Validity and arguments
6.3 Multi-step arguments again
slavishly follow all our pre-theoretical intuitions, especially our intuitions about unusual cases. The aim of an official definition of validity is to give a tidy ‘rational reconstruction’ of our ordinary notion which smoothly cap- tures the uncontentious cases; the classical definition does that particularly neatly. So we should bite the bullet and accept the mild oddity of the verdict on C.
Likewise there are two responses to the observation that the classical definition warrants argument D. We could either say this just goes to show that our defini- tion allows more gross fallacies of irrelevance. Or we could bite the bullet again and deem this result to be a harmless oddity (we already need to know that a conclusion C is necessarily true before we are in a position to say that it is classi- cally entailed by arbitrary premisses – and if we already know that C is neces- sary, then extra valid arguments for it like D can’t extend our knowledge).
The majority of modern logicians opt for the second responses. They hold that the cost of trying to construct a plausible notion of ‘relevant’ inference turns out to be too high in terms of complexity to make it worth abandoning the neat sim- plicity of the idea of classical validity. But there is a significant minority who insist that we really should be going back to the drawing board, and hold that there are well-motivated and workable systems of ‘relevant logic’. We won’t be able to explore their arguments and their alternative logical systems here – and in any case, these variant logics can really only be understood and appreciated in contrast to classical systems. We just have to note frankly that the classical defi- nition of validity is not beyond challenge.
Still, our basic aim is to model certain forms of good reasoning; and models can be highly revealing even if some of their core constructs idealize. The idea of a ‘classically valid argument’ – like e.g. the idea of a ‘ideal gas’ – turns out at least to be a highly useful and illuminating idealization: and this can be so even if it would be wrong to claim that it is in the end the uniquely best and most accu- rate idealization. Anyway, we are going to stick to the classical definition in our introductory discussions in this book.
6.3 Multi-step arguments again
We characterized deductive validity as, in the first place, a property of individual
inference steps. We then fell in with the habit of calling a one-step argument
valid if it involves a valid inference step from initial premisses to final conclu- sion. In this section, we’ll see the significance of that qualification ‘one-step’.
Consider the following mini-argument:
F (1) All philosophers are logicians. So (2) All logicians are philosophers.
The inference is obviously fallacious. It just doesn’t follow from the assumption that all philosophers are logicians that only philosophers are logicians. (You might as well argue ‘All women are human beings, hence all human beings are women’.) Here’s another really bad inference:
48 Validity and arguments
G (1) All existentialists are philosophers. (2) All logicians are philosophers. So (3) Hence, all existentialists are logicians.
It plainly doesn’t follow from the claims that the existentialists and logicians are both among the philosophers that any of the existentialists are logicians, let alone that all of them are. (You might as well argue ‘All women are human beings, all men are human beings, hence all women are men’.)
But now imagine someone who chains this pair of rotten inference steps together into a two-step argument as follows. He starts from the premisses
H (1) All existentialists are philosophers. (premiss) (2) All philosophers are logicians. (premiss) He then cheerfully makes the same fallacious inference as in F, and infers that
(3) All logicians are philosophers. (from 2!!) Then he compounds the sin by committing the same inferential howler as in G, and supposes that it follows from H(1) and the newly inferred H(3) that
(4) All existentialists are logicians. (from 1, 3!!) Here our reasoner has got from the initial premisses H(1) and H(2) to his final conclusion H(4) by two quite terrible moves: it would therefore be extremely odd to dignify his supposed ‘proof’ as deductively cogent.
Note, however, that in this case the two howlers by luck happen to cancel each other out; there are no possible circumstances in which the initial premisses H(1) and H(2) are both true and yet the final conclusion H(4) is false. If the existen- tialists are all philosophers, and all philosophers are logicians, then the existen- tialists must of course be logicians. Hence, the inferential jump from initial premisses to final conclusion is in fact valid.
So if we were to say (without any qualification) that an argument is valid if the initial premisses absolutely guarantee the final conclusion, then we’d have to count the two-step argument H as ‘valid’. Which is, to say the least, a very unhappy way of describing the situation, given that H involves a couple of nasty fallacies!
The previous chapter reminds us that many arguments involve more than one inference step. To be sure, we don’t very often set them out as semi-formal proofs; but multi-step arguments are common enough in any form of extended enquiry – indeed our initial informal definition of an argument (§1.1) was as a
chain of reasoning. The point of multi-step chains is to show that you can legiti-
mately infer some (perhaps highly unobvious) conclusion by breaking down the big inferential leap from initial premisses to final conclusion into a series of smaller, more evident steps. But if an argument goes astray along the way, then it will be mere good luck if it happens to get back on track to a conclusion that does genuinely follow via some other route from the starting premisses.
The moral here is a simple one: we need to make a distinction. Suppose we have a step-by-step argument from premisses A1, A2, …, An to conclusion C.
6.3 Multi-step arguments again 49
Then it is one thing to say that the overall, big-leap, inference from A1, A2, …,
An to C is valid. It is another thing to say that the argument we give from A1,
A2, …, An to C is deductively cogent. Of course, in the case where the argument has only one step, then the argument is deductively cogent if and only if the overall inference is valid. But as example H shows, in the case of multi-step arguments, this two-way link breaks down. The inferential leap from the prem- isses to conclusion might still be valid, even if some step-by-step argument we give to defend the inference is not deductively cogent.
You might think that we are making much ado about very little here. For doesn’t this just show that we must define a genuine proof – a deductively virtu- ous multi-step argument, where luck doesn’t enter into it – to be an argument
whose individual inferential steps are all valid? (In fact, that was our first-shot
definition in §5.1.)
But things aren’t quite that easy. Not only must individual steps be valid, but they must be chained together in the right kind of way.
To illustrate, consider this one-step inference:
I (1) Socrates is a philosopher.
(2) All philosophers have snub noses.
So (3) Socrates is a philosopher and all philosophers have snub noses. That’s trivially valid (if you are given A and B, you can infer A-and-B). Here is another equally trivial but valid inference:
J (1) Socrates is a philosopher and all philosophers have snub noses. So (2) All philosophers have snub noses.
From A-and-B, you can infer B. And thirdly, this too is plainly valid:
K (1) Socrates is a philosopher.
(2) All philosophers have snub noses. So (3) Socrates has a snub nose.
Taken separately, then, those three little inferences are quite unproblematic. But now imagine someone chains them together, not in the usual way (where each step depends on something that has gone before) but in the following tangle:
L (1) Socrates is a philosopher. (premiss)
(2) Socrates is a philosopher and (from 1, 3: as in I) all philosophers have snub noses.
(3) All philosophers have snub noses. (from 2: as in J) So (4) Socrates has a snub nose. (from 1, 3: as in K) By separately valid steps we seem to have deduced the shape of Socrates’ nose just from the premiss that he is a philosopher! What has gone wrong?
The answer is obvious enough. In the middle of the argument we have gone round in a circle. L(2) is derived from L(3), and then L(3) is derived from L(2). Circular arguments can’t take us anywhere.
So that suggests we need an improved account of what makes for a deduc- tively cogent proof: the individual inferential steps must all be valid, and the
50 Validity and arguments
steps must be chained together in a non-circular way, with steps only depending on what’s gone before in the argument.
That’s an improvement, but more still needs to be said. For recall, we need to allow for multi-step arguments like the ‘reductio’ arguments we met in §5.4 – i.e. arguments where we can introduce a new temporary supposition that doesn’t follow from what’s gone before (and then later ‘discharge’ the assumption by an inference step that relies, not on prior propositions, but on a prior ‘sub-proof’). Then there will be more complications when we introduce ‘downward-branch- ing tree proofs’ (see §20.1). Recognizing such various argument-styles as all being in good order complicates the task of framing a definition of what makes for a deductively cogent proof in general. Fortunately, for our purposes in this book, we don’t need to come up with a general story here: so we won’t attempt to do so.
6.4 Summary
• Our classical definition of validity counts as valid any inference which has inconsistent premisses, or has a necessarily true conclusion. Despite being counter-intuitive, these are upshots we will learn to live with.
• It is unhappy to say that a multi-step argument is ‘valid’ just if there is no way the initial premisses can be true and the conclusion false. For we don’t want to endorse arguments where there are inferential steps which are invalid but where it happens that the fallacies cancel each other out.
What has been covered so far?
• We have explained, at least in an introductory way, the notion of a classi- cally valid inference-step.
• We have explained the corresponding notion of a deductively valid one- step argument.
• We have distinguished deductive validity from other virtues that an argu- ment might have (like being a good inductive argument).
• We have noted how different arguments may share the same pattern of inference. And we exploited this fact when we developed the counterexample technique for demonstrating invalidity.
• We have seen some simple examples of direct and indirect multi-step proofs, where we show that a conclusion really can be validly inferred from certain premisses by filling in the gap between premisses and conclusion with evi- dently valid intermediate inference steps.
What haven’t we done so far?
• We still haven’t given a very sharp characterization of the notion of deduc- tive validity (we invoked an intuitive notion of ‘possible situation’ without really spelling out what that came to).
• Nor have we given a really clear story about what counts as a ‘pattern’ of inference.
• Nor have we given a sharp characterization of what counts as a legitimate ‘well-built’ form of multi-step proof.
• Indeed, we even set aside the question of the nature of the basic constitu- ents of informal arguments, namely propositions (so in a sense, we don’t really know the first thing about valid arguments, i.e. what they are made of).
At this point, then, we could go in a number of directions. We could remain content with an informal level of discussion, and first explore more techniques