Fundamentals

Argument Strength

If you’ll think back to one of the very early lectures you’ll recall that I gave a very general theory of arguments in which an argument is good if it possess a quality that we called ‘effectiveness’; which is to say that an argument is good iff it causes the respondent to feel the so-called Argument Intuition, so that the respondent is more disposed to accept the conclusion if the other statements are accepted. I mentioned at the time that it follows from such a conception that arguments may be good to different degrees. If an argument is such that the respondent is disposed to some degree to accept the conclusion if the other statements are accepted, then we consider the argument to be good to that degree.  

The Likelihood of Truth

At the time we took our judgments of the truth or falsity of statements to be equivalent to our judgment of the acceptability or unacceptability statements. We say that it is the same thing to say that a statement is true and that it’s an acceptable thing to believe. (And we made the necessary comment that this ‘acceptability’ is to be understood in the context of our looking for the right sorts of thing to be in our store of statements about how the world is.) It was therefore natural for us to talk about a subset of effective arguments in which the acceptability of the statements is interpreted in terms of the truth of the statements. In that case the scale of argument strengths could be interpreted as referring to the likelihood (or the perceived likelihood if you prefer) that the conclusion is true given that the premisses are true. I then gave you an arbitrary division of arguments by strength on a scale that looks like this:

1.                   A useless argument is one in which the truth of the premisses has no effect at all on the truth of the conclusion.

2.                   A weak argument is one in which the likelihood of the conclusion’s being true is not much affected by the truth or falsity of the premisses.

3.                   A moderate argument is one in which the likelihood of the conclusion being false if the premisses are true is quite low.

4.                   A strong argument is one in which the likelihood of the conclusion being false if the premisses are true is very low.

5.                   A valid argument is one in which it is just impossible for the conclusion to be false if the premisses are true.

So far we’ve looked only at the very last level, and, in fact, we’ve only looked at a subset of those arguments, because we’ve been concerned with arguments which are valid because of the form of the statements that occur in them. Such arguments we called formally valid. They are also sometimes called deductively valid because there’s a way to create a procedure (called a deduction) by which one can go from the premisses to the conclusion. That needn’t concern us now, however, because it’s the sort of thing that a course in logic is concerned with, and this isn’t a course in logic.

Rather, what we’re going to be interested in now is all the other arguments that aren’t valid and that aren’t quite useless. We’ve already seen a few of these in the course of the course, but we didn’t pay much attention to them. You’ll recall, for example

                Bob is an Australian

                Most Australians are happy

                ----------------------------------

                Bob is happy

but a more common sort of example would be

The sun has risen every morning for (at least) the past 500 years

                -----------------------------------------------------------------------------

                The sun will rise tomorrow

The general name for such arguments is inductive, and we’ll define them thus:

D1.          An inductive argument is an argument that claims that the likelihood of the conclusion is increased by the truth of the premisses, but is not made certain by their truth.

And the value of such arguments will be described in terms of their inductive strength, thus:

D2.          An argument is Inductively Strong if and only whenever the premises are true the conclusion is highly likely.

Some people have proposed that we can simplify our discussion of inductive arguments by finding some appropriate terms and giving them a technical meaning in this context similar to the terms ‘valid’ and ‘sound’ for deductive arguments. Soccio and Barry propose the following definitions:

D3.          An inductive argument is Justified if and only whenever the premises are true the conclusion is highly likely.

D4.          An argument is Cogent if and only it is justified and the premisses are true.

Obviously, the most important question for an inductive argument is going to be whether the premisses – even if they are true – provide sufficient support for the conclusion. Unlike the case of deductive arguments there are no hard and fast rules about what is going to count as sufficient evidence, and many of the rules of thumb that we’ll encounter are specific to particular types of inductive argument. In fact the problem of sufficient evidence can even be applied to the possibility of induction at all.

There is a notorious philosophical problem — Hume's Problem — centering on the justification of inductive reasoning in general, which is a part of broader concerns focusing on justification in epistemology and Philosophy more generally. (Knowledge requires justification of our beliefs and inductive reasoning is typically invoked as a means of justifying certain beliefs. But the question is whether inductive reasoning itself can be relied on or justified. Those interested might like to read A. Chalmers, What Is This Thing Called Science?, UQ Press, Ch 2.) We are not concerned with its general justification, but rather, supposing its general use can be justified, we shall be concerned with its characteristics and criteria for its legitimate employment in particular cases. We now turn to consider inductive arguments in more detail.

Induction and Deduction Compared and Contrasted

In order to bring the general characteristics of inductive arguments into sharp relief, consider the following contrast between a deductively valid argument and an inductively strong argument:

Misconceptions

a.                   Valid arguments are better than inductively strong ones.

One might well ask why we should employ arguments or reasoning which can only, at best, give strong yet not conclusive grounds for the conclusion. Aren't valid arguments always better?

No. They are better only in the sense that if the premises are true then a valid argument will ensure the conclusion — whereas an inductively strong one will only strongly support it.

But getting true premises that are strong or powerful enough to ensure the conclusion may be difficult or even impossible. Inductively strong arguments have the advantage of only needing premises powerful enough to strongly support the conclusion.

All ravens are black                                             All observed ravens have been black

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                Any raven on Pikes Peak will be black             Any raven on Pikes Peak will be black

The left-hand-side (valid) argument has a very strong connection between premises and conclusion but requires acceptance of a very strong and therefore a very contentious premise. The right-hand-side (inductively strong) argument, however, in spite of its weaker connection between premises and conclusion, requires acceptance of a less contentious premise. Consequently, the left-hand-side argument will be of little use as a means of getting someone to believe the conclusion because justifying the premise will be very difficult (if not impossible) whereas the right-hand-side argument would be useful.

Certainty is a virtue if obtainable, but where it is not obtainable a high degree of probability is better than none at all!

b.                   Valid arguments go from the general to the particular, whereas inductively strong ones go from the particular to the general.

No. Whilst this is true of some cases — for example:

                Valid Argument                                        Inductively Strong Argument

                with general premise and                         with particular premises and

                particular conclusion                                 general conclusion

                All ravens are black                                   Raven1 is black

                ------------------------                                      Raven2 is black

                That raven is black                                     :

                                                                                      Ravenn is black                                  

                                                                                      ------------------------

                                                                                      All ravens are black

it need not be — for example:

                Valid Argument                                         Inductively Strong Argument

                with particular premise and                       with general premise and

                particular conclusion                                  particular conclusion

                John and Phil are extremists                      All ravens we have seen have been black

                ----------------------------------                         --------------------------------------------------

                John is an extremist                                    The next raven we will see will be black

Varieties

You might imagine that this is going to be a fairly amorphous category of arguments because you might suppose there are many different ways in which the premisses might be so related to the conclusion that the likelihood is increased. To some extent you would be correct, but there are only a few really important styles of inductive argument. We’ll look at several types of argument: the argument from analogy, the inference to the best explanation, the sampling argument, and the statistical argument.

The Argument from Analogy

You may recall that I talked about fallacies of analogy before, but arguments from analogy can’t really be dismissed so easily, because very many of the common arguments that we use – especially when we’re looking for counterarguments to someone else’s fallacious argument – take this form, and they don’t all appear to be ‘bad’ arguments. Here’s a famous example.

The Argument from Design for the Existence of God (aka ‘The Teleological Argument’)

Consider a watch. A watch exhibits (a) complexity of parts; (b) suitability to fulfil a certain function (i.e. telling the time); and (c) its complexity is what enables it to fulfil this function. These three features are extremely unlikely to have come about by accident. No one on seeing a watch would think it the product of chance. Even seeing it for the first time, one would conclude that it is the product of design by some intelligent being.

But many things in nature we observe (e.g. the eye) are similarly complex, fulfil a function (e.g. seeing) and their complexity enables them to fulfil this function.

            So it is reasonable to suppose that they too are made by an intelligent being.

This argument is a paradigmatic argument from analogy, its important parts can be summarized thus:

                A watch has (a), (b), (c).

The world has (a), (b), (c).

Watches require a watch-maker     

                -----------------------------------------

                The world requires a world-maker

But that’s just one argument: the more general definition of an argument from analogy looks like this:

1.        It is claimed that the Object (an argument, or a natural phenomenon, or an idea, or what you will) has properties P₁, P₂, …, P_n.

2.        The Analogue also has properties P₁, P₂, …, P_n.

3.        The analogue has property P.

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4.        Therefore the object has property P.

We can see from this that an argument of this form could be treated as a type of enthymematic valid argument with the hidden premiss that:

[5.    If two objects share properties P₁, P₂, …, P_n, they will also share property P.]

As was said before, arguments like this are fallacious if the hidden premiss is not true or if it is not obviously true and yet is not argued for.

For example:

                Bob is a blue-eyed, blonde male

                So is Henry

                Bob is a criminal

                ---------------------------------------

                So is Henry

The hidden premiss here is that anyone who is a blue-eyed, blonde male is also a criminal. There would need to be some reason given to believe this because the connection isn’t obvious. But, as I say, this isn’t yet an inductive argument. On our definition it only becomes a type of inductive argument if the conclusion is said to be made more likely by the truth of the premisses.

How about:

                Alan, Bob, Carl, David, …, and Xavier are blue-eyed, blonde males

                So is Zach.

                Alan, Bob, Carl, David, …, and Xavier are criminals                                               

                ----------------------------------------------------------------------------------

                So is Zach

Evaluating Arguments from Analogy

1. Obviously, a necessary condition for the argument to be cogent is that the premises must be true.

2. The analogy itself must be strong. This can be undermined by pointing to:

1. the fact that the similarities P₁, P₂, …, P_n cited are either irrelevant or unimportant in relation to P (e.g. consider the argument for Zach’s criminality above)

2. the fact that there are relevant disanalogies.

The philosopher David Hume, in his Dialogues Concerning Natural Religion (1779) thought the analogy between the world and a machine was rather weak. In support of this we could point to relevant disanalogies like the world's containing objects with features apparently not well suited to fulfil their functions (cf. Stephen Jay Gould's The Panda's Thumb)

3. The strength of the conclusion must be considered. Such arguments might support only a rather weak conclusion and the argument itself will be weakened by trying to draw overly strong conclusions.

The philosopher David Hume, in his Dialogues Concerning Natural Religion (1779) pointed to the fact that, even supposing the analogy were a strong one, it would only strongly support the very limited, much weaker conclusion that there are Gods (not necessarily one as required by advocates of the argument), that are very powerful (not necessarily all-powerful, all-good, all-knowing).

The Inference to Best Explanation

This is sometimes called Abductive Inference and is treated as an entirely different form of inference only marginally related to inductive inference. I have much sympathy for that point of view, but I’ll let you be the judge. First you’ll want to see an example of the sort of thing that’s meant by this.

You return home to find your door broken and some valuable items missing. This cries out for explanation. Possible explanations include: 

                1.                A meteorite struck your door and vaporised your valuables.

                2.                Friends are playing a joke on you.

                3.                A police Tactical Response Group entered your house mistakenly.

                4.                You were robbed.

                Explanation 4 seems the best, so you conclude

                -------------------------------------------------------------------------------------------

you were robbed.

More generally, inferences to best explanation take the following form:

                Phenomenon C is observed

                Explanation A explains C and does so better than any rival explanation

                --------------------------------------------------------------------------------------

                A

The underlying assumption is that the best explanation of a phenomenon is likely to be true.

1. It is an inductive form of argument — the conclusion is merely rendered probable and can be undermined by further evidence.

2. We use such inferences all the time. (Gil Harman claims, in fact, that all inductive inferences are inferences to best explanation — see Harman, Thought, Princeton Uni. Press (1973), p. 130f.)

My car stops after a complete service by a reliable mechanic. Best explanation (in this context) might be that it is out of petrol. It is possible, of course, that someone broke into my car overnight and meddled with the engine but this is less plausible. In this situation you might reasonably conclude it has run out of petrol.

                This conclusion can be seen as following from inference to best explanation and, if it is the                 best explanation, then it seems a rational thing to believe since it is probably true.

Evaluating Inferences to Best Explanation

1. What do we mean by "best explanation"?

I.e. under what conditions can we be confident in the truth of the second premise and thus be confident that the explanation is probably true?

1. Firstly, how do we evaluate them for strength? (See Text, pp. 267-70.)

                                                               i.       They should actually explain the event in question as opposed to merely shifting the burden of explanation onto something else itself needing explaining.

                                                              ii.      They should be powerful (i.e. widely applicable).

                                                            iii.      The simpler the better.

                                                            iv.      They should be conservative with respect to prior beliefs.

(Compare creationism and evolutionary theory as rival explanations of the diversity of the biological world.)

2. Secondly, we want to be as confident as possible that we have to hand all the "rival explanations" there might be ... or at least, all strong rivals.

Consider the case of Nile floods discussed on p. 269 of the text. The problem here was that the conclusion that floods were due to the Gods was perhaps the best of the rivals considered but one strong rival — the best, in fact — was not considered. The conclusion was flawed because premise 2 was flawed.

Thinking up strong rival explanations is a challenging task. A lot of effort is expended in scientific research trying to develop strong rival explanations of a phenomenon for consideration. Phenomena are often riddles in the sense of their being puzzling facts.

                        Example 1.

A man, when alone, always rides the lift to the 10^th floor and then walks the last two floors to his 12^th floor flat. He never does this when someone else is in the lift. WHY? The best explanation can reasonably be said to be probably true by inference to best explanation. But what are the candidates?

                        Example 2.

The buried ruins of a town believed to be that of the builders of the pyramids, was unearthed in Egypt in the late twentieth century and archaeological research — including the newly available method of DNA analysis — established:

(i)                   the workers were well-fed (bone fragments in floors suggested they ate fish and high-quality beef)

(ii)                 they lived in family groups (with 50% male, 50% female and  ~24% children)

(iii)                their bones showed clear signs of having been subject to medical treatment for breaks and fractures.

Yet:

(iv)               The ancient Greek historian Herodotus had claimed that the pyramid builders were slaves.

How can we explain (i)-(iii) given (iv)?

Archaeologists concluded that we cannot and so rejected Herodotus's claim.

The best explanation of (i)-(iii) is that the pyramid builders were not slaves!

2. Is the claimed phenomenon C real or illusory?

People often believe things since they are thought necessary to explain some illusory phenomenon. They draw an inference to the best explanation of what they take to be a real phenomenon requiring explanation. In fact, they believe a falsehood inferred from the false claim that some phenomenon obtains (and is best explained by that which they come to believe).

                        Example.

I think that Phil always acts strangely in my presence. I take this to be a puzzling phenomenon requiring explanation. In the circumstances I reasonably suppose, say, that the best explanation of this is the idea that Phil dislikes me ... and so I go on to infer that Phil doesn't like me.                                                
In fact, Phil does not always act strangely in my presence. I have a partial and defective recall of social situations we have been in together.

Inductive Generalisation

In spite of the fact, noted in earlier lectures, that inductive reasoning is not characterized by its moving from the particular to the general, common examples of inductive reasoning are like this — namely, inductive generalisations. So let us turn to look at this form of inductive reasoning.

An argument is an inductive generalisation if and only if a generalised conclusion about the character some class as a whole is drawn from characteristics of a sample of the class.Such arguments are obviously generalisations, and they are inductive because not all members of the class in question need necessarily have the characteristics of the sample (as the conclusion suggests they do). It is, at best, highly likely.

a.                    Strong Inductive Generalisation

Consider the inductive generalisation:

                A Canadian quarter did not work in the American telephone on occasion 1.

                A Canadian quarter did not work in the American telephone on occasion 2.

                :

                A Canadian quarter did not work in the American telephone on occasion 20.

                Canadian quarters do not (ever) work in American telephones.

This generalization claims that all of the members of the target class possess a certain property. Arguments like this are called Strong Inductive Generalisations.

b.                    Weak Inductive Generalisation

Other generalizations claim no more than that some of the members of the target class possess a certain property. Arguments like this are called Weak Inductive Generalisations.

For example:

Many of the people I know who didn’t graduate from college have gone into real estate.

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                People who don’t graduate from college tend to go into real estate

c.                    Statistical Generalisation

Our final class of generalizations claims that some specific proportion of the members of the target class possess a certain property. Arguments like this are called Statistical Generalisations.

For example:

10% of people in this sample of the general population indicated they would eat Zowie Bars.

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                10% of the general population would eat Zowie Bars.

Evaluating Inductive Generalisation

1.                    Are the premises true?

This question is always important to address. Did the sample really substantiate claims made about it in the premises? Does the premise, perhaps, merely report hearsay or popular opinion and not fact?

2.                    Is the sample large enough?

A small sample may be unrepresentative and so suggest a general characteristic which does not apply.

This leads to the Fallacy of Hasty Generalisation

For example:To try one coin and then generalise would be overly hasty and fallacious because we cannot assume the one coin is truly representative of all coins..

Cognitive psychologists have noted our unreasonable tendency to see a small sample as truly representative (thus explaining our tendency to hastily generalise):

"We submit that people view a sample randomly drawn from a population as highly representative, that is, similar to the population in all essential characteristics. Consequently, they expect any two samples drawn from a particular population to be more similar to one another and to the population than sampling theory predicts, at least for small samples."

A. Tversky & D. Kahneman, 'Belief in the Law of Small Numbers', Psychological Bulletin 76 (1971): 105.

On the other hand, it is not generally the case that the substantial results of a survey will be affected by the size of the sample. What will change is the margin of error.

3.                    Is the sample biased in some other way?

The representativeness of the sample is what makes it possible (by definition) for the result of testing the sample to be true of the population. If there is a bias in the sample then the sample will not be representative. The best way to try to make sure there are no biases is to select the sample randomly from the population. (If there is no systematicity in the selection there can be no systematic bias.) It is not easy to do this.

A sample may be unrepresentative due to the method of sampling having been such as to select for particular characteristics which go unnoticed. This leads to the Fallacy of Biased Sampling.

Bias can arise due to:

—                 Insufficient variation in the sample

A large sample might still be unrepresentative of the whole population because it is badly chosen.

1.            Using the same Canadian quarter in all the many trials.

                                               Using the same American telephone in all the many trials.

                                2.            Using a phone book to get a large sample of the population.

                                3.            Birth-order as a behavioural predictor

Suppose (as was actually done) you sample scientists to see what factors might pre-dispose someone to accept radical new theories.

You discover that, of the people sampled, socio-economic factors do not vary with such a disposition yet birth-order does (i.e. the first born of the sample-set tend to be so disposed and the later born do not).               

Conclusion:                 First-borns are more likely to accept radical new theories in general — a biological or "nature" factor, contra an account which might suggest the determining factor is socio-economic — "nurture".

Problem:                Cannot draw conclusion that birth-order is a more important  determinant of willingness to accept radical new theories per se since sample of scientists might not be representative of socio-economic variation.

4.            Basing inferences concerning the general behaviour of native pigeons, say, on their behaviour when you've observed them. (They always seem very nervous when I observe them and so I am tempted to conclude that they are quite nervous birds, but of course my initial data may simply reflect the fact that they're very nervous when being watched by a ÿÿ pÿÿential predator (namely, me)!) ÿÿ

—                Ignoring or emphasising characteristics of a sample due to

                    Prejudice and Stereotypes

—                Eliciting a particular characteristic from the sample by Slanted Questions

4.                    Is the inference justified?

There are three factors that need to be considered.

                                       i.      The sample size: the proportion of the population that is tested.

                                      ii.      The level of confidence: the accuracy level of the extrapolation of the sample’s result to the population. At a 95% level, the result for the population will be within the margin of error 95% of the time. (ie. If you did 100 samplings you would get only 5 whose results would indicate a value for the population that was outside the margin of error.)

                                    iii.      The margin of error: the precision of the result. Generally expressed as ± y %. (eg. Voters are projected to vote Whig at 45% ± 3 %, which means that we have a 95% level of confidence, say, that the result would have been between 42% and 48%).

These three factors are interdependent; changing one affects both the others.

For a result to justify an inference the margin of error has to be narrow enough to make the result non-trivial, and the level of confidence has to be high enough to make the result significant.