I ‘Supposition’ explained: and how to handle simple cases
In this chapter we deal with a distinctive kind of reasoning – suppositional reasoning. Most informal logic/critical thinking texts make no mention of it at all (although there are some notable exceptions, for example Stephen Thomas’s Practical Reasoning in Natural Language). This is surprising since this kind of reasoning is elegant, powerful, and extremely common, as we shall illustrate in the next three chapters.
The arguments considered in most texts employ only assertions: in speak-ing of reasons and conclusions they are always talkspeak-ing about asserted propo-sitions – propopropo-sitions which their authors have put forward as being true (cf. our remarks on assertion in Chapter 2, p. 23). However, some arguments reach their conclusion not by asserting their starting points, but by assuming or supposing something ‘for the sake of argument’ as it is often described.
If someone begins an argument by saying ‘Suppose that oxygen does not burn’ he is not asserting that oxygen does not burn – he is not presenting this as true. Indeed he may well know that oxygen burns and he may be setting out on a reductio ad absurdum argument to prove that it does. Suppositions then are not assertions.
An atheist who begins to argue her case by saying, ‘Suppose there is an omniscient Being of the sort in which Christians believe’, is not asserting (claiming) that there is a Christian God (because she doesn’t believe that there is one). She could just as well have said, ‘If there is an omniscient Being of the sort in which Christians believe’ and as we pointed out earlier (p. 23), someone who uses such a hypothetical does not assert its antecedent.
A mathematician who presents the standard Euclidean proof that there are infinitely many prime numbers begins by supposing that there are only finitely many. He is not asserting (telling us) that there are only finitely many primes (because he knows full well that this is false) but he is asking us to consider the proposition with a view to drawing out its implications.
Several of the arguments with which we began this book employed suppositions: the Galileo argument supposed that ‘the heavier a body is the faster it falls’; Pascal’s Wager considered alternative suppositions about one’s beliefs and actions; Hume’s argument began ‘Suppose four-fifths of all the money in Great Britain to be annihilated in one night’. The reasoning
employed by John Stuart Mill in the example of Chapter 5 was based on a supposition: ‘Let us suppose . . . that a general reduction of the hours of factory labour, say from ten to nine, would be for the advantage of the work-people’. Similarly with Ayer’s argument in Chapter 8.
The essential thing about a supposition is that it is not presented as being true – it is not asserted – it is put forward so that we may consider its implications. Arguments employing suppositions are common enough in theoretical contexts – in mathematics, in the physical sciences, the biological sciences, social studies and philosophy – to name some obvious ones so we must explain carefully how to handle suppositions in argument analysis if we are not to leave a serious gap.
We shall generally speak of suppositions in this book, though for most people it would probably be more natural to speak of assumptions. In many contexts the words are interchangeable, but many arguments contain implicit assumptions – propositions the author takes for granted as true – with-out bothering to mention them: although the author does not assert such assumptions (because he doesn’t mention them) he would be prepared to, or would have to, if they were drawn to his attention. For example argu-ments about nuclear deterrence usually assume – without actually saying it – that your alleged opponent wants to attack and dominate you. This implicit assumption is usually taken for granted and needs no explicit men-tion. We have already encountered various implicit assumptions in Chapter 1 and subsequently. However, in this chapter we are not especially interested in such implicit assumptions. We are interested in the case where someone assumes or supposes something ‘for the sake of the argument’ in the sense we just explained, so in order to focus attention on this case we shall use the less familiar term supposition.
In Chapter 10 we shall deal with scientific hypotheses. These are obviously closely related to suppositions in the way they function in reasoning, but we shall reserve the term ‘hypothesis’ until then, again in order to focus attention at this stage on ‘supposing something for the sake of argument’.
We begin to explain how to handle suppositions by looking at a simple example. Consider the following piece of reasoning:
Suppose Darwin’s theory of evolution is true. Then there should be fossil evidence which shows species changing and evolving, but this evidence simply doesn’t exist so Darwin’s theory must be wrong.
If we now attempt to extract the argument in accordance with the instructions in Chapter 2 (pp. 21f.), it is clear that we must circle
so and underline what is obviously the main conclusion,
C Darwin’s theory must be wrong.
(If the reader is also tempted to circle
then he or she will see in the course of this chapter both what is the source of this temptation and why it should be resisted in a simple case like this and whilst the instructions of Chapter 2 remain unrevised.)
When we ask, ‘What immediate reasons are presented in the text for accepting C?’ we clearly have one reason in,
(2) this evidence simply doesn’t exist
but we may hesitate before seeing how to mark up the remainder. A moment’s reflection will show that its meaning is captured correctly if we construe the supposition as the antecedent of a hypothetical, so the argument becomes,
(1) If Darwin’s theory of evolution is true then there ought to be fossil evidence which shows species changing and evolving
and (2) This evidence simply doesn’t exist therefore
C Darwin’s theory must be wrong
and the result of attempting to mark up this simple example according to the instructions of Chapter 2 will look something like this,
(1) SupposeIfDarwin’s theory of evolution is true. Then there should be fossil evidence which shows species changing and evolving, but 1 + 2
C↓
(2)this evidence simply doesn’t exist
so Darwin’s theory must be C wrong.
This example shows how to handle a supposition in a simple case; however, this will not always be the best way to proceed. In more complicated cases it may prove unnatural and laborious to treat suppositions as the antecedents of hypotheticals. We now illustrate this with another example as a prelim-inary to presenting an alternative way of handling suppositions. Consider the following piece of reasoning, taken from Stephen Thomas’s Practical Reasoning in Natural Language (2nd edn).
Suppose that only good researchers can be effective college teachers.
In that case it follows that a faculty member will be an effective teacher only if he or she is a good researcher. From this it fol-lows that if a faculty member is an effective teacher, then he or she must be a good researcher. Therefore every effective college teacher must be a good researcher. So, if only good researchers can be effec-tive college teachers then every effeceffec-tive college teacher must be
a good researcher. Therefore we could ensure that the university will excel in research by basing tenure decisions solely on teaching effectiveness.
If we attempt to treat the supposition here as part of an hypothetical and then re-construe the argument accordingly we get something like,
If only good researchers can be effective college teachers then a faculty member will be an effective teacher only if he or she is a good researcher. And if a faculty member will be an effective teacher only if he or she is a good researcher then if a faculty member is an effective teacher he or she must be a good researcher.
This is awful to read and rapidly obscures the logic of quite simple moves which follow the initial supposition! Not only that, if we are still trying to fol-low the instructions in Chapter 2, we circle the occurrences of
it follows that as conclusion indicators, which means that we take the sentences to which they refer to be asserted, but they are not asserted; they are in effect the consequent of a hypothetical.
We shall not continue to describe ways in which standard methods of argument and analysis, including those of Chapter 2, are inadequate for dealing with what we shall call ‘suppositional contexts’. Instead we shall revise our method in such a way that we can still do everything we want to do with ordinary (non-suppositional) contexts but which also copes with suppositional contexts. The result clarifies our thinking in both contexts.
It is clear that reasoning does take place with the aid of suppositions – within the ‘scope’ of suppositions – and any proposed method of argument analysis must cope with this. In fact suppositional contexts are very important and instructive. In traditional logic and in most informal logic texts they have been given too little attention, with the result that conditionals have been misunderstood, though this is not the place to elaborate on such matters.
We now revise our method of argument analysis before applying the revised method to an illuminating example.
II The method of informal argument analysis revised
The key to the revised method is the distinction between an asserted and an unasserted proposition. We already encountered this distinction in Chapter 2 (p. 23) and at the beginning of the present chapter. To recap briefly what we need, the proposition ‘oxygen burns’ may be presented as being true or it may be used in a compound proposition like ‘if oxygen burns then the phlogiston theory is wrong’, or ‘either oxygen burns or nitrogen burns’, and
in these cases it is not presented as being true. If a proposition is presented as being true logicians say (following Gottlob Frege (1848–1925), the founder of modern logic) that it is asserted. Otherwise it is not asserted. This is the distinction we need.
Given this distinction the method of Chapter 2 does not need much revi-sion. We shall need to extend our lists of reason and conclusion indicators (pp. 16, 17). We shall need to re-interpret R→ C (p. 19). We shall need to put our earlier remarks about hypotheticals into our new context (cf. pp. 23f.).
And we shall have to revise the requirement that the premisses of an argu-ment have to be true in order to establish its conclusion (p. 25).
The language of reasoning: some revisions
In informal logic books it is normal to say that reasoning or arguing consists in giving reasons for conclusions, but the only reasons and conclusions usually considered are asserted. The simplest and most economical way of coping with reasoning which proceeds from suppositions is to call suppositions reasons (or premisses) and, similarly, to call what follows from them conclusions (which are in turn reasons for their conclusions etc.) and to recognise that in suppositional contexts reasons and conclusions are not necessarily asserted – and hence that the occurrence of reason and conclusion indicators does not necessarily imply that what they relate to is asserted. (This may involve some slight distortion of normal usage – in calling a supposition a reason (or premiss) – but let this chapter and the next show whether the resulting simplification is justified.) If we do this our reason and conclusion indicators will include all the ones we gave earlier but the list will need extension in the two following ways.
Firstly, we need a list of ‘supposition indicators’ to add to our list of reason indicators. These will be such words or phrases as these:
Supposition indicators suppose that . . .
let us assume (for the sake of the argument) that . . . imagine that . . .
consider the hypothesis/theory that . . . let us postulate that . . .
As with the usual lists of reason and conclusion indicators, we are not saying that whenever these phrases are used a supposition is present. They are markers which have to be used intelligently in the light of our interests and our explanation of what a supposition is.
Secondly, a supposition is presented so that we may consider its impli-cations and it is equally natural to write after it ‘it follows that’ or ‘then’
(cf. some of our earlier examples). So we now need to include ‘then’ among our conclusion indicators – this explains the temptation we mentioned on
p. 117 – and to circle it as such when it occurs in a suppositional context to signal that a conclusion is being drawn from a supposition. Of course if we decide it is simpler to handle this particular context by means of hypotheticals we do not circle ‘then’.
All the reservations which were expressed in Chapter 2 about using argu-ment indicators apply with equal force to the extended lists.
The structure of reasoning: some revisions Some conventions and terminology
We need to be able to mark the distinction between asserted and unasserted propositions now, and we shall ‘flag’ the occurrence of an unasserted propo-sition which is functioning as a reason or a conclusion by means of a small raised letter‘u’(for ‘unasserted’) placed before it. Thus our example from p. 117 will be marked,
Suppose thatuonly good researchers can be effective college teachers.
In that case it follows thatua faculty member will be an effective teacher only if he or she is a good researcher. [etc.]
The simplest way to revise what we said in Chapter 2 about ‘→’ is as follows. We shall now construe the arrow ‘→’ to stand for the logical rela-tionship which is assumed to obtain between a reason R and its conclusion C in the context in which it occurs. If a speaker asserts R and also believes that C follows from R, or equivalently that R implies C, then he or she naturally says, ‘R therefore C’ relying on the assumed logical relationship between R and C to justify saying ‘therefore C’. We shall still represent such a case thus,
R→ C
and read it ‘R therefore C’ or some idiomatically appropriate equivalent. We might call this a ‘categorical’ context to distinguish it from a suppositional context.
If, on the other hand, the speaker says, ‘Suppose R. Then C will be true’
he or she is asserting neither R nor C and we shall represent this either as the hypothetical ‘if R then C’ or as follows,
(Suppose)uR
u↓C
The ‘(Suppose)’ is to remind us that this is the beginning of a suppositional argument. The arrow now stands for the logical relationship which is pre-sented by the speaker as obtaining between R and C and is read ‘then’ or ‘it
follows that’ or whatever seems idiomatically appropriate: a lengthy suppo-sitional argument,
(Suppose)uR→uC1. . . →uCn
might be read, ‘Suppose R. Then C1 follows. So C2. Therefore C3. In that case C4follows. (Etc. up to the nth conclusion Cn.)’
Note that there is clearly a very close relationship between saying ‘if R then C’ and saying ‘Suppose R. Then C.’ For our purposes we take them to be equivalent, and which way to construe a piece of natural language reasoning depends entirely on which seems simplest and most natural.
Except for these revisions everything which is said under the heading
‘Some conventions and terminology’, p. 19, about reasons and conclusions still obtains: reasons may still be independent or joint and conclusions may still be intermediate or final etc. But there are two important additions to argument diagrams. (Again, those who hate notation and diagrams need to grasp the underlying ideas.)
Clearly suppositions can be combined with assertions in argument. The following is a simple example, correctly marked up,
(1)
Supposeuthe Government wants to raise bank interest rates.
(2)
Sincethe Government also wants to keep mortgage rates down
C uit will clearly have to issue directives to the building societies.
We write the argument diagram for this argument as follows,
(Devising a clear linear form is left for the moment as an exercise especially for the reader who dislikes diagrams.)
Notice that, in general, if a reason R is unasserted in the course of some piece of reasoning this unasserted character will, so to speak, ‘infect’ every proposition P which is taken to follow from it (it will infect it in the sense that the truth and assertibility of P are conditional on the truth and assertibility of R) except in an important case which we must now explain. This is inference by ‘conditionalisation’.
Conditionalisation
Suppose we have an argument which proceeds from some supposition R to the conclusion C by logically sound steps (i.e. the conclusion at each step follows from the reasons given for it) then the soundness of the argument entitles us to infer the conditional (hence the name ‘conditionalisation’),
if R then C.
We have already seen an example of such an inference by conditionalisation:
here it is again, marked up for present purposes, (1)
Supposeuonly good researchers can be effective college teachers.
In that case it follows thatua faculty member will be an effective (2)
(3)
(4) C
teacher only if he or she is a good researcher.
From this it follows
thatuif a faculty member is an effective teacher then he or she must be a good researcher.
Thereforeuevery effective college teacher must be a good researcher.
So if only good researchers can be effective college teachers then every effective college teacher must be a good researcher.
This argument begins with the supposition (1); on this basis alone it reaches the conclusion (4) by logically sound steps; it then draws the conditional conclusion C, i.e. ‘if (1) then (4)’. Notice that, as we mentioned above, the unasserted character of (1) infects all the conclusions which follow from it except the conclusion C, ‘if (1) then (4)’. If the argument steps from (1) to (4) are sound, ‘if (1) then (4)’ must be true, whether (1) is true or not, and we can assert it simply because the soundness of the reasoning guarantees it.
In general if an argument proceeds from supposition R to conclusion C and then concludes ‘if R then C’ we shall represent this process of condition-alisation in an argument diagram as follows,
The arrow which is drawn out from the side of the arrow to C serves to remind us that the justification for ‘if R then C’ is the argument to C (not C itself). Thus our previous example is diagrammed as follows,
There are two concluding cases we need to mention. An argument may proceed from two (or more) joint reasons only one of which is unasserted and it may then conditionalise on that unasserted premiss. Such an argument is diagrammed,
(Suppose) u1 + 2 + (…) If 1 then C uC
Alternatively we may have an argument which contains two (or more) unasserted reasons among its joint basic reasons and which then condition-alises on only one of them. We diagram such a case as follows,
and the conclusion ‘if (1) then C’ is still unasserted because it depends on the unasserted (2). We need develop technicalities of this representation no further here.
The method of extracting arguments revised
The method outlined in Chapter 2 (pp. 21f.) needs only slight revision to cope with suppositional contexts. Inference indicators are circled just as before, except that we must now circle supposition indicators too (since they are rea-son indicators). In underlining conclusions and bracketing rearea-sons we should now mark those which are clearly unasserted thusu . . . , etc. Otherwise everything is as before. (It might be worth mentioning in connection with the use of the Assertibility Question that the answer to the question ‘What
The method outlined in Chapter 2 (pp. 21f.) needs only slight revision to cope with suppositional contexts. Inference indicators are circled just as before, except that we must now circle supposition indicators too (since they are rea-son indicators). In underlining conclusions and bracketing rearea-sons we should now mark those which are clearly unasserted thusu . . . , etc. Otherwise everything is as before. (It might be worth mentioning in connection with the use of the Assertibility Question that the answer to the question ‘What