Thinking Skills: Critical Thinking and Problem Solving

7.2 Soundness and validity: a taste of logic 257

that express them. As we have seen, symbols,

diagrams and formulas can be used to show

the form of an argument. The symbols may

stand for individuals, like Socrates; or classes

of things, like birds. Or they can stand for

whole sentences, like ‘Parrots can fly,’ or

‘Whales are not fish.’

Another way to reach the conclusion of

[5], for example, is as follows:

[8] If sharks are fish, they have gills. Sharks

are fish, so they do have gills.

As we saw in the last chapter, the form of

conditional sentences can be given by

replacing each of the simple sentences with a

letter. If the letter f stands for ‘sharks are fish’,

and g for ‘sharks have gills’, [8] can be written:

If f then g

f

g

This form of argument is always valid,

whatever sentences you substitute for f

and g (or whatever symbols you use). In fact,

[8] is so obviously valid that it hardly

needs saying.

Here is another simple but valid argument.

It has the same first premise as before, but this

time the second premise is a denial of g –

written ‘Not-g’ – and the conclusion is ‘Not-f’.

[9] If f then g

Not-g

Not-f

The validity of [9] is not quite as obvious as [8],

but it is a valid form of argument. For example:

[9a] If (f) whales are fish, then (g) whales

have gills. Whales do not have gills

(Not-g), so whales are not fish (Not-f).

Or in more natural language:

[9b] If whales were fish they’d have gills; but

they don’t, so they’re not.

Activity

Discuss why [7] is not valid. What is wrong

with it, and how is it different from [6]?

Commentary
In [6] we are told Socrates is a man and that all
men are mortals. That tells us that Socrates is
also mortal. In [7] we are again told that all
men are mortal, and that Socrates is mortal
too. But that would not tell us that Socrates is a
man, if we did not already know it. There are
many other classes of mortals besides men:
women, children, parrots and penguins, to
name just a few. Therefore the premises in [7]
do not themselves establish that the Socrates
referred to in the argument is a man. (If we
didn’t know differently ‘Socrates’ could be the
name of a parrot.)
Here is a Euler diagram showing the
invalidity of [7]:

men

mortalsparrots

Socrates

The diagram shows that men, Socrates and
parrots are all mortal, but does not establish
that all individuals called ‘Socrates’ are men
(or parrots). As it happens Socrates was a man,
so the conclusion is true; but it does not follow
from the reasons.

Formal logic
Because logicians are primarily concerned
with different forms of argument, they are less
concerned with the meanings of the sentences