THE PHILOSOPHER’S TOOLKIT - A Compendium of Philosophical Concepts and Methods - SECOND EDITION - BY JULIAN BAGGINI AND PETER S. FOSL - CHAPTER 1

back to ...

https://sites.google.com/site/philosophygreekancient/the-philosopher-s-toolkit---a-compendium-of-philosophical-concepts-and-methods---second-edition---by-julian-baggini-and-peter-s-fosl---table-of-contents-and-preface

===

1 Basic Tools for Argument

1.1 Arguments, premises and conclusions 1

1.2 Deduction 6

1.3 Induction 8

1.4 Validity and soundness 13

1.5 Invalidity 17

1.6 Consistency 19

1.7 Fallacies 23

1.8 Refutation 26

1.9 Axioms 28

1.10 Definitions 31

1.11 Certainty and probability 34

1.12 Tautologies, self-contradictions and the law of non-contradiction 38

1.1 Arguments, premises and conclusions

Philosophy is for nit-pickers. That’s not to say it is a trivial pursuit. Far from

it. Philosophy addresses some of the most important questions human beings

ask themselves. The reason philosophers are nit-pickers is that they are concerned

with the ways in which beliefs we have about the world either are or

are not supported by rational argument. Because their concern is serious, it

is important for philosophers to demand attention to detail. People reason in

a variety of ways using a number of techniques, some legitimate and some

not. Often one can discern the difference between good and bad arguments

only if one scrutinizes their content and structure with supreme diligence.

Argument

What, then, is an argument? For many people, an argument is a contest or

conflict between two or more people who disagree about something. An

argument in this sense might involve shouting, name-calling and even a bit

of shoving. It might – but need not – include reasoning.

Philosophers, by contrast, use the term ‘argument’ in a very precise and

narrow sense. For them, an argument is the most basic complete unit of

reasoning, an atom of reason. An ‘argument’ is an inference from one or

more starting points (truth claims called a ‘premise’ or ‘premises’) to an end

point (a truth claim called a ‘conclusion’).

Argument vs. explanation

‘Arguments’ are to be distinguished from ‘explanations’. A general rule to keep

in mind is that arguments attempt to demonstrate that something is true,

while explanations attempt to show how something is true. For example, consider

encountering an apparently dead woman. An explanation of the woman’s

death would undertake to show how it happened. (‘The existence of water

in her lungs explains the death of this woman.’) An argument would undertake

to demonstrate that the person is in fact dead (‘Since her heart has

stopped beating and there are no other vital signs, we can conclude that she is

in fact dead.’) or that one explanation is better than another (‘The absence of

bleeding from the laceration on her head combined with water in the lungs

indicates that this woman died from drowning and not from bleeding.’)

The place of reason in philosophy

It is not universally realized that reasoning comprises a great deal of what

philosophy is about. Many people have the idea that philosophy is essentially

about ideas or theories about the nature of the world and our place in it.

Philosophers do indeed advance such ideas and theories, but in most cases

their power and scope stems from their having been derived through rational

argument from acceptable premises. Of course, many other regions of human

life also commonly involve reasoning, and it may sometimes be impossible to

draw clean lines distinguishing philosophy from them. (In fact, whether or

not it is possible to do so is itself a matter of heated philosophical debate.)

The natural and social sciences are, for example, fields of rational inquiry

that often bump up against the borders of philosophy (especially in inquiries

into the mind and brain, theoretical physics and anthropology). But

theories composing these sciences are generally determined through certain

formal procedures of experimentation and reflection to which philosophy

has little to add. Religious thinking sometimes also enlists rationality

and shares an often-disputed border with philosophy. But while religious

thought is intrinsically related to the divine, sacred or transcendent – perhaps

through some kind of revelation, article of faith or religious practice

– philosophy, by contrast, in general is not.

Of course, the work of certain prominent figures in the Western philosophical

tradition presents decidedly non-rational and even anti-rational

dimensions (for example, that of Heraclitus, Kierkegaard, Nietzsche,

Heidegger and Derrida). Furthermore, many include the work of Asian

(Confucian, Taoist, Shinto), African, Aboriginal and Native American

thinkers under the rubric of philosophy, even though they seem to make

little use of argument.

But, perhaps despite the intentions of its authors, even the work of nonstandard

thinkers involves rationally justified claims and subtle forms of

argumentation. And in many cases, reasoning remains on the scene at least

as a force to be reckoned with.

Philosophy, then, is not the only field of thought for which rationality is

important. And not all that goes by the name of philosophy is argumentative.

But it is certainly safe to say that one cannot even begin to master the

expanse of philosophical thought without learning how to use the tools of

reason. There is, therefore, no better place to begin stocking our philosophical

toolkit than with rationality’s most basic components, the subatomic

particles of reasoning – ‘premises’ and ‘conclusions’.

Premises and conclusions

For most of us, the idea of a ‘conclusion’ is as straightforward as a philosophical

concept gets. A conclusion is, literally, that with which an argument

concludes, the product and result of an inference or a chain of

inferences, that which the reasoning justifies and supports.

What about ‘premises’? In the first place, in order for a sentence to serve

as a premise, it must exhibit this essential property: it must make a claim

that is either true or false. Sentences do many things in our languages, and

not all of them have that property. Sentences that issue commands, for

example (‘Forward march, soldier!’), or ask questions (‘Is this the road to

Edinburgh?’), or register exclamations (‘Holy cow!’), are neither true nor

false. Hence it is not possible for them to serve as premises.

This much is pretty easy. But things can get sticky in a number of ways.

One of the most vexing issues concerning premises is the problem of

implicit claims. That is, in many arguments key premises remain unstated,

implied or masked inside other sentences. Take, for example, the following

argument: ‘Socrates is a man, so Socrates is mortal.’ What’s left implicit is

the claim that ‘all men are mortal’. Such unstated premises are called

enthymemes, and arguments which employ them are enthymemetic.

In working out precisely what the premises are in a given argument, ask

yourself first what the claim is that the argument is trying to demonstrate.

Then ask yourself what other claims the argument relies upon (implicitly or

explicitly) in order to advance that demonstration. Sometimes certain words

and phrases will indicate premises and conclusions. Phrases like ‘in conclusion’,

‘it follows that’, ‘we must conclude that’ and ‘from this we can see that’

often indicate conclusions. (‘The DNA, the fingerprints and the eyewitness

accounts all point to Smithers. It follows that she must be the killer.’) Words

like ‘because’ and ‘since’, and phrases like ‘for this reason’ and ‘on the basis of

this’, often indicate premises. (For example, ‘Since the DNA, the fingerprints

and the eyewitness accounts all implicate Smithers, she must be the killer.’)

Premises, then, compose the set of claims from which the conclusion is

drawn. In other sections, the question of how we can justify the move from

premises to conclusion will be addressed (see 1.4 and 4.7). But before we get

that far, we must first ask, ‘What justifies a reasoner in entering a premise in

the first place?’

Grounds for premises?

There are two basic reasons why a premise might be acceptable. One is that

the premise is itself the conclusion of a different, solid argument. As such,

the truth of the premise has been demonstrated elsewhere. But it is clear

that if this were the only kind of justification for the inclusion of a premise,

we would face an infinite regress. That is to say, each premise would have to

be justified by a different argument, the premises of which would have to be

justified by yet another argument, the premises of which … ad infinitum.

(In fact, sceptics – Eastern and Western, modern and ancient – have pointed

to just this problem with reasoning.)

So, unless one wishes to live with the infinite regress, there must be another

way of finding sentences acceptable to serve as premises. There must be, in

short, premises that stand in need of no further justification through other

arguments. Such premises may be true by definition, such as ‘all bachelors

are unmarried.’ But the kind of premises we’re looking for might also include

premises that, though conceivably false, must be taken to be true for there to

be any rational dialogue at all. Let’s call them ‘basic premises’.

Which sentences are to count as basic premises depends on the context in

which one is reasoning. One example of a basic premise might be, ‘I exist.’ In

most contexts, this premise does not stand in need of justification. But if, of

course, the argument is trying to demonstrate that I exist, my existence cannot

be used as a premise. One cannot assume what one is trying to argue for.

Philosophers have held that certain sentences are more or less basic for

various reasons: because they are based upon self-evident or ‘cataleptic’

perceptions (Stoics), because they are directly rooted in sense data (positivists),

because they are grasped by a power called intuition or insight

(Platonists), because they are revealed to us by God (religious philosophers),

or because we grasp them using cognitive faculties certified by God

(Descartes, Reid, Plantinga). In our own view, a host of reasons, best

described as ‘context’ will determine them.

Formally, then, the distinction between premises and conclusions is clear.

But it is not enough to grasp this difference. In order to use these philosophical

tools, one has to be able both to spot the explicit premises and to

make explicit the unstated ones. And aside from the question of whether or

not the conclusion follows from the premises, one must come to terms with

the thornier question of what justifies the use of premises in the first place.

Premises are the starting points of philosophical argument. As in any edifice,

however, intellectual or otherwise, the construction will only stand if

the foundations are secure.

SEE ALSO

1.2 Deduction

1.3 Induction

1.9 Axioms

1.10 Definitions

3.6 Circularity

7.1 Basic beliefs

7.8 Self-evident truths

READING

★ Nigel Warburton, Thinking From A to Z, 2nd edn (2000)

★ Graham Priest, Logic: A Very Short Introduction (2001)

Patrick J. Hurley, A Concise Introduction to Logic, 10th edn (2007)

1.2 Deduction

The murder was clearly premeditated. The only person who knew where

Dr Fishcake would be that night was his colleague, Dr Salmon. Therefore,

the killer must be …

Deduction is the form of reasoning that is often emulated in the formulaic

drawing-room denouements of classic detective fiction. It is the

most rigorous form of argumentation there is, since in deduction, the

move from premises to conclusions is such that if the premises are true,

then the conclusion must also be true. For example, take the following

argument:

1. Elvis Presley lives in a secret location in Idaho.

2. All people who live in secret locations in Idaho are miserable.

3. Therefore Elvis Presley is miserable.

If we look at our definition of a deduction, we can see how this argument

fits the bill. If the two premises are true, then the conclusion must also be

true. How could it not be true that Elvis is miserable, if it is indeed true that

all people who live in secret locations in Idaho are miserable, and Elvis is

one of these people?

You might well be thinking there is something fishy about this, since you

may believe that Elvis is not miserable for the simple reason that he no

longer exists. So, all this talk of the conclusion having to be true might

strike you as odd. If this is so, you haven’t taken on board the key word at

the start of this sentence, which does such vital work in the definition of

deduction. The conclusion must be true if the premises are true. This is a

big ‘if’. In our example, the conclusion is, we confidently believe, not true,

because one or both (in this case both) premises are not true. But that

doesn’t alter the fact that this is a deductive argument, since if it turned out

that Elvis does live in a secret location in Idaho and that all people who lived

in secret locations in Idaho are miserable, it would necessarily follow that

Elvis is miserable.

The question of what makes a good deductive argument is addressed in

more detail in the section on validity and soundness (1.4). But in a sense,

everything that you need to know about a deductive argument is contained

within the definition given: a (successful) deductive argument is one where,

if the premises are true, then the conclusion is definitely true.

But before we leave this topic, we should return to the investigations of

our detective. Reading his deliberations, one could easily insert the vital,

missing word. The killer must surely be Dr Salmon. But is this the conclusion

of a successful deductive argument? The fact is that we can’t answer

this question unless we know a little more about the exact meaning of the

premises.

First, what does it mean to say the murder was ‘premeditated’? It could

mean lots of things. It could mean that it was planned right down to the last

detail, or it could mean simply that the murderer had worked out what she

would do in advance. If it is the latter, then it is possible that the murderer

did not know where Dr Fishcake would be that night, but, coming across

him by chance, put into action her premeditated plan to kill him. So, it

could be the case (1) that both premises are true (the murder was premeditated,

and Dr Salmon was the only person who knew where Dr Fishcake

would be that night) but (2) that the conclusion is false (Dr Salmon is, in

fact, not the murderer). Therefore the detective has not formed a successful

deductive argument.

What this example shows is that, although the definition of a deductive

argument is simple enough, spotting and constructing successful ones is

much trickier. To judge whether the conclusion really must follow from the

premises, we have to be sensitive to ambiguity in the premises as well as to

the danger of accepting too easily a conclusion that seems to be supported

by the premises but does not in fact follow from them. Deduction is not

about jumping to conclusions, but crawling (though not slouching) slowly

towards them.

SEE ALSO

1.1 Arguments, premises and conclusions

1.3 Induction

1.4 Validity and soundness

READING

Fred R. Berger, Studying Deductive Logic (1977)

★ John Shand, Arguing Well (2000)

A. C. Grayling, An Introduction to Philosophical Logic (2001)

1.3 Induction

I (Julian Baggini) have a confession to make. Once, while on holiday in

Rome, I visited the famous street market, Porta Portese. I came across a

man who was taking bets on which of the three cups he had shuffled around

was covering a die. I will spare you the details and any attempts to justify my

actions on the grounds of mitigating circumstances. Suffice it to say, I took

a bet and lost. Having been budgeted so carefully, the cash for that night’s

pizza went up in smoke.

My foolishness in this instance is all too evident. But is it right to say my

decision to gamble was ‘illogical’? Answering this question requires wrangling

with a dimension of logic philosophers call ‘induction’. Unlike deductive

inferences, induction involves an inference where the conclusion follows

from the premises not with necessity but only with probability (though even

this formulation is problematic, as we will see).

Defining induction

Often, induction involves reasoning from a limited number of observations

to wider, probable generalizations. Reasoning this way is commonly called

‘inductive generalization’. It is a kind of inference that usually involves reasoning

from past regularities to future regularities. One classic example is

the sunrise. The sun has risen regularly so far as human experience can

recall, so people reason that it will probably rise tomorrow. (The work of

the Scottish philosopher David Hume [1711–76] has been influential on

this score.) This sort of inference is often taken to typify induction. In the

case of my Roman holiday, I might have reasoned that the past experiences

of people with average cognitive abilities like mine show that the probabilities

of winning against the man with the cups is rather small.

But beware: induction is not essentially defined as reasoning from the specific

to the general.

An inductive inference need not be past-future directed. And it can

involve reasoning from the general to the specific, the specific to the specific

or the general to the general.

I could, for example, reason from the more general, past-oriented claim

that no trained athlete on record has been able to run 100 metres in under

9 seconds, to the more specific past-oriented conclusion that my friend had

probably not achieved this feat when he was at university, as he claims.

Reasoning through analogies (see 2.4) as well as typical examples and

rules of thumb are also species of induction, even though none of them

involves moving from the specific to the general.

The problem of induction

Inductive generalizations are, however, often where the action is. Reasoning

in experimental science, for example, often depends on them in so far as

scientists formulate and confirm universal natural laws (e.g. Boyle’s ideal

gas law) on the basis of a relatively small number of observations. Francis

Bacon (1561–1626) argued persuasively for just this conception of induction.

The tricky thing to keep in mind about inductive generalizations,

however, is that they involve reasoning from a ‘some’ in a way that only

works definitely or with necessity for an ‘all’. This type of inference makes

inductive generalization fundamentally different from deductive argument

(for which such a move would be illegitimate). It also opens up a

rather enormous can of conceptual worms. Philosophers know this

conundrum as the ‘problem of induction’. Here’s what we mean. Take the

following example:

1. Almost all elephants like chocolate.

2. This is an elephant.

3. Therefore, this elephant likes chocolate.

This is not a well-formed deductive argument, since the premises could be

true and the conclusion still be false. Properly understood, however, it may

be a strong inductive argument – if the conclusion is taken to be probable,

rather than certain.

On the other hand, consider this rather similar argument:

1. All elephants like chocolate.

2. This is an elephant.

3. Therefore, this elephant likes chocolate.

Though similar in certain ways, this one is, in fact, a well-formed deductive

argument, not an inductive argument at all. The problem of induction is the

problem of how an argument can be good reasoning as induction but be poor

reasoning as a deduction. Before addressing this problem directly, we must

take care not to be misled by the similarities between the two forms.

A misleading similarity

Because of the kind of general similarity one sees between these two arguments,

inductive arguments can sometimes be confused with deductive

arguments. That is, although they may actually look like deductive arguments,

some arguments are actually inductive. For example, an argument

that the sun will rise tomorrow might be presented in a way that might easily

be taken for a deductive argument:

1. The sun rises every day.

2. Tomorrow is a day.

3. Therefore the sun will rise tomorrow.

Because of its similarity with deductive forms, one may be tempted to read

the first premise as an ‘all’ sentence:

The sun rises on all days (every 24-hour period) that there ever have

been and ever will be.

The limitations of human experience, however (the fact that we can’t

experience every single day), justify us in forming only the less strong ‘some’

sentence:

The sun has risen on every day (every 24-hour period) that humans have

recorded their experience of such things.

This weaker formulation, of course, enters only the limited claim that the

sun has risen on a small portion of the total number of days that have ever

been and ever will be; it makes no claim at all about the rest.

But here’s the catch. From this weaker ‘some’ sentence one cannot construct

a well-formed deductive argument of the kind that allows the conclusion

to follow with the kind of certainty characteristic of deduction. In

reasoning about matters of fact, one would like to reach conclusions with

the certainty of deduction. Unfortunately, induction will not allow it.

The uniformity of nature?

Put at its simplest, the problem of induction can be boiled down to the problem

of justifying our belief in the uniformity of nature across space and time.

If nature is uniform and regular in its behaviour, then events in the observed

past and present are a sure guide to unobserved events in the unobserved past,

present and future. But the only grounds for believing that nature is uniform

are the observed events in the past and present. (Perhaps to be precise we should

only count observed events in the present, especially when claims about the

past also rely on assumptions about the uniform operations of nature, for

example memory.) We can’t then it seems go beyond observed events without

assuming the very thing we need to prove – that is, that unobserved parts of the

world operate in the same way as the parts we observe. (This is just the problem

to which Hume points.) Believing, therefore, that the sun may possibly not

rise tomorrow is, strictly speaking, not illogical, since the conclusion that it

must rise tomorrow does not inexorably follow from past observations.

A deeper complexity

Acknowledging the relative weakness of inductive inferences (compared to

those of deduction), good reasoners qualify the conclusions reached

through it by maintaining that they follow not with necessity but only with

probability. But does this fully resolve the problem? Can even this weaker,

more qualified formulation be justified? Can we, for example, really justify

the claim that, on the basis of uniform and extensive past observation, it is

more probable that the sun will rise tomorrow than it won’t?

The problem is that there is no deductive argument to ground even this

qualified claim. To deduce this conclusion successfully we would need the

premise ‘what has happened up until now is more likely to happen tomorrow’.

But this premise is subject to just the same problem as the stronger claim that

‘what has happened up until now must happen tomorrow’. Like its stronger

counterpart, the weaker premise bases its claim about the future only on

what has happened up until now, and such a basis can be justified only if we

accept the uniformity (or at least general continuity) of nature. But again the

uniformity (or continuity) of nature is just what’s in question.

A groundless ground?

Despite these problems, it seems that we can’t do without inductive generalizations.

They are (or at least have been so far!) simply too useful to refuse. Inductive

generalizations compose the basis of much of our scientific rationality, and they

allow us to think about matters concerning which deduction must remain

silent. In short, we simply can’t afford to reject the premise that ‘what we have

so far observed is our best guide to what is true of what we haven’t observed’,

even though this premise cannot itself be justified without presuming itself.

There is, however, a price to pay. We must accept that engaging in inductive

generalization requires that we hold an indispensable belief which itself,

however, must remain in an important way ungrounded.

SEE ALSO

1.1 Arguments, premises and conclusions

1.2 Deduction

1.7 Fallacies

2.4 Analogies

5.4 Hume’s fork

READING

★ Francis Bacon, Novum Organum (1620)

★ David Hume, A Treatise of Human Nature (1739–40), Bk 1

Colin Howson, Hume’s Problem: Induction and the Justification of Belief (2003)

1.4 Validity and soundness

In his book The Unnatural Nature of Science the eminent British biologist

Lewis Wolpert (b. 1929) argued that the one thing that unites almost all of

the sciences is that they often fly in the face of common sense. Philosophy,

however, may exceed even the sciences on this point. Its theories, conclusions

and terms can at times be extraordinarily counter-intuitive and contrary

to ordinary ways of thinking, doing and speaking.

Take, for example, the word ‘valid’. In everyday speech, people talk about

someone ‘making a valid point’ or ‘having a valid opinion’. In philosophical

speech, however, the word ‘valid’ is reserved exclusively for arguments.

More surprisingly, a valid argument can look like this:

1. All blocks of cheese are more intelligent than any philosophy student.

2. Meg the cat is a block of cheese.

3. Therefore Meg the cat is more intelligent than any philosophy

student.

All utter nonsense, you may think, but from a strictly logical point of view

it is a perfect example of a valid argument. What’s going on?

Defining validity

Validity is a property of well-formed deductive arguments, which, to recap,

are defined as arguments where the conclusion in some sense (actually,

hypothetically, etc.) follows from the premises necessarily (see 1.2). Calling

a deductive argument ‘valid’ affirms that the conclusion actually does follow

from the premises in that way. Arguments that are presented as or

taken to be successful deductive arguments but where the conclusion does

not in fact definitely follow from the premises are called ‘invalid’ deductive

arguments.

The tricky thing, in any case, is that an argument may possess the property

of validity even if its premises or its conclusion are not in fact true.

Validity, as it turns out, is essentially a property of an argument’s structure.

And so, with regard to validity, the content or truth of the statements composing

the argument is irrelevant. Let’s unpack this.

Consider structure first. The argument featuring cats and cheese given

above is an instance of a more general argumentative structure, of the

form:

1. All Xs are Ys.

2. Z is an X.

3. Therefore Z is a Y.

In our example, ‘block of cheese’ is substituted for X, ‘things that are more

intelligent than all philosophy students’ for Y, and ‘Meg’ for Z. That makes

our example just one particular instance of the more general argumentative

form expressed with the variables X, Y and Z.

What you should notice is that you don’t need to attach any meaning to

the variables to see that this particular structure is a valid one. No matter

what we replace the variables with, it will always be the case that if the

premises are true (although in fact they might not be), the conclusion must

also be true. If there’s any conceivable way possible for the premises of an

argument to be true but its conclusion simultaneously be false, then it is an

invalid argument.

What this boils down to is that the notion of validity is content-blind (or

‘topic-neutral’). It really doesn’t matter what the content of the propositions

in the argument is – validity is determined by the argument having a

solid, deductive structure. Our example is then a valid argument because if

its ridiculous premises were true, the ridiculous conclusion would also have

to be true. The fact that the premises are ridiculous is neither here nor there

when it comes to assessing the argument’s validity.

The truth machine

From another point of view we might consider that arguments work a bit

like sausage machines. You put ingredients (premises) in, and then you get

something (conclusions) out. Deductive arguments may be thought of as

the best kind of sausage machine because they guarantee their output in the

sense that when you put in good ingredients (all true premises), you get out

a quality product (true conclusions). Of course if you don’t start with good

ingredients, deductive arguments don’t guarantee a good end product.

Invalid arguments are not generally desirable machines to employ. They

provide no guarantee whatsoever for the quality of the end product. You

might put in good ingredients (true premises) and sometimes get a highquality

result (a true conclusion). Other times good ingredients might yield

a poor result (a false conclusion).

Stranger still (and very different from sausage machines), with invalid

deductive arguments you might sometimes put in poor ingredients (one or

more false premises) but actually end up with a good result (a true conclusion).

Of course, in other cases with invalid machines you put in poor

ingredients and end up with rubbish. The thing about invalid machines is

that you don’t know what you’ll get out. With valid machines, when you put

in good ingredients (though only when you put in good ingredients), you

have assurance. In sum:

Invalid argument

Put in false premise(s) → get out either a true or false conclusion

Put in true premise(s) → get out either a true or false conclusion

Valid argument

Put in false premise(s) → get out either a true or false conclusion

Put in true premise(s) → get out only a true conclusion

Soundness

To say an argument is valid, then, is not to say that its conclusion must be

accepted as true. The conclusion is established as true only if (1) the argument

is valid and (2) the premises are true. This combination of valid argument

plus true premises (and therefore a true conclusion) is called

approvingly a ‘sound’ argument. Calling it sound is the highest endorsement

one can give for an argument. If you accept an argument as sound,

you are really saying that one must accept its conclusion. This can be shown

by the use of another especially instructive valid, deductive argument:

1. If the premises of the argument are true, then the conclusion must also

be true. (That is to say, you’re maintaining that the argument is valid.)

2. The premises of the argument are true.

If you regard these two as premises, you can advance a deductive argument

that itself concludes with certainty:

3. Therefore, the conclusion of the argument must also be true.

For a deductive argument to pass muster, it must be valid. But being valid is

not sufficient to make it a sound argument. A sound argument must not

only be valid; it must have true premises, as well. It is, strictly speaking,

only sound arguments whose conclusions we must accept.

Importance of validity

This may lead you to wonder why, then, the concept of validity has any

importance. After all, valid arguments can be absurd in their content and

false in their conclusions – as in our cheese and cats example. Surely it is

soundness that matters.

Keep in mind, however, that validity is a required component of soundness,

so there can be no sound arguments without valid ones. Working out

whether or not the claims you make in your premises are true, while important,

is simply not enough to ensure that you draw true conclusions. People

make this mistake all the time. They forget that you can begin with a set of

entirely true beliefs but reason so poorly as to end up with entirely false

conclusions. The problem is that starting with truth doesn’t guarantee ending

up with it.

Furthermore in launching criticism, it is important to grasp that understanding

validity gives you an additional tool for evaluating another’s position.

In criticizing a specimen of reasoning you can either

1. attack the truth of the premises from which he or she reasons,

2. or show that his or her argument is invalid, regardless of whether or

not the premises deployed are true.

Validity is, simply put, a crucial ingredient in arguing, criticizing and thinking

well, even if not the only ingredient. It is an indispensable philosophical

tool. Master it.

SEE ALSO

1.1 Arguments, premises and conclusions

1.2 Deduction

1.5 Invalidity

READING

Aristotle (384–322 bce), Prior Analytics

Fred R. Berger, Studying Deductive Logic (1977)

★ Patrick J. Hurley, A Concise Introduction to Logic, 10th edn (2007)

1.5 Invalidity

Given the definition of a valid argument, it may seem obvious what an invalid

one looks like. Certainly, it is simple enough to define an invalid argument: it

is one where the truth of the premises does not guarantee the truth of the

conclusion. To put it another way, if the premises of an invalid argument are

true, the conclusion may still be false. Invalid arguments are unsuccessful

deductions and therefore, in a sense, are not truly deductions at all.

To be armed with an accurate definition of invalidity, however, may not

be enough to enable you to make use of this tool. The man who went looking

for a horse equipped only with the definition ‘solid-hoofed, herbivorous,

domesticated mammal used for draught work and riding’ (Collins

English Dictionary) discovered as much, to his cost. In addition to the definition,

you need to understand the definition’s full import. Consider this

argument:

1. Vegetarians do not eat pork sausages.

2. Gandhi did not eat pork sausages.

3. Therefore Gandhi was a vegetarian.

If you’re thinking carefully, you’ll have probably noticed that this is an

invalid argument. But it wouldn’t be surprising if you and a fair number of

readers required a double take to see that it is in fact invalid. And if one can

easily miss a clear case of invalidity in the midst of an article devoted to a

careful explanation of the concept, imagine how easy it is not to spot invalid

arguments more generally.

One reason why some fail to notice that this argument is invalid is because

all three propositions are true. If nothing false is asserted in the premises of

an argument and the conclusion is true, it’s easy to think that the argument

is therefore valid (and sound). But remember that an argument is valid only

if the truth of the premises guarantees the truth of the conclusion in the

sense that the conclusion is never false when the premises are true. In this

example, this isn’t so. After all, a person may not eat pork sausages yet not

be a vegetarian. He or she may, for example, be an otherwise carnivorous

Muslim or Jew. He or she simply may not like pork sausages but frequently

enjoy turkey or beef.

So, the fact that Gandhi did not eat pork sausages does not, in conjunction

with the first premise, guarantee that he was a vegetarian. It just so

happens that he was. But, of course, since an argument can only be sound if

it is valid, the fact that all three of the propositions it asserts are true does

not make it a sound argument.

Remember that validity is a property of an argument’s structure. In this

case, the structure is

1. All Xs are Ys.

2. Z is a Y.

3. Therefore Z is an X.

where X is substituted for ‘vegetarian’, Y for ‘person who does not eat pork

sausages’ and Z for ‘Gandhi’. We can see why this structure is invalid by

replacing these variables with other terms that produce true premises, but

a clearly false conclusion. (Replacing terms creates a new ‘substitution

instance’ of the argument form.) If we substitute X for ‘Cat’, Y for ‘meat

eater’ and Z for ‘the president of the United States’, we get:

1. All cats are meat eaters.

2. The president of the United States is a meat eater.

3. Therefore the president of the United States is a cat.

The premises are true but the conclusion clearly false. Therefore this cannot

be a valid argument structure. (You can do this with various invalid argument

forms. Showing that an argument form is invalid by substituting sentences

into that form in a way that results in true premises but a false

conclusion is called showing invalidity by ‘counterexample’. See 3.8.)

It should be clear therefore that, as with validity, invalidity is not determined

by the truth or falsehood of the premises but by the logical relations

among them. This reflects a wider, important feature of philosophy.

Philosophy is not just about saying things that are true; it is about making

true claims that are grounded in good arguments. You may have a particular

viewpoint on a philosophical issue, and it may just turn out by sheer luck

that you are right. But, in many cases, unless you can show you are right by

the use of good arguments, your viewpoint is not going to carry any weight

in philosophy. Philosophers are not just concerned with the truth, but with

what makes it the truth and how we can show that it is the truth.

SEE ALSO

1.2 Deduction

1.4 Validity and soundness

1.7 Fallacies

READING

★ Irving M. Copi, Introduction to Logic, 10th edn (1998)

★ Harry Gensler, Introduction to Logic (2001)

★ Patrick J. Hurley, A Concise Introduction to Logic, 10th edn (2008)

1.6 Consistency

Ralph Waldo Emerson may have written that ‘a foolish consistency is the

hobgoblin of little minds’, but of all the philosophical crimes there are,

the one you really don’t want to get charged with is inconsistency.

Consistency is the cornerstone of rationality. What then, exactly, does

consistency mean?

‘Consistency’ is a property characterizing two or more statements. If you

hold two or more inconsistent beliefs, then, at root, this means you face a

logically insurmountable problem with their truth. More precisely, the

statements of your beliefs will be found to be somehow either to ‘contradict’

one another or to be ‘contrary’ to one another, or together imply contradiction

or contrariety. Statements are ‘contradictory’ when they are opposite in

‘truth value’: when one is true the other is false, and vice versa. Statements

are ‘contrary’ when they can’t both be true but, unlike contradictories, can

both be false. (A single sentence can be ‘self-contradictory’ when it makes

an assertion that is necessarily false – often by conjoining two inconsistent

sentences).

Tersely put, then, two or more statements are consistent when it is possible

for them all to be true in the same sense and at the same time. Two or

more statements are inconsistent when it is not possible for them all to be

true in the same sense and at the same time.

Apparent and real inconsistency: the abortion example

At its most flagrant, inconsistency is obvious. If I say, ‘All murder is wrong’

and ‘That particular murder was right’, I am clearly being inconsistent,

because the second assertion is clearly contrary to the first. On a more general

level it would be a bald contradiction to assert both that ‘all murder is

wrong’ and ‘not all murder is wrong’.

But sometimes inconsistency is difficult to determine. Apparent inconsistency

may actually mask a deeper consistency – and vice versa.

Many people, for example, agree that it is wrong to kill innocent human

beings. And many of those same people also agree that abortion is morally

acceptable. One argument against abortion is based on the claim that these

two beliefs are inconsistent. That is, critics claim that it is inconsistent to

hold both that ‘It is wrong to kill innocent human beings’ and that ‘It is

permissible to destroy living human embryos and fetuses.’

Defenders of the permissibility of abortion, on the other hand, may

retort that properly understood the two claims are not inconsistent.

A defender of abortion could, for example, claim that embryos are not

human beings in the sense normally understood in the prohibition (e.g.

conscious or independently living or already-born human beings). Or a

defender might change the prohibition itself to make the point more clearly

(e.g. by claiming that it’s wrong only to kill innocent human beings that

have reached a certain level of development, consciousness or feeling).

Exceptions to the rule?

But is inconsistency always undesirable? Some people are tempted to say it

is not. To support their case, they present examples of beliefs that intuitively

seem perfectly acceptable yet seem to match the definition of inconsistency

given. Two examples might be:

It is raining, and it is not raining.

My home is not my home.

In the first case, the inconsistency may be only apparent. What one may

really be saying is not that it is raining and not raining, but rather that it’s

neither properly raining nor not raining, since there is a third possibility –

perhaps that it is drizzling, or intermittently raining – and that this other,

fuzzy possibility most accurately describes the current situation.

What makes the inconsistency only apparent in this example is that the

speaker is shifting the sense of the terms being employed. Another way of

saying the first sentence, then, is that, ‘In one sense it is raining, but in another

sense of the word it is not.’ For the inconsistency to be real, the relevant terms

being used must retain precisely the same meaning throughout.

This equivocation in the meanings of the words shows that we must be

careful not to confuse the logical form of an inconsistency – asserting both

X and not-X – with ordinary language forms that appear to match it but

really don’t. Many ordinary language assertions that both X and not-X are

true turn out, when analysed carefully, not to be inconsistencies at all. So, be

careful before accusing someone of inconsistency.

But, when you do unearth a genuine logical inconsistency, you’ve

accomplished a lot, for it is impossible to defend the inconsistency without

rejecting rationality outright. Perhaps, however, there are poetic, religious

and philosophical contexts in which this is precisely what people

find it proper to do.

Poetic, religious or philosophical inconsistency?

What about the second example we present above – ‘My home is not my

home.’ Suppose that the context in which the sentence is asserted is in the

diary of someone living under a horribly violent and dictatorial regime –

perhaps a context like the one George Orwell’s character Winston Smith

endures in 1984. Literally, the sentence is self-contradictory, internally

inconsistent. It seems to assert both that ‘This is my home’ and that ‘This

is not my home.’ But the sentence also seems to carry a certain poetic

sense, which conveys how absurd the world has come to seem to the

speaker, how alienated he or she feels from the world in which he or she

exists.

The Danish existentialist philosopher Søren Kierkegaard (1813–55)

maintained that the Christian notion of the incarnation (‘Jesus is God,

and Jesus was a man’) is a paradox, a contradiction, an affront to reason,

but nevertheless true. Existentialist philosopher Albert Camus (1913–60)

maintained that there is something fundamentally ‘absurd’ (perhaps inconsistent?)

about human existence.

Perhaps, then, Emerson was right, and there are contexts in which inconsistency

and absurdity paradoxically make sense.

Consistency ≠ truth

Be this as it may, inconsistency in philosophy is generally a serious vice.

Does it follow from this that consistency is philosophy’s highest virtue? Not

quite. Consistency is only a minimal condition of acceptability for a philosophical

position. Since it is often the case that one can hold a consistent

theory that is inconsistent with another, equally consistent theory, the consistency

of any particular theory is no guarantee of its truth. Indeed, as

French philosopher-physicist Pierre Maurice Marie Duhem (1861–1916)

and the American philosopher Willard Van Orman Quine (1908–2000)

have maintained, it may be possible to develop two or more theories that

are (1) internally consistent, yet (2) inconsistent with each other, and also

(3) perfectly consistent with all the data we can possibly muster to determine

the truth or falsehood of the theories.

Take as an example the so-called problem of evil. How do we solve the

puzzle that God is supposed to be good but that there is also awful suffering

in the world? As it turns out, you can advance a number of theories that

may solve the puzzle but remain inconsistent with one another. You can

hold, for instance, that God does not exist. Or you can hold that God allows

suffering for a greater good. Although each solution may be perfectly consistent

with itself, they can’t both be right, as they are inconsistent with each

other. One theory asserts God’s existence, and the other denies it. Establishing

the consistency of a position, therefore, may advance and clarify philosophical

thought, but it probably won’t settle the issue at hand. We often need to

appeal to more than consistency if we are to decide between competing

positions. How we do this is a complex and controversial subject of its

own.

SEE ALSO

1.12 Tautologies, self-contradictions and the law of non-contradiction

3.25 Sufficient reason

READING

Pierre M. M. Duhem, La théorie physique, son objet et sa structure (1906)

★ Fred R. Berger, Studying Deductive Logic (1977)

★ José L. Zalabardo, Introduction to the Theory of Logic (2000)

1.7 Fallacies

The notion of ‘fallacy’ will be an important instrument to draw from your

toolkit, for philosophy often depends upon identifying poor reasoning, and

a fallacy is nothing other than an instance of poor reasoning – a faulty inference.

Since every invalid argument presents a faulty inference, a great deal

of what one needs to know about fallacies has already been covered in the

entry on invalidity (1.5). But while all invalid arguments are fallacious, not

all fallacies involve invalid arguments. Invalid arguments are faulty because

of flaws in their form or structure. Sometimes, however, reasoning goes

awry for reasons not of form but of content.

All fallacies are instances of faulty reasoning. When the fault lies in the

form or structure of the argument, the fallacious inference is called a

‘formal’ fallacy. When it lies in the content of the argument, it is called an

‘informal’ fallacy. In the course of philosophical history philosophers

have been able to identify and name common types or species of fallacy.

Oftentimes, therefore, the charge of fallacy calls upon one of these

types.

Formal fallacies

One of the most common types of inferential error attributable to the form

of argument has come to be known as ‘affirming the consequent’. It is an

extremely easy error to make and can often be difficult to detect. Consider

the following example:

1. If Fiona won the lottery last night, she’ll be driving a red Ferrari

today.

2. Fiona is driving a red Ferrari today.

3. Therefore Fiona won the lottery last night.

Why is this invalid? It is simply that, as with any invalid argument, the truth

of the premises does not guarantee the truth of the conclusion. Drawing

this conclusion from these premises leaves room for the possibility that the

conclusion is false, and if any such possibility exists, the conclusion is not

guaranteed.

You can see that such a possibility exists in this case by considering that

it is possible that Fiona is driving a Ferrari today for reasons other than her

winning the lottery. Fiona may, for example, have just inherited a lot of

money. Or she may be borrowing the car, or perhaps she stole it.

Note, however, that her driving the Ferrari for other reasons does not

render the first premise false. Even if she’s driving the car because she in fact

inherited a lot of money, it still might be true that if she had instead won the

lottery she would have gone out and bought a Ferrari just the same. Hence

the premises and conclusion might all be true, but the conclusion will not

follow with necessity from the premises.

The source of this fallacy’s persuasive power lies in an ambiguity in ordinary

language concerning the use of ‘if’. The word ‘if’ is sometimes used to

imply ‘if and only if’ (‘iff’ in philosophical jargon) but sometimes means

simply ‘if’. Despite their similarity, these two phrases have very different

meanings.

As it turns out, the argument would be valid if the first premise were

stated in a slightly different way. Strange as it may seem, while the argument

about Fiona above is deductively invalid, substituting either of the following

statements for the first premise in that argument will yield a perfectly

valid argument.

1′. If Fiona is driving a red Ferrari today, then she won the lottery last

night.

1″. Iff Fiona won the lottery last night is she driving a red Ferrari today.

Because ‘if’ and ‘if and only if’ are ordinarily used in rather vague ways (that

don’t distinguish the usages above), philosophers redefine them in a very

precise sense (see 4.5).

In addition, because fallacies can be persuasive and are so prevalent, it

will be very useful for you to acquaint yourself with the most common fallacies.

(Equivocation [3.10], false cause fallacies [3.12], the masked man

fallacy [3.16] and others have their own entries in this book. More are

delineated in the texts listed below.) Doing so can inoculate you against

being taken in by bad reasoning. It can also save you some money.

Informal fallacies

The ‘gambler’s fallacy’ is both a dangerously persuasive and a hopelessly

flawed species of inference. The fallacy occurs when someone is, for

example, taking a bet on the tossing of a fair coin. The coin has landed

heads up four times in a row. The gambler therefore concludes that the

next time it is tossed, the coin is more likely to come up tails than heads

(or the reverse). But what the gambler fails to realize is that each toss of

the coin is unaffected by the tosses that have come before it. No matter

what has been tossed beforehand, the odds remain roughly 50–50 for

every single new toss. The odds of tossing eight heads in a row are rather

low. But if seven heads in a row have already been tossed, the chances of

the sequence of eight in a row being completed (or broken) on the next

toss is still 50–50.

What makes this an informal rather than a formal fallacy is that we can

actually present the reasoning here using a valid form of argument.

1. If I’ve already tossed seven heads in a row, the probability that the

eighth toss will yield a head is less than 50–50 – that is, I’m due for a

tails.

2. I’ve already tossed seven heads in a row.

3. Therefore the probability that the next toss will yield a head is less than

50–50.

The flaw here is not with the form of the argument. The form is perfectly

valid; logicians call it modus ponens, the way of affirmation. It’s the same

form we used in the valid Fiona argument above. Formally, modus ponens

looks like this:

1. If P, then Q.

2. P.

3. Therefore, Q.

The flaw rendering the gambler’s argument fallacious instead lies in the

content of the first premise – the first premise is simply false. The probability

of the next individual toss (like that of all individual tosses) is and

remains 50–50 no matter what toss or tosses preceded it. But people mistakenly

believe that past flips of coins somehow affect future flips. There’s

no formal problem with the argument, but because this factual error

remains so common and so easy to commit, it has been classified as a fallacy

and given a name. It is a fallacy, but only informally speaking.

Sometimes ordinary speech deviates from these usages. Sometimes any

widely held, though false, belief is described as a fallacy. Don’t worry. As

the philosopher Ludwig Wittgenstein (1889-1951) said, language is like a

large city with lots of different avenues and neighbourhoods. It’s alright to

adopt different usages in different parts of the city. Just keep in mind where

you are.

SEE ALSO

1.5 Invalidity

3.19 Question-begging

3.13 Genetic fallacy

4.5 Conditional/biconditional

READING

★ S. Morris Engel, With Good Reason: An Introduction to Informal Fallacies, 5th edn

(1974)

★ Irving M. Copi, Informal Fallacies (1986)

★ Patrick J. Hurley, A Concise Introduction to Logic, 10th edn (2007)

1.8 Refutation

Samuel Johnson was not impressed by Bishop George Berkeley’s argument

that matter does not exist. In his Life of Johnson (1791) James Boswell

reported that, when discussing Berkeley’s theory with him, Johnson once

kicked a stone with some force and said, ‘I refute it thus.’

Any great person is allowed one moment of idiocy to go public, and

Johnson’s attempt at a refutation must be counted as just such a moment,

because he wildly missed Berkeley’s point. The bishop would never have

denied that one could kick a stone; he denied that stones properly understood

can be conceived to be matter. But Johnson’s refutation also failed

even to be the kind of thing a true refutation is.

To refute an argument is to show that its reasoning is bad. If you, however,

merely register your disagreement with an argument, you are not

refuting it – even though in everyday speech people often talk about refuting

a claim in just this way. So, how can one really refute an argument?

Refutation tools

There are two basic ways of doing this, both of which are covered in more

detail elsewhere in this book. You can show that the argument is invalid: the

conclusion does not follow from the premises as claimed (see 1.5). You can

show that one or more of the premises are false (see 1.4).

A third way is to show that the conclusion must be false and that therefore,

even if you can’t identify what is wrong with the argument, something

must be wrong with it (see 3.25). This last method, however, isn’t strictly

speaking a refutation, as one has failed to show what is wrong with the

argument, only that it must be wrong.

Inadequate justification

Refutations are powerful tools, but it would be rash to conclude that in

order to reject an argument only a refutation will do. You may be justified in

rejecting an argument even if you have not strictly speaking refuted it. You

may not be able to show that a key premise is false, for example, but you

may believe that it is inadequately justified. An argument based on the

premise that ‘there is intelligent life elsewhere in our universe’ would fit this

model. We can’t show that the premise is actually false, but we can argue

that we have both no good reasons for believing it to be true and good

grounds for supposing it to be false. Therefore we can regard any argument

that depends on this premise as dubious and rightly ignore it.

Conceptual problems

More contentiously, you might also reject an argument by arguing that it

utilizes a concept inappropriately. This sort of problem is particularly clear

in cases where a vague concept is used as if it were precise. For instance,

consider the claim that the government is obliged to provide assistance only

to those who do not have enough to live on. But given that there can be no

precise formulation of what ‘enough to live on’ is, any argument must be

inadequate that concludes by making a sharp distinction between those

who have enough and those who don’t. The logic of the argument may be

impeccable and the premises may appear to be true. But if you use vague

concepts in precise arguments you inevitably end up with distortions.

Using the tool

There are many more ways of legitimately objecting to an argument without

actually refuting it. The important thing is to know the clear difference

between refutation and other forms of objection and to be clear what form

of objection you are offering.

SEE ALSO

1.4 Validity and soundness

1.5 Invalidity

3.3 Bivalence and the excluded middle

READING

★ Jamie Whyte, Crimes Against Logic: Exposing the Bogus Arguments of Politicians,

Priests, Journalists and Other Serial Offenders (2005)

★ Theodore Schick, Jr, and Lewis Vaughn, How to Think about Weird Things: Critical

Thinking for a New Age, 5th edn (2007)

★ Julian Baggini, The Duck That Won the Lottery and 99 Other Bad Arguments

(2008)

1.9 Axioms

Obtaining a guaranteed true conclusion in a deductive argument requires

both (1) that the argument be valid, and (2) that the premises be true.

Unfortunately, the procedure for determining whether or not a premise is

true is much less determinate than the procedure for assessing an argument’s

validity.

Defining axioms

Because of this indeterminacy, the concept of an ‘axiom’ becomes a useful

philosophical tool. An axiom is a proposition that acts as a special kind of

premise in a specific kind of rational system. Axiomatic systems were first

formalized by the geometer Euclid (fl. 300 bce) in his famous work the

Elements. In these kinds of systems axioms function as initial claims that

stand in no need of justification – at least from within the system. They are

simply the bedrock of the theoretical system, the basis from which, through

various steps of deductive reasoning, the rest of the system is derived. In

ideal circumstances, an axiom should be such that no rational agent could

possibly object to its use.

Axiomatic vs. natural systems of deduction

It is important to understand, however, that not all conceptual systems are

axiomatic – not even all rational systems. For example, some deductive systems

try simply to replicate and refine the procedures of reasoning that seem

to have unreflectively or naturally developed among humans. This type of

system is called a ‘natural system’ of deduction; it does not posit any axioms

but looks instead for its formulae to the practices of ordinary rationality.

First type of axiom

As we have defined them, axioms would seem to be pretty powerful premises.

Once, however, you consider the types of axiom that there are, their power

seems to be somewhat diminished. One type of axiom comprises premises

that are true by definition. Perhaps because so few great philosophers have

been married, the example of ‘all bachelors are unmarried men’ is usually

offered as the paradigmatic example of this. The problem is that no argument

is going to be able to run very far with such an axiom. The axiom is

purely tautological, that is to say, ‘unmarried men’ merely restates in different

words the meaning that is already contained in ‘bachelor’. (This sort of

proposition is sometimes called – following Immanuel Kant – an ‘analytic’

proposition. See 4.3.) It is thus a spectacularly uninformative sentence

(except to someone who doesn’t know what ‘bachelor’ means) and is therefore

unlikely to help yield informative conclusions in an argument.

Second type of axiom

Another type of axiom is also true by definition, but in a slightly more

interesting way. Many regions of mathematics and geometry rest on their

axioms, and it is only by accepting these basic axioms that more complex

proofs can be constructed within those regions. (You might call these

propositions ‘primitive’ sentences within the system; see 7.7.) For example,

it is an axiom of Euclidean geometry that the shortest distance

between any two points is a straight line. But while axioms like these are

vital in geometry and mathematics, they merely define what is true within

the particular system of geometry or mathematics to which they belong.

Their truth is guaranteed, but only in a limited way – that is, only in the

context within which they are defined. Used in this way, axioms’ acceptability

rises or falls with the acceptability of the theoretical system as a

whole.

Axioms for all?

Some may find the contextual rendering of axiom we’ve given rather unsatisfactory.

Are there not any ‘universal axioms’ that are both secure and

informative in all contexts, for all thinkers, no matter what? Some philosophers

have thought so. The Dutch philosopher Baruch (also known as

Benedictus) Spinoza (1632–77) in his Ethics (1677) attempted to construct

an entire metaphysical system from just a few axioms, axioms that he

believed were virtually identical with God’s thoughts. The problem is that

most would agree that at least some of his axioms seem to be empty, unjustifiable

and parochial assumptions.

For example, one of Spinoza’s axioms states that ‘if there be no determinate

cause it is impossible that an effect should follow’ (Ethics, Bk 1, Pt 1,

axiom 3). But as John Locke (1632–1704) pointed out, this claim is, taken

literally, pretty uninformative since it is true by definition that all effects

have causes. What the axiom seems to imply, however, is a more metaphysical

claim – that all events in the world are effects that necessarily follow

from their causes.

Hume, however, points out that we have no reason to accept this claim

about the world. That is to say, it’s not senseless to hold that an event might

occur without a cause, and we have no reason to believe that events can’t

occur without causes (Treatise, Bk 1, Pt 3, §14). Certainly, by definition, an

effect must have a cause. But for any particular event, we have no reason

to believe it has followed necessarily from some cause. Medieval Islamic

philosopher al-Ghazali (1058–1111) advanced a similar line (The Incoherence

of the Philosophers, ‘On Natural Science’, Question 1ff.).

Of course, Spinoza seems to claim that he has grasped the truth of his axioms

through a special form of intuition (scientia intuitiva), and many philosophers

have held that there are ‘basic’ and ‘self-evident’ truths that may

serve as axioms in our reasoning. (See 7.1.) But why should we believe them?

In many contexts of rationality, therefore, axioms seem to be a useful

device, and axiomatic systems of rationality often serve us well. But the

notion that those axioms can be so secure that no rational person could in

any context deny them seems to be rather dubious.

SEE ALSO

1.1 Arguments, premises and conclusions

1.10 Definitions

1.12 Tautologies, self-contradictions and the law of non-contradiction

7.8 Self-evident truths

READING

★ Euclid, Elements

Al-Ghazali, The Incoherence of the Philosophers

Benedictus Spinoza, Ethics (1677)

1.10 Definitions

If, somewhere, there lie written on tablets of stone the ten philosophical

commandments, you can be sure that numbered among them is the injunction

to ‘define your terms’. In fact, definitions are so important in philosophy

that some have maintained that definitions are ultimately all there is to

the subject.

Definitions are important because without them, it is very easy to argue

at cross-purposes or to commit fallacies involving equivocation. As the

experience of attorneys who questioned former US president Bill Clinton

show, if you are, for example, to interrogate someone about extramarital sex,

you need to define what precisely you mean by ‘sex’. Otherwise, much argument

down the line, you can bet someone will turn around and say, ‘Oh, well,

I wasn’t counting that as sex.’ Much of our language is vague and ambiguous,

but if we are to discuss matters in as precise a way as possible, as philosophy

aims to do, we should remove as much vagueness and ambiguity as possible,

and adequate definitions are the perfect tool for helping us do that.

Free trade example

For example, consider the justice of ‘free trade’. In doing so, you may define

free trade as ‘trade that is not hindered by national or international law’. But

note that with this rendering you have fixed the definition of free trade for the

purposes of your discussion. Others may argue that they have a better, or

alternative, definition of free trade. This may lead them to reach different conclusions

about its justice. You might respond by adopting the new definition,

defending your original definition, or proposing yet another definition. And

so it goes. That’s why setting out definitions for difficult concepts and reflecting

on their implications comprises a great deal of philosophical work.

Again, the reason why it is important to lay out clear definitions for difficult

or contentious concepts is that any conclusions you reach properly

apply only to those concepts (e.g. ‘free trade’) as defined. A clear definition

of how you will use the term thereby both helps and constrains discussion.

It helps discussion because it gives a determinate and non-ambiguous

meaning to the term. It limits discussion because it means that whatever

you conclude does not necessarily apply to other uses of the term. As it

turns out, much disagreement in life results from the disagreeing parties,

without their realizing it, meaning different things by their terms.

Too narrow or too broad?

That’s why it’s important to find a definition that does the right kind of

work. If one’s definition is too narrow or idiosyncratic, it may be that one’s

findings cannot be applied as broadly as could be hoped. For example, if

one defines ‘man’ to mean bearded, human, male adult, one may reach

some rather absurd conclusions – for example, that many Native American

males are not men. A tool for criticism results from understanding this

problem. In order to show that a philosophical position’s use of terms is

inadequate because too narrow, point to a case that ought to be covered by

the definitions it uses but clearly isn’t.

If, on the other hand, a definition is too broad, it may lead to equally erroneous

or misleading conclusions. For example, if you define wrongdoing as

‘inflicting suffering or pain upon another person’ you would have to count

the administering of shots by physicians, the punishment of children and

criminals, and the coaching of athletes as instances of wrongdoing. Another

way, then, of criticizing someone’s position on some philosophical topic is

to indicate a case that fits the definition he or she is using but which should

clearly not be included under it.

A definition is like a property line; it establishes the limits marking those

instances to which it is proper to apply a term and those instances to which

it is not. The ideal definition permits application of the term to just those

cases to which it should apply – and to no others.

A rule of thumb

It is generally better if your definition corresponds as closely as possible to

the way in which the term is ordinarily used in the kinds of debates to which

your claims are pertinent. There will be, however, occasions where it is

appropriate, even necessary, to coin special uses. This would be the case

where the current lexicon is not able to make distinctions that you think are

philosophically important. For example, we do not have a term in ordinary

language that describes a memory that is not necessarily a memory of

something the person having it has experienced. Such a thing would occur,

for example, if I could somehow share your memories: I would have a

memory-type experience, but this would not be of something that I had

actually experienced. To call this a memory would be misleading. For this

reason, philosophers have coined the special term ‘quasi-memory’ (or

q-memory) to refer to these hypothetical memory-like experiences.

A long tradition

Historically many philosophical questions are, in effect, quests for adequate

definitions. What is knowledge? What is beauty? What is the good? Here, it

is not enough just to say, ‘By knowledge I mean …’ Rather, the search is for

a definition that best articulates the concept in question. Much of the philosophical

work along these lines has involved conceptual analysis or the

attempt to unpack and clarify the meanings of important concepts. What is

to count as the best articulation, however, requires a great deal of debate.

Indeed, it is a viable philosophical question as to whether such concepts

actually can be defined. For many ancient and medieval thinkers (like Plato

and Aquinas), formulating adequate definitions meant giving verbal expression

to the very ‘essences’ of things – essences that exist independently of us.

Many more recent thinkers (like some pragmatists and post-structuralists)

have held that definitions are nothing more than conceptual instruments

that organize our interactions with each other and the world, but in no way

reflect the nature of an independent reality.

Some thinkers have gone so far as to argue that all philosophical puzzles

are essentially rooted in a failure to understand how ordinary language

functions. While, to be accurate, this involves attending to more than just

definitions, it does show just how deep the philosophical preoccupation

with getting the language right runs.

SEE ALSO

1.9 Axioms

3.4 Category mistakes

3.9 Criteria

READING

★ Plato (c.428–347 bce), Meno, Euthyphro, Theaetetus, Symposium

J. L. Austin, Sense and Sensibilia (1962)

Michel Foucault, The Order of Things (1966)

1.11 Certainty and probability

Seventeenth-century French philosopher René Descartes (1596–1650) is

famous for claiming he had discovered the bedrock upon which to build a

new science that could determine truths with absolute certainty. The bedrock

was an idea that could not be doubted, the cogito (‘I think’) – or, more

expansively, as he put it in Part I, §7 of the 1644 Principles of Philosophy,

‘I think therefore I am’ (‘cogito ergo sum’). Descartes reasoned that it is

im possible to doubt that you are thinking, for even if you’re in error or being

deceived or doubting, you are nevertheless thinking; and if you are thinking,

you exist.

Ancient Stoics like Cleanthes (c.331–c.232 bce) and Chrysippus (c.280–c.207

bce) maintained that there are certain experiences of the world and of morality

that we simply cannot doubt – experiences they called ‘cataleptic impressions’.

Later philosophers like the eighteenth century’s Thomas Reid (1710–96)

believed that God guarantees the veracity of our cognitive faculties. His contemporary

Giambattista Vico (1688–1744) reasoned that we can be certain

about things human but not about the non-human world. More recently the

Austrian philosopher Ludwig Wittgenstein (1889–1951) tried to show how it

simply makes no sense to say that one doubts certain things.

Others have come to suspect that there may be little or nothing we can

know with certainty and yet concede that we can still figure things out with

some degree of probability. Before, however, you go about claiming to have

certainly or probably discovered philosophical truth, it will be a good idea

to give some thought to what each concept means.

Types of certainty

‘Certainty’ is often described as a kind of feeling or mental state (perhaps as

a state in which the mind believes some X without any doubt at all), but

doing so simply renders a psychological account of the concept. It fails to

define when we are warranted in feeling this way. A more philosophical

account of certainty would add the claim that a proposition is certainly true

when it is impossible for it to be false – and certainly false when it is impossible

for it to be true. Sometimes propositions that are certain in this way

are called ‘necessarily true’ and ‘necessarily false’.

The sceptical problem

The main problem, philosophically speaking, thinkers face is in establishing

that it is in fact impossible for any candidate for certainty to have a different

truth value. Sceptical thinkers have been extremely skilful in showing how

9781405190183_4_001.indd 35 1/29/2010 5:38:51 PM

36 BASIC TOOLS FOR ARGUMENT

virtually any claim might possibly be false even though it appears to be true

(or possibly true though it appears to be false). In the wake of sceptical

scrutiny, most would agree that absolute certainty in advancing truth claims

remains unattainable. Moreover, even if achieving this sort of certainty

were possible, while it may be that all that’s philosophically certain is true,

clearly not all that’s true is certain.

But if you can’t have demonstrable certainty, what is the next best thing?

To give a proper answer to this question would require a much larger study

of the theory of knowledge. But for the sake of our concerns here, consider

the answer that’s most commonly advanced: probability.

Probability is the natural place to retreat to if certainty is not attainable.

As a refuge, however, it is rather like the house of sticks the pig flees to from

his house of straw. The problem is that probability is a precise notion that

cannot always be assumed to be the next best thing to certainty.

Objective and subjective probability

We can distinguish between objective and subjective probability. Objective

probability is where what will happen is genuinely indeterminate.

Radioactive decay could be one example. For any given radioactive atom,

the probability of it decaying over the period of its half-life is 50–50. This

means that, if you were to take ten such atoms, it is likely that five will decay

over the period of the element’s half-life and five will not decay. On at least

some interpretations, it is genuinely indeterminate which atoms will fall

into which category.

Subjective probability, on the other hand, refers to cases where there may

be no actual indeterminacy, but some particular mind or set of minds

makes a probability judgement about the likelihood of some event. These

subjects do so because they lack complete information about the causes

that will determine the event. Their ignorance requires them to make a

probabilistic assessment, usually by assigning a probability based on the

number of occurrences of each outcome over a long sequence in the past.

So, for example, if we toss a coin, cover it and ask you to bet on heads or

tails, the outcome has already been determined. Since you don’t know what

it is, you have to use your knowledge that heads and tails over the long run

fall 50–50 to assign a 50 per cent probability that it is a head and a 50 per cent

probability that it is a tail. If you could see the coin, you would know that, in

fact, it is 100 per cent certain that it is whichever side is facing up.

The odds set by gamblers and handicappers at horse races are also species

of subjective probability. The posted odds record simply what the many

people betting on the race subjectively believe about the outcome.

Certainty and validity

If you have a sound deductive argument, then its conclusion is often said to

follow from the premises with certainty. Many inquirers, however, demand

not only that conclusions follow with certainty but that the conclusions

themselves be true. Consider the difference between the following arguments:

1. If it rained last night, England will probably win the match.

2. It rained last night.

3. Therefore, England will probably win the match.

1. All humans are mortal.

2. Socrates was a human.

3. Therefore, Socrates was mortal.

The conclusion of the first argument clearly enters only a probable claim.

The conclusion of the second argument in contrast to the first, enters a

much more definite claim. But here’s the rub: both examples present valid

deductive arguments. Both arguments possess valid forms. Therefore in

both arguments the conclusion follows with certainty – i.e. the truth of the

premises guarantees the truth of the conclusion – even though the content

of one conclusion is merely probable while that of the other is not.

You must therefore distinguish between (1) whether or not the conclusion

of an argument follows from the premises with certainty or some probability,

and (2) whether or not the conclusion of an argument advances a

statement the content of which concerns matters of probability.

Philosophical theories

But what about philosophical theories? It would seem that if certainty in

philosophical theories were attainable, there would be little or no dispute

among competent philosophers about which are true and which false – but,

in fact, there seems to be a lot of dispute. Does this mean that the truth of

philosophical theories is essentially indeterminate?

Some philosophers would say no. For example, they would say that

although there remains a great deal of dispute, there is near unanimous

agreement among philosophers on many things – for example, that Plato’s

theory of forms is false and that Cartesian mind–body dualism is untenable.

Others of a more sceptical bent are, if you’ll pardon the pun, not so certain

about the extent to which anything has been proven, at least with certainty,

in philosophy. Accepting a lack of certainty can from their point of

view be seen as a matter of philosophical maturity.

SEE ALSO

1.1 Arguments, premises and conclusions

1.2 Deduction

1.4 Validity and soundness

1.5 Invalidity

1.9 Axioms

READING

Giambattista Vico, Scienza nuova (1725)

Ludwig Wittgenstein, On Certainty (1969)

★ Brad Inwood and Lloyd P. Gerson, Hellenistic Philosophy: Introductory Readings,

2nd edn (1988)

1.12 Tautologies, self-contradictions and

the law of non-contradiction

Tautology and self-contradiction fall at opposite ends of a spectrum: the

former is a sentence that’s necessarily true and the latter a sentence that’s

necessarily false. Despite being in this sense poles apart, they are actually

intimately related.

In common parlance, ‘tautology’ is a pejorative term used to deride a

claim because it purports to be informative but in fact simply repeats the

meaning of something already understood. For example, consider: ‘A criminal

has broken the law.’ This statement might be mocked as a tautology

since it tells us nothing about the criminal to say he has broken the law. To

be a lawbreaker is precisely what it means to be a criminal.

In logic, however, ‘tautology’ has a more precisely defined meaning. A

tautology is a statement in logic such that it will turn out to be true in every

circumstance – or, as some say, in every possible world. Tautologies are in

this sense ‘necessary’ truths.

Take, for example:

P or not-P.

If P is true the statement turns out to be true. But if P is false, the statement

still turns out to be true. This is the case for whatever one substitutes for P:

‘today is Monday’, ‘atoms are invisible’ or ‘monkeys make great lasagna’. One

can see why tautologies are so poorly regarded. A statement that is true

regardless of the truth or falsehood of its components can be considered to

be empty; its content does no work.

This is not to say that tautologies are without philosophical value.

Understanding tautologies helps one to understand the nature and function

of reason and language.

Valid arguments as tautologies

As it turns out, all valid arguments can be restated as tautologies – that is, hypothetical

statements in which the antecedent is the conjunction of the premises

and the consequent the conclusion. In other words, every valid argument may

be articulated as a statement of this form: ‘If W, X, Y are true, then C is true’,

where W, X and Y are the argument’s premises and C is its conclusion. When

any valid argument is substituted into this form, a tautology results.

Law of non-contradiction

In addition, the law of non-contradiction – a cornerstone of philosophical

logic – is also a tautology. The law may be formulated this way.

Not (P and not-P).

The law is a tautology since, whether P is true or false, the complete

statement will turn out to be true.

The law of non-contradiction can hardly be said to be uninformative,

since it is the foundation upon which all logic is built. But, in fact, it is not

the law itself that’s informative so much as any attempt to break it.

Attempts to break the law of non-contradiction are themselves contradictions,

and they are obviously and in all circumstances wrong. A contradiction

flouts the law of non-contradiction, since to be caught in a

contradiction is to be caught asserting both that something is true and

something is false in the same sense and at the same time – asserting both P

and not-P. Given that the law of non-contradiction is a tautology, and thus

in all circumstances true, there can be nothing more clearly false than something

that attempts to break it.

The principle of non-contradiction has also been historically important

in philosophy. The principle underwrote ancient analyses of change and

plurality and is crucial to Parmenides of Elea’s sixth-century bce proclamation

that ‘what is is and cannot not be’. It also seems central to considerations

of identity – for example in Leibniz’s claim that objects that are

identical must have all the same properties.

Self-refuting criticism

One curious and useful feature of the law of non-contradiction is that any

attempt to refute it presupposes it. To argue that the law of non-contradiction

is false is to imply that it is not also true. In other words, the critic presupposes

that what he or she is criticizing can be either true or false but not both

true and false. But this presupposition is just the law of non- contradiction

itself – the same law the critic aims to refute. In other words, anyone who

denies the principle of non-contradiction simultaneously affirms it. It is a

principle that cannot be rationally criticized, because it is presupposed by

all rationality.

To understand why a tautology is necessarily, and in a sense at least, uninformatively

true and why a self-contradiction is necessarily false is to understand

the most basic principle of logic. The law of non-contradiction is

where those two concepts meet and so is perhaps best described as the keystone,

rather than cornerstone, of philosophical logic.

SEE ALSO

1.4 Validity and soundness

1.6 Consistency

3.24 Self-defeating arguments

5.6 Leibniz’s law of identity

7.5 Paradoxes

READING

Aristotle, Interpretation, esp. Chs 6–9

Aristotle, Posterior Analytics, Bk 1, Ch. 11:10

★ Patrick J. Hurley, A Concise Introduction to Logic, 10th edn (2007)

===

continued in ...

...