in Fitch format. The symbol .·. (read “therefore”) is used to indicate the conclusion:
All men are mortal.
Socrates is a man.
.·. Socrates is mortal.
There is a huge variety of formal deductive systems, each with its own notation. We can’t possibly cover all of these alternatives, though we describe one, the resolution method, in Chapter 17.