We should, however, pause over one point. Talk of ‘logical form’ is associated with a popular slogan: Valid arguments are valid in virtue of their form. We will conclude this chapter with some deflationary remarks.
We have seen that valid arguments typically exemplify a number of different inferential patterns, some good, some bad. It is therefore more than a bit mis- leading to say baldly that an argument is valid in virtue of its form (as if it has only got one). The idea behind the popular slogan is presumably that for every valid argument there is some relevant inferential ‘form’ which captures what makes it valid.
But that idea is empty if we are allowed to take an argument as the one and only instance of its own entirely specific inference pattern. Of course a valid argument always exemplifies some reliable pattern if you count the argument as a one-off pattern for itself! So, to avoid triviality, the claim had better be some- thing like this: every valid argument belongs to a family of valid arguments cut to the same pattern, a family which also has other members, and the argument is valid in virtue of the inference pattern it shares with the other members of the family.
Well, that’s true of all our examples of valid arguments so far. Is it true in every case? Take for example the following one-step argument:
J (1) Everyone is female. So (2) Any siblings are sisters.
This is valid. Any situation in which the premiss is true – for instance, a plague having killed off all the men – must be a situation without brothers. But what more general but still reliable form does the inference exemplify? Certainly, at the surface level, there is no way of replacing various parts of the argument with
3.7 Summary 27
place-holding letters so that you end up representing a schematic pattern whose instances are all valid. (You might reply: ‘Still, “sister” means female sibling so at the level of propositions expressed the argument is of the valid general form
Everyone is F.
So, anyone who is G is F and G.’
This riposte, however, takes us back to troublesome questions about proposi- tions and their constituents. For a start, what makes it the case that the concept of a sister is to be analysed into the constituents female and sibling, rather than the concept of a sibling being the complex one, to be analysed as brother-or-
sister? Let’s not open up these murky issues!)
As we will see later, e.g. in §13.6, there are good uses for crisply defined tech- nical notions of being ‘valid in virtue of logical form’. These notions will sharpen up the idea of owing validity to the way in which all-purpose, topic- neutral, logical concepts like ‘and’ and ‘not’ or ‘all’ and ‘some’ distribute between the premisses and conclusion. However, these crisp notions are put to work in contexts where we want to contrast arguments which are valid in virtue of ‘logical form’ (in some narrow, technical sense) with other kinds of valid arguments like examples F and J (which depend for their validity on something other than the presence of topic-neutral logical notions like ‘not’ and ‘all’). Hence many valid arguments are not valid-in-virtue-of-form in the technical senses to be later explained.
In fact, it is very unclear whether there is any useful notion of being valid-in- virtue-of-form which is applicable to all valid inferences.
3.7 Summary
• Arguments with valid inference steps are often – but not always – valid in virtue of the way that various key logical concepts (like ‘all’, ‘no’ and ‘most’) are distributed in a patterned way between the premisses and con- clusion.
• Inferential patterns can be conveniently represented using schematic vari- ables, whose use is governed by the obvious convention that a given vari- ables represents the same name or predicate whenever it appears within an argument schema.
• Intuitively, inferential patterns can be shared even when the surface linguis- tic form differs (e.g. it is natural to say that various arguments beginning ‘Every man …’, ‘Any man …’, ‘Each man …’ may yet exemplify the same inferential move).
• There is strictly no such thing as the unique form exemplified by an argu- ment. Valid arguments will typically be instances of reliable forms; but they can also be instances of other, more general, unreliable forms.
• Our main concern henceforth will be with rather general, ‘topic-neutral’, patterns of inference.
28 Patterns of inference
Exercises 3
A Which of the following types of inference step are valid (i.e. are such that all their instances are valid)? If you suspect an inference-type is invalid, find an instance which obviously fails because it has plainly true premisses and a false conclusion.
1. Some F are G; no G is H; so, some F are not H. 2. Some F are G; some F are H; so, some G are H. 3. All F are G; some F are H; so, some H are G. 4. No F is G; some G are H; so, some H are not F. 5. No F is G; no G is H; so, some F are not H.
(Arguments of these kinds, with two premisses and a conclusion, with each
proposition being of one of the kinds ‘All … are …’, ‘No … is …’ or ‘Some … are/are not …’, and each predicate occurring twice, are the traditional
syllogisms first discussed by Aristotle.)
B What of the following patterns of argument? Are these valid? 1. All F are G; so, nothing that is not G is F.
2. All F are G; so, at least one thing is F and G.
3. All F are G; no G are H; some J are H; so, some J are not F. 4. There is an odd number of F, there is an odd number of G; so
there is an even number of things which are either F or G. 5. m is F; n is F; so, there are at least two F.
6. All F are G; no G are H; so, all H are H.
C Consider the following argument: Dogs have four legs. Fido is a dog. So Fido has four legs.
In this chapter we exploit the fact that many arguments come in families which share an inferential structure (Chapter 3) and put that together with the invalid- ity principle (Chapter 2) to give a technique for showing that various invalid arguments are indeed invalid.
4.1 The technique illustrated
Recall the following argument (§1.4):
A (1) Most Irish are Catholics.
(2) Most Catholics oppose abortion. So (3) At least some Irish oppose abortion.
We quickly persuaded ourselves that the inference is invalid by imagining how the premisses could be true and conclusion false. But let’s now proceed a bit more methodically. Notice first that this argument exemplifies the argumentative pattern
M Most F are G
Most G are H
So, at least some F are H
And the ‘Irish’ argument surely stands or falls with this general pattern of infer- ence – for if this pattern isn’t reliable, what else can the argument depend on?
So, next ask: is this inference pattern M in fact a reliable one? Are inferences of this shape always valid? Consider:
A′ (1) Most chess masters are men. (2) Most men are no good at chess.
So (3) At least some chess masters are no good at chess.
The premisses of this argument are true. Chess is a minority activity, and still (at least at the upper levels of play) a predominantly male one. But the conclusion is crazily false. So we have true premisses and a plainly false conclusion: hence by the invalidity principle, argument A′ is invalid. We have shown, then, that the inference pattern is not reliable: an inference of this type can have true premisses