Think: A Compelling Introduction to Philosophy (27 page)

Obviously the attitude one takes to the'fideism'that simply lets
particular religious beliefs walk free from reason may depend
heavily on what has recently been happening when they do so.
Hume was horn less than twenty years after the last legal religious
executions in Britain, and himself suffered from the enthusiastic
hostility of believers. If in our time and place all we see are church
picnics and charities, we will not he so worried. But enough people
come down the mountain carrying their own practical certainties
to suggest that we ought to be.

Maybe some day something will be found that answers to the
needs without pandering to the bad desires, but human history
suggests that it would be unwise to bank on it.

THIS CHAPTER GIVES us an acquaintance with some basic categories to use when we think about reasoning. We want our reasonings to be good. We want to follow reliable methods for sifting
truth from falsehood, and forming beliefs about our world. But
which are these reliable methods, and what are their credentials? In
this chapter we take a very brief glance at formal logic, and then we
cone upon the problems of inductive reasoning, and some of the
elements of scientific reasoning.

A LITTLE LOGIC

The working parts of an argument are, first, its premises. These are
the starting point, or what is accepted or assumed, so far as the argument is concerned. An argument can have one premise, or several. From the premises an argument derives a conclusion. If we are reflecting on the argument, perhaps because we are reluctant to accept the conclusion, we have two options. First, we might reject one
or more of the premises. But second, we might reject the way the
conclusion is drawn from the premises. The first reaction is that
one of the premises is untrue. The second is that the reasoning is invalid. Of course, an argument may be subject to both criticisms: its
premises are untrue, and the reasoning from them is invalid. But
the two criticisms are distinct (and the two words, untrue and invalid, are well kept for the distinction).

In everyday life, arguments are criticized on other grounds
again. The premises may not be very sensible. It is silly to make an
intricate argument from the premise that I will win next week's lottery, if it hasn't a dog's chance of happening. It is often inappropriate to help ourselves to premises that are themselves controversial.
It is tactless and tasteless in some circumstances to argue some
things. But `logical' is not a synonym for `sensible'. Logic is interested in whether arguments are valid, not in whether it is sensible
to put them forward. Conversely, many people called `illogical'
may actually be propounding valid arguments, but be dotty in
other ways.

Logic has only one concern. It is concerned whether there is no
way that the premises could be true without the conclusion being
true.

It was Aristotle (384-322 Bc) who first tried to give a systematic
taxonomy of valid and invalid arguments. Aristotle realized that
any kind of theory would need to classify arguments by the patterns of reasoning they exhibit, or what is called their form. One of
the most famous forms of argument, for instance, rejoicing in the title `modus ponendo ponens, or modus ponens for short, just
goes:

P;

If p then q;

So, q.

Here p and q stand for any piece of information, or proposition,
that you like. The form of the argument would remain the same
whether you were talking of cows or philosophers. Logic then
studies forms of information, not particular examples of it. Particular arguments are instances of the forms, but the logician is interested in the form or structure, just as a mathematician is interested
in numerical forms and structure, but not interested in whether
you are counting bananas or profits.

We want our reasonings to be valid. We said what this means: we
want there to be no way that our conclusion could be false, if our
premises are true. So we need to study whether there is `any way'
that one set of things, the premises, can be true without another
thing, the conclusion, also being true. To investigate this we need to
produce a science of the ways things can be true. For some very
simple ways of building up information, we can do this.

TRUTH-TABLES

The classical assumptions are first that every proposition (p, q ...)
has just one of two truth-values. It must be either true or false, and
it cannot be both. ('But suppose I don't grant that?' Patience.) The second assumption is that the terms the logic is dealing with-
centrally,'and','not,'or, and 'If ... then ..: -can be characterized
in terms of what they do to truth-values. ('But suppose I don't
grant that?' Patience, again.)

Thus, consider 'not-pt Not-p, which is often written -'p, is the
denial or negation of p: it is what you say when you disagree with p.
Whatever it is talking about, p, according to our first assumption,
is either true (T), or false (F). It is not both. What does'not' do? It
simply reverses truth-value. If pis true, then -p is false. If pis false,
then -'p is true. That is what'not' does. We can summarize the result as a truth-table:

The table gives the result, in terms of truth or falsity, for each assignment of truth-value to the components (such an assignment is
called an interpretation). A similar table can be written for 'and',
only here there are more combinations to consider. We suppose
that 'and' conjoins two propositions, each of which can he true or
false. So there are four situations or interpretations to consider:

We are here given the truth-value for the overall combination, the conjunction, as a function of the combination of truth-values of
the components: the four different interpretations of the formula.

The fact that we can give these tables is summed up by saying
that conjunction, and negation, are truth-junctional, or that they
are truth-functional operators. Elementary propositional logic
studies the truth-functions. Besides 'not' and'and, they include'or'
(p or q, regarded as true except when both p and q are false); and a
version of'If p then q', regarded as true except in the case where p is
true yet q false. If we write this latter as'p - q, its truth-table is:

These are also called Boolean operators. People familiar with databases and spreadsheets will know about Boolean searches, which
implement exactly the same idea. A search for widgets over five
years old held in the warehouse in York returns a hit when it finds
a widget meeting both conditions. A search for customers not paid
up on i December returns just the reverse hits from a search for
customers paid up on i December. A search for customers who either bought a washing machine or a lawnmower turns up those
who bought one and those who bought the other.

We can now see a rationale for some rules of inference. Consider
the rule that from' p & q' we can derive p (or equally q). You cannot
thereby get from truth to falsity, because the only interpretation
(the top line) that has `p & q' true also has each ingredient true. So this is a good rule. We can also see why modus ponendo ponens, introduced above, is a good rule. It has two premises, `p', and `If p
then q'. Can we find an interpretation (a'way') in which both these
are true without q being true? No. Because given that p is true, the
only interpretation of p - q that allows it to be true also displays q
as true.

There are some interesting animals in this jungle. One is that of
a contradiction. Consider this formula: