The counterexample technique
5.4 Reductio arguments
Plainly the premiss that whatever is ever-moving is immortal doesn’t by itself logically entail the desired conclusion. We need some additional assumption. The intended extra assumption in this case is obvious: Plato is assuming that every soul is ever-moving (whatever exactly that means!).
Another quick example. Consider the argument
E The constants of nature have to take values in an extremely narrow range (have to be ‘fine-tuned’) to permit the evolution of intelligent life. So the universe was intelligently designed.
For this to work as a deductively valid inference, we presumably need to add two more premisses. One is uncontroversial: intelligent life has evolved. The other is much more problematic: intelligent design is needed in order for the uni- verse to be fine-tuned for the evolution of intelligent life. Only with some such additions can we get a cogent argument (exercise: construct a fully annotated argument with versions of these additional premisses).
Arguments like D and E with missing premisses left to be understood (that is, with ‘suppressed premisses’) are traditionally called enthymemes. And the exer- cise of trying to regiment informal reasoning into semi-formal annotated proofs can often help us in spotting an enthymeme and finding a suitable repair. Note too that the same exercise can also help identify redundancies, when it turns out that a premiss given at the outset needn’t actually be used in getting to the conclusion.
5.4 Reductio arguments
Suppose someone couldn’t see why in the ‘crocodile argument’ A′ it was correct to infer that babies cannot manage a crocodile (given that nobody is despised who can manage a crocodile, and babies are despised). How might we convince them? We might amplify that step, as follows:
F (1) Nobody is despised who can wrestle a crocodile. (premiss)
(2) Babies are despised. (premiss)
Suppose temporarily, for the sake of argument,
(3) Babies can manage a crocodile. (supposition)
(4) Babies are not despised. (from 1, 3)
(5) Contradiction! (from 2, 4)
So the supposition that leads to this absurdity must be wrong.
(6) Babies cannot manage a crocodile (RAA)
Here we are arguing ‘indirectly’. We want to establish (6). But instead of aiming directly for the conclusion, we branch off by temporarily supposing the exact opposite is true, i.e. we suppose (3). This supposition is very quickly shown to lead to something that contradicts an earlier claim. Hence the supposition (3) is ‘reduced to absurdity’: its opposite must therefore be true after all. The label ‘(RAA)’ indicates that the argument terminates with a reductio ad absurdum inference.
40 Proofs
Another quick example. We’ll demonstrate the validity of the inference ‘No girl loves any sexist pig; Caroline is a girl who loves whoever loves her; Henry loves Caroline; hence Henry is not a sexist pig’.
G (1) No girl loves any sexist pig. (premiss) (2) Caroline is a girl who loves whoever loves her. (premiss)
(3) Henry loves Caroline. (premiss)
(4) Caroline is a girl who loves Henry. (from 2, 3)
(5) Caroline is a girl. (from 4)
(6) Caroline loves Henry. (from 4)
Suppose temporarily, for the sake of argument,
(7) Henry is a sexist pig. (supposition)
(8) No girl loves Henry. (from 1, 7)
(9) Caroline does not love Henry. (from 5, 8)
(10) Contradiction! (from 6, 9)
So the supposition that leads to this absurdity must be wrong.
(11) Henry is not a sexist pig (RAA)
And we are done.
Let’s say – at least for now (compare §12.1) – that a contradiction is a pair of propositions, C together with its exact opposite not-C (or what comes to the same thing, a single proposition of the type C and not-C). The principle under- lying the last inference move in F and G can then be summarized as follows:
Reductio ad absurdum If the propositions A1, A2, …, An plus the tem- porary supposition S logically entail a contradiction then, keeping A1, A2, …, An as premisses, we can validly infer that not-S.
Why does this principle hold? Suppose that a bunch of premisses plus S do entail a contradiction. Then there can be no situation in which those other premisses plus S are all true together (or else this would be a situation in which the entailed contradiction would also have to be true, and it can’t be). So any situation in which the other premisses are all true is one in which S has to be false. Hence those premisses logically entail it isn’t the case that S, i.e. they entail not-S.
It is worth noting that the reductio principle continues to apply in the special case where the number of original premisses is zero. In other words, if the sup- position S by itself leads to a contradiction, then it can’t be true.
As an illustration of the last point, here’s a third – and rather more interesting – example of an RAA argument. We’ll prove that the square root of 2 is an irra- tional number, i.e. √2 is not a fraction mn, where m, n are integers.
If√2 is a fraction mn, then we can put it in lowest terms (that is to say, divide out any common factors from m and n). So we’ll start by make the following ini- tial assumption:
H Suppose temporarily, for the sake of argument,
(1) √2 = mn where m and n are integers with no common factor.
5.5 Limitations 41
We now show that this leads to contradiction, so it can’t be true:
(2) 2 = m2n2 (squaring 1)
(3) m2= 2n2 (rearranging 2)
(4) m2 is even. (from 3)
(5) m is even. (from 4)
That last move depends on the easily checked point that odd integers always have odd squares: so if m2 is even, m cannot be odd. Continuing:
(6) m= 2r, for some integer r. (from 5)
(7) (2r)2= 2n2 (from 3, 6)
(8) n2= 2r2 (rearranging 7)
(9) n2 is even. (from 8)
(10) n is even. (from 9)
(11) m and n are both even. (from 5, 10)
(12) m and n have a common factor (i.e. 2). (from 11)
(13) Contradiction! (from 1, 12)
So the supposition (1) that leads to (13) must be wrong.
(14) √2 ≠ mn where m and n are integers with (RAA) no common factor.
Despite the relative simplicity of the argument, the result is deep. (It shows that there can be ‘incommensurable’ quantities, i.e. quantities which lack a common measure. Take a right-angled isosceles triangle. Then there can’t be a unit length that goes into one of the equal sides exactly n times and into the hypotenuse exactly m times.)
Finally, two quick remarks. The first concerns the way we have displayed these arguments involving RAA. When we are giving proofs where temporary suppositions are made en route, we need to have some way of clearly indicating which steps of the argument are being made while that temporary supposition is in play. A fairly standard way of doing this is to indent the argument to the right while the supposition is in play; and then to go back left when the supposition is ‘discharged’ and is finished with – we’ve also used vertical lines to help signal the indentation. That gives us a neat visual display of the argument’s structure.
Second, allowing suppositions to be made temporarily complicates the range of permissible structures for arguments. However, well-formed proofs will still march steadily on in the sense that each new step of the proof that isn’t a new assump-
tion will depend on what has gone before. It is just that now what goes before
can not only be previous propositions but also whole indented subproofs. For example, the conclusion in G is derived from the subproof from lines (7) through to (10); the conclusion in H is derived from the subproof from (1) to (12).
5.5 Limitations
The styles of direct and indirect proof we’ve been illustrating can be used infor- mally and can also be developed into something much more rigorous. However, there is a certain limitation to the technique of trying to warrant inferences by
42 Proofs
finding step-by-step proofs. If we are lucky/clever, and spot a proof that takes us from the given premisses to the target conclusion, that will show that a putative inference is indeed valid. But we may be unlucky/dim, and not spot the proof. In the general case, even if a proof exists, proof-discovery requires ingenuity, and failing to find a proof doesn’t show that the original inference from premisses to conclusion is invalid. (Compare the counterexample technique: if we are lucky/ clever, deploying that technique will show us that an invalid inference is invalid. But we may miss the counterexample.)
It would be nice to have less hit-and-miss techniques for establishing validity and invalidity. Ideally, we would like a mechanical general technique for work- ing out whether an inference is or is not valid that doesn’t need any imaginative searching for proofs or counterexamples. Can we ever (as it were) calculate the validity or invalidity of inferences in as mechanical a way as we calculate how much five pints of beer will cost, given the price per pint?
Profound results (mostly discovered in the 1930s) show that, in general, a mechanically applicable calculus will not in fact be available. But in a limited class of cases, we can pull off the trick. In the next part of this book, starting in
Chapter 7, we will be exhibiting one such calculus.
5.6 Summary
• To establish validity we can use many-step deductions, i.e. proofs, which chain together little inference steps that are obviously valid and that lead from the initial premisses to the desired conclusion.
• Some common methods of proof, however, are ‘indirect’, like reductio ad absurdum.
• In the general case, even if a proof exists, proof-discovery requires ingenuity.
Exercises 5
Which of the following arguments are valid? Where an argument is valid, pro- vide a proof. Some of the examples are enthymemes that need repair.
1. No philosopher is illogical. Jones keeps making argumentative blunders. No logical person keeps making argumentative blunders. All existentialists are philosophers. So, Jones is not an existentialist.
2. Jane has a first cousin. Jane’s father had no siblings. So, if Jane’s mother had no sisters, she had a brother.
3. Every event is causally determined. No action should be punished if the agent isn’t responsible for it. Agents are only responsible for actions they can avoid doing. Hence no action should be punished.
4. Something is an elementary particle only if it has no parts. Nothing which has no parts can disintegrate. An object that cannot be destroyed must con- tinue to exist. So an elementary particle cannot cease to exist.
Exercises 5 43
competent person is always blundering. So, Jenkins is inexperienced. 6. Only logicians are good philosophers. No existentialists are logicians. Some
existentialists are French philosophers. So, some French philosophers are not good philosophers.
7. Either the butler or the cook committed the murder. The victim died from poison if the cook did the murder. The butler did the murder only if the vic- tim was stabbed. The victim didn’t die from poison. So, the victim was stabbed.
8. Promise-breakers are untrustworthy. Beer-drinkers are very communicative. A man who keeps his promises is honest. No one who doesn’t drink beer runs a bar. One can always trust a very communicative person. So, no one who keeps a bar is dishonest.
9. When I do an example without grumbling, it is one that I can understand. No easy logic example ever makes my head ache. This logic example is not arranged in regular order, like the examples I am used to. I can’t understand these examples that are not arranged in regular order, like the examples I am used to. I never grumble at an example, unless it gives me a headache. So, this logic example is difficult.
Before turning to explore some formal techniques for evaluating arguments, we should pause to say a little more about the ‘classical’ conception of validity for individual inference steps, and also say more about what makes for cogent multi-step arguments.