Chapter 4
Deep Diagramming: Reasons For and Against
In this chapter we will go a bit further in our capacity to accurately
diagram reasoning. Until this point in our discussion of how to
represent the ways in which reasons relate to conclusions we have only
been concerned with reasons insofar as they are intended to support
conclusions. But as we all realize when we are presented with someone
else’s process of reasoning or when we present our own arguments, we
also take into account counter-considerations (reasons against the truth
of reasons, reasons against the truth of conclusions and reasons that
attack inferences). Indeed this is a fundamental dimension of any
extended argument for it can provide a balanced presentation of the
issues.
When we talk of being able to represent in diagrammatic form the
nature of reasons against we are speaking of three possible forms of
relationship. There are reasons which counter the truth of other
reasons, there are reasons that count against the truth of conclusions
and there are reasons which attack the nature of inferences in inductive
arguments. In the latter case we assume the truth of the reasons (for the
sake of the argument) and we try to think of other reasons which attack
the reliability of reaching the conclusion—reasons that weaken the
support of those reasons assumed to be true. In all three forms of attack
we are thinking ‘outside the box’ since we are trying to see problems in
the argument.
Let us begin at the simplest level by looking at single premise
arguments.
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Attacks Upon the Truth of Reasons
Let us suppose we have been presented with the following argument:
Example 4.1
It has been argued that (1) the University library contains very few
philosophy books. Therefore, (2) the University library should
start buying philosophy books.
Diagram:
(1)
(2)
But then it might be objected that we ought to doubt whether (1) is in
fact true because (3) students at the University are so enthralled with
philosophy that (4) the substantial collection of philosophy books in the
library are always checked out. The reasoning now has an objection
against the truth of (1) that is itself now supported by a reason. The
reasoning now reads:
Example 4.2
It has been argued that the University library contains very few
philosophy books. Therefore, (2) the University library should
start buying philosophy books. But this is not the case because (3)
students at the University are so enthralled with philosophy that
(4) the substantial collection of philosophy books in the library are
always checked out.
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Diagrammatically we arrive at:
(3)
(1)
(4)
(2)
One way to think of this is to ask ourselves whether the truth of (4)
makes (1) less likely to be true. Here the attack upon the truth of the
reason is signaled by the broken inference arrow.
Attacks upon the Truth of Conclusions
Sometimes instead of attacking the truth of a reason we want to directly
deny the truth of a conclusion. When this happens we may remain
agnostic about the reasons given for the conclusion (we may want to
attack the reasons as well but we don’t need to).
Example 4.3
It has been argued that the University library contains very few
philosophy books. Therefore, (2) the University library should
start buying philosophy books. But (3) the library’s budget is
already spent, and moreover (4) philosophy is not even studied at
this University.
Here we can remain agnostic about the reason given to support (2) and
we are simply pointing out that there are reasons which count against
the truth of (2).
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Diagram:
(1)
(3)
(4)
(2)
Once again the attacks represented by (3) and (4) are represented by the
broken inference arrow but now the two convergent reasons are drawn
directly against the conclusion (2).
Attacks upon Inferences
Another way in which we can present counter-considerations involves
neither attacking the truth of reasons nor necessarily attacking the truth
of conclusions but rather attacking the relationship between reasons
and conclusions. In this case one can remain agnostic about the truthvalue of reasons and conclusions but can point out that in the internal
structure of the argument, even if we assume (for the sake of argument)
that the reasons are true, the conclusion is not firmly established. It is
important to note here that attacks upon inferences can only occur in
inductive arguments and never in valid arguments. This is because in
inductive arguments the conclusion is supported with varying levels of
strength whereas in valid arguments the reasons, if true, give 100%
support for the conclusion. So there can be no successful attack against
a valid inference.
Suppose in a single premise argument we have:
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Example 4.4
(1) Something can exist only if it is a material thing. Therefore, (2)
God does not exist.
Diagram:
(1)
(2)
This little argument immediately strikes us as odd. The reason for this
is that there is a missing premise, namely that God is not a material
thing. Nevertheless, let us imagine that (3) God is, in fact, a material
thing. In this case if we assume (for the sake of argument) that premise
(1) is indeed true, we will have to admit that the conclusion no longer
follows.
The argument then looks like this:
Example 4.5
(1) Something can exist only if it is a material thing. Therefore, (2)
God does not exist. But (3) God is a material thing.
In order to diagram an attack upon an inference a broken arrow is
drawn from the counter-consideration (or objection) to the unbroken
inference arrow.
Diagram:
(1)
(3)
(2)
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Deep Diagramming: Reasons For and Against
A heuristic for testing this approach is to assume (1) and (3) are true
and then ask ourselves whether the support of (1) to (2) is weakened by
(3). If the answer is yes then we have a successful attack upon the
inference.
Example 4.6
Alice: (1) Nearly all Singaporeans are Chinese. (2) Nora is
Singaporean. So probably (3) Nora is Chinese.
Diagram:
(1) + (2)
(3)
This inference is inductively strong. But now suppose that Bob objects
to Alice’s argument:
Bob: But that can’t be right. (4) Nora is a Singaporean Muslim. (5)
Nearly all Singaporean Muslims are not Chinese. Notice that the
truth of (4)-plus-(5) does not reduce the likelihood that (1) is true.
Nor does (4)-plus-(5) count against (2) or (3). So the principle of
charity tells us that we should not see Bob as mounting an attack on
Alice’s reason or her conclusion. However, assuming that (1) and (2)
are true, the truth of (4)-plus-(5) will weaken the inference from (1)plus-(2) to (3). In fact, if (4) is true, this will weaken Alice’s inference
greatly, turning it from a strong inference into a weak one.
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Diagram:
(1) + (2)
(4) + (5)
(3)
In this book we will not deal with reasons for an inference, in other
words, a reason given with the intention of strengthening an inference.
This means that we will never draw a diagram such as
Diagram:
(1)
(3)
(2)
How Do We Know What A Reason Attacks?
Suppose that you are sure that a reason attacks part of an argument
but you are not sure whether it attacks a reason, a conclusion or an
inference. Our heuristic is:
Suppose that the attacking reason is true. Does this make a
reason in the argument attacked less likely to be true? If so then
that reason is attacked.
Suppose that the attacking reason is true. Does this make a
conclusion in the argument attacked less likely to be true? If so
then that conclusion is attacked.
Suppose that the attacking reason is true and that a reason in
the argument attacked is also true. Does the reason in the
argument attacked now support the conclusion less strongly? If so
then the inference from the reason in the argument attacked to its
conclusion, is attacked.
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All-Out Assaults
Typically, counter-considerations that attack the truth of reasons or the
truth of conclusions or attacks upon inferences are themselves argued
for and sometimes in great detail. One of the great benefits of learning
how to diagram arguments is that when we start to deal with complex
and long chains of reasoning (such as we might find in an article or
book) we always know where we are and how we got there because
diagramming itself charts the overall flow of reasoning in a transparent
way. Let us see how we can apply this by building a longer piece of
reasoning step by step.
Consider the following short argument:
Example 4.7
(1) If I buy a car then I’ll be able to drive from Singapore across the
causeway and travel around Malaysia. (2) I have always wanted to
visit the Taman Negara National park and (3) to view one of the
last remaining pristine jungle environments in Asia. Therefore, (4)
I should buy a car.
Diagram:
(1) + (2) + (3)
(4)
Let us suppose now that we want to attack the truth of proposition (1).
For example, I might think that buying a car in Singapore is too
expensive and if I buy a car I will not be able to do any traveling as I’ll
need a second job to make the repayments on the car. So we can extract
from these considerations two central reasons that attack the truth of ‘If
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I buy a car I’ll be able to drive across the causeway and travel around
Malaysia.’ They are: ‘Buying a car in Singapore is too expensive’ and ‘If
I buy a car I will not be able to do much traveling anyway.’ Strictly
speaking we do not have two reasons which oppose the truth of (1) but
one reason against the truth of (1) which is itself backed up by a reason.
Let us look at this carefully:
Example 4.8
(1) If I buy a car then I’ll be able to drive from Singapore across the
causeway and travel around Malaysia. (2) I have always wanted to
visit the Taman Negara National park and (3) to view one of the
last remaining pristine jungle environments in Asia. Therefore, (4)
I should buy a car. On the other hand, (5) if I buy a car I will not be
able to do much traveling because (6) I’ll need a second job to
make the repayments because (7) buying a car is so expensive in
Singapore.
Diagram:
(7)
(6)
(5)
(1) + (2) + (3)
(4)
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The attack upon the truth of (1) is represented diagrammatically by the
broken inference arrow. Notice that we have a pattern of serial
reasoning for proposition (5); while (5) itself directly attacks the truth of
(1). Since propositions (1) + (2) + (3) are linked, reason (5) which directly
attacks the truth of (1) also indirectly attacks the conjunction of reasons
(1) + (2) + (3), because they need each other to support (4). So (5) attacks
the support for (4).
One could of course mount the attack on different grounds. One might
suggest that even though it may well be true that if I buy a car then I’ll
be able to drive from Singapore across the causeway and travel around
Malaysia, the conclusion that I should buy a car doesn’t follow because
I could hire a car to drive into Malaysia to see the National Park.
Example 4.9
(1) If I buy a car then I’ll be able to drive from Singapore across the
causeway and travel around Malaysia. (2) I have always wanted to
visit the Taman Negara National park and (3) to view one of the
last remaining pristine jungle environments in Asia. Therefore, (4)
I should buy a car. On the other hand, (5) if I buy a car I’ll not be
able to do much traveling because (6) I’ll need a second job to
make the repayments because (7) buying a car is so expensive in
Singapore. Then again, (8) I could always hire a car to drive into
Malaysia.
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Diagram:
(7)
(6)
(5)
(1) + (2) + (3)
(8)
(4)
We might go even further and argue that I am prohibited by law from
buying a car in Singapore. This directly attacks the truth of the
conclusion, because being prohibited by law from buying a car is a
good reason for not buying a car (assuming that I don’t want to break
the law).
Example 4.10
(1) If I buy a car then I’ll be able to drive from Singapore across the
causeway and travel around Malaysia. (2) I have always wanted to
visit the Taman Negara National park and (3) to view one of the
last remaining pristine jungle environments in Asia. Therefore, (4)
I should buy a car. On the other hand, (5) if I buy a car I’ll not be
able to do much traveling because (6) I’ll need a second job to
make the repayments because (7) buying a car is so expensive in
Singapore. Then again, (8) I could always hire a car to drive into
Malaysia. But as a matter of fact (9) I am prohibited by law from
buying a car in Singapore.
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Diagram:
(7)
(6)
(5)
(1)+ (2) + (3)
(8)
(4)
(9)
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Extended Examples
Let us look at two further examples to consolidate our understanding of
what we have learned in this chapter.
Example 4.11
(1) We must stop treating juveniles differently from adult
offenders. (2) Justice demands it, since (3) justice implies that
people should be treated equally. Besides, (4) the social effects of
pampering
juvenile
offenders
include
sinister
social
consequences. (5) The record shows that juveniles who have been
treated leniently for offences have subsequently committed
serious crimes. Yet (6) it is also quite obvious that juveniles, unlike
adults, lack some of the moral understanding necessary to see the
consequences of their actions. Aside from that, (7) there is nothing
in the concept of justice that insists on treating everybody equally.
Diagram:
(7)
(3)
(5)
(2)
(4)
(6)
(1)
Having produced a diagram, we should now use it to evaluate the
argument. This will include evaluating how well it holds up against the
reasons that attack it. We would use the same methods explained in
chapter 3. Consider reasons against reasons. The inference from the
attacking reason against the reason attacked is weak if the truth of the
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attacking reason is largely irrelevant to the falsehood of the reason
attacked. It is moderate if the truth of the attacking reason only makes
the reason attacked more likely to be false than true, but not make it
very likely to be false. It is strong if the truth of the attacking reason
makes the reason attacked very likely to be false. It is 100% or in other
words, valid, if the truth of the attacking reason makes it a certainty
that the reason attacked is false. In such a case it is impossible for the
reason attacked to be true if the attacking reason is true. The same goes
for reasons against conclusions.
How do we evaluate the strength of reasons against an inference? The
inference from the attacking reason against the inference attacked is
weak if the truth of the attacking reason has little or no effect in
weakening the inference. The inference from the attacking reason
against the inference attacked is moderate if the truth of the attacking
reason makes the inference attacked moderate. The inference from the
attacking reason against the inference attacked is strong if the truth of
the attacking reason makes the inference attacked weak. The inference
from the attacking reason against the inference attacked is 100% if the
truth of the attacking reason makes the inference attacked have no
strength at all.
How strong is the inference from (3) to (2)? To answer this question, we
must note the hanging pronoun ‘it’ in ‘justice demands it’ in (2).
Demands what? (1) provides the answer—the full meaning of (2) is
‘Justice demands that we stop treating juveniles differently from adult
offenders.’ This shows that we often need to look at other parts of the
passage in order to read an assertion in its proper context. We may now
see that the inference from (3) to (2) is 100%. If justice implies or in other
words, demands, that people be treated equally, then it must be true
that justice demands that we stop treating juveniles differently from
adult offenders, because juveniles and adults are both people. The
inference from (2) to (1) is also 100%, assuming that we care about
justice. If justice demands that we stop treating juveniles differently
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from adult offenders, then we must stop treating juveniles differently
from adult offenders, assuming that we must do what justice demands
of us. The strength of the attack upon (3) by (7) is 100%. If it is true that
there is nothing in the concept of justice that insists on treating
everybody equally, then it must be false that this is what justice implies.
(7) is just an outright denial of (3). The inference from (5) to (4) is weak.
We might allow that committing serious crimes is to be described as
‘sinister.’ We might even be charitable enough to allow that treating
offenders leniently counts as ‘pampering.’ But to claim that the
commission of serious crimes is an effect of treating offenders leniently
is not supported by the fact that offenders who received lenient
treatment subsequently committed serious crimes. Just because they
first received lenient treatment and then committed serious crimes does
little to show that lenient treatment is the cause of serious crime. This is
the fallacy of ‘after this, so because of this.’ Perhaps the offenders of
serious crime would have committed these crime whatever treatment
they received. We need more evidence and this has not been supplied.
The inference from (4) to (1) is also weak. In the context of (1), (4) tells
us that juvenile offenders are treated leniently, while adult offenders
are not. But while we are told that treating juvenile offenders leniently
results in bad consequences for society, we are not told anything about
the consequences of stricter treatment for adult offenders. For all we
know, this might result in worse consequences for society. The
inference also makes the deep background assumption that deciding
how to treat offenders is a matter of looking at consequences. Not
everyone will agree. Some will say that we should not do what has the
best consequences if doing it is unjust. The author himself seems
committed to this objection. The attack from (6) upon the inference from
(4) to (1) is 100% given the background assumption that we should give
more lenient treatment to offenders who are incapable of seeing the
harm that they produce. Many, although by no means all, will agree
with this assumption.
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Granting the background assumptions implicit in both the argument
and in the objections to it, we may now add the evaluations to our
diagram:
Diagram:
valid
(7)
(3)
valid
(2)
valid
(5)
weak
(4)
weak
(6)
valid
(1)
We are now ready to attempt the difficult job of deciding what is true.
We can see that trying to establish the truth of (1) on the basis of (4) is
hopeless. Thus it is largely irrelevant whether (4) is true. For the same
reason, (5) is largely irrelevant as well, because its role is to support (4),
which is itself largely irrelevant. Of course none of this means that (1) is
false, only that we have at this point, little reason to think that it is true.
Our best hope of establishing that (1) is true is to look at (2). If (3) is true
then (2) must be true, and if (2) is true then (1) must be true, assuming
that we must do what justice demands of us. So if (3) is true then (1)
must be true, assuming that we must do what justice demands of us. Is
(3) true? It is at least plausible that justice implies that people should be
treated equally. Indeed many will say that the idea of equality is part of
meaning of justice. On this view of it (7) is false. But not everyone will
be willing to grant the assumption that we must always do what justice
demands of us. Utilitarians, in other words, those who think that we
must always do whatever produces the best consequences in the long
run, will be prepared to act unjustly if doing so has the best overall
consequences. On the other side of the controversy, (1) is bound to be
false if (7) is true, given the assumption that we should give more
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lenient treatment to offenders who are incapable of seeing the harm that
they produce. Is it true that juveniles, unlike adults, lack some of the
moral understanding necessary to see the consequences of their actions?
To decide this mater we might research the findings of developmental
psychologists, but from a commonsense perspective it is surely true that
as children develop into adulthood, they gain a greater appreciation of
the effects of what they do.
Example 4.12
(1) The eighteenth-century philosopher David Hume was
undoubtedly a finer thinker than his even more celebrated
successor Immanuel Kant. (2) Hume was by far the more lucid
writer. (3) His contributions were more diverse than Kant's, for (4)
he was a first-rate historian as well as a philosopher. Further, (5)
Hume's ethical thought did not suffer from the rigidity of Kant's.
(6) Hume, unlike Kant, would never have said the duty not to lie is
so absolute that we should answer truthfully even when a wouldbe murderer asks where his intended victim is hiding. But this
argument is nonsense. (7) Being a better writer does not mean
you’re a better thinker. Moreover, (8) being a historian does not
mean that you are a better philosopher. And finally, (9) being rigid
could be one of the merits of a fine thinker, since (10) strictness is
something to be cherished in philosophical thought.
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Diagram:
(4)
(6)
(3)
(5)
(10)
(8)
(2)
(7)
(9)
(1)
First we evaluate the inferences. The inference from (4) to (3) is weak.
Reading (3) in the context of (4), we may assume that Kant was a
philosopher but not a historian. But the fact that Hume was both, does
not itself give much support to the claim that Hume’s contributions
were more diverse than Kant’s. For all we are told, Kant may have
made contributions in fields other than history that Hume did not, such
as mathematics or music. The inference from (3) to (1) is weak. The
context of the (9) and (10) shows that ‘thinker’ in (1) is used to mean
‘philosopher.’ But as the objection in (8) shows, the fact that Hume’s
contributions were more diverse than Kant’s in being a fine historian as
well as a philosopher need not add anything to the quality of Hume’s
philosophy. Since the truth of (8) would make the inference from (3) to
(1) weak, (8) is a strong objection. Quite generally, just because a person
makes contributions in many different disciplines is poor evidence that
he excels in any particular one of them. The inference from (2) to (1) is
at best moderate. All other things being equal, a philosopher who
writes much more lucidly is better than one who does not. But all we
are told, things might not be equal. Suppose that although Kant was
much less lucid than Hume, he was also more profound, wider in
scope, more rigorous, addressed more important questions, and
changed the history of philosophy more dramatically. In that case we
might well judge Kant a finer philosopher than Hume. The objection in
(7) against the inference from (2) to (1) is strong. If it is true that being a
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better writer does not make one a better philosopher then the fact that
Hume is a better writer than Kant does not make him a better
philosopher. Since the truth of (7) would make the inference from (2) to
(1) weak, (7) is a strong objection. This of course is not to say that (7) is
true. The inference from (6) to (5) is 100% or in other words, valid. To
claim that we have a duty to refrain from lying in absolutely all
circumstances is a perfectly rigid or inflexible view of the matter. Since
Kant but not Hume makes this claim, it follows with certainty that
Kant’s ethical thinking in this respect is rigid in a way that Hume’s is
not. The inference from (5) to (1) is weak, because there is a missing
connection between rigidity of thought and its quality. ‘Rigid’ sounds
like a defect, but as the objection in (9) observes, this might not be so.
Since the truth of (9) makes the inference from (5) to (1) weak, (9) is a
strong objection. The inference from (10) to (9) may be seen as 100%, in
other words, valid, if we supply the assumption that being rigid is a
form of strictness. For if any form of strictness in philosophical thought
is valuable and being rigid in philosophical thought is a form of
strictness in philosophical thought, then it must be true that being rigid
in philosophical thought is valuable. Granting any needed assumptions
in the name of charity, we may add our evaluations to our diagram:
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Diagram:
(4)
(8)
strong
(2)
moderate
(7)
strong
weak
(6)
valid
(3)
weak
weak
(1)
(10)
valid
(5)
strong
(9)
Now we try to decide what is true. The prospects of establishing that (1)
is true look dim. (3) or (5) offer little hope because even if they are true,
there is little chance that truth will transmitted down to (1). The next
best option is (2). Since the (7) is a strong objection to the inference from
(2) to (1), if (7) is true then the inference will be weakened even more.
But (7) is false. Surely all other things being equal, being a better writer
does indeed make one a better philosopher. A good deal of philosophy
consists not just in discussion, careful reading and reflection, but
philosophical writing. The better the writing is, the clearer and more
persuasive it will be, and this is surely a benefit. Nonetheless the
inference from (2) to (1) remains moderate at best for reasons we have
given above that are different from (7). Is (2) true? The best way to find
out whether Hume was by far the more lucid writer would be to read
both Kant and Hume. That would be an enormous amount of work and
it would be difficult even for a trained philosopher. The next best
option is to see whether there is a consensus on the matter by
philosophers who are authorities on both philosophers. In fact such
philosophers will all agree that whereas Hume is clear, Kant is
notoriously obscure. So in the last analysis, if we were forced at the
point of a gun to bet the final conclusion, we should think that it is true,
but we certainly should not place any confidence in it being true that
we may derive from the argument. Note by the way that once our
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diagram was fully evaluated, we did not need to try to decide whether
every proposition in it is true. To make a point we have made
elsewhere, thinking things through often saves us work.
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