William

The counterexample method

• A method to show that an argument is invalid.

• There is a need to show that there exists a possible situation in which all premises are true, the conclusion is false.

What happens if we do not find such a situation: If there is no situation where the above applies, the argument is valid.

However, remember that it could be that we ”didn’t look hard enough”

How can we show that no such derivation wherein an argument is valid doesn’t exist?

So come up with counterexamples!

• Not a complete solution to the subjectivity problem, but will at least be practically useful for us as philosopher.

Truth preservation perspective (the samantic perspective)

Valid: All situations in which the premises are true… (All)

Invalid: There exists a situation … (There exists)

Derivation perspective (syntactical perspective)

Valid: There exists a derivation… (There exists)

Invalid: There does not exist a derivation… (All)

Ideally a good argument is valid if they are valid in both of these senses.

There is an assymetry between the two perspectives.

The assymetry, however, feeds into each other quite nicely, they are complementary in their assymetry.

eg.

Some logicians are Belgians (T)

Lorenz is a logician (T)

so Lorenz is a Belgian (T)

:

Invalid

The form is invalid because if we make the subjects into variables, we can find situations in which the same kind of argument is invalid.

We move from an argument in which we cannot exploit the dependence of validity from truth to one in which we can. Ie. We retain the form of the argument, but change the subjects of it.

Most X are Y

Most Y are Z

So there exist some X that are Y

:

Invalid

Eg.

Most KU leuven students live in Belgium

Most people who live in Belgium, do not study at KU Leuven

so there exist KU leuven students who do not study at KU Leuven

When making a counterexample, we have to replace the words in the argument.

However, we can also find a different situation using the same words as the argument, in which we find that an argument has true premises and a false conclusion. (this uses the definition of validity in order to show that the argument is invalid)

You never need to both change the situation and the words.

Disadvantages to using this method:

• Coming up with a possible counterexample requires imagination (Subjectivity problem)

• You have to be very careful with argument forms for the invalidity of the counter-example’s truth value to travel to the original example.

Advantages:

• Possible situations are murky (what can coherently be imagined?)

• All you have to think about is words

• De facto truth is completely clear

Invalid argument-forms can however have valid instances.

So be careful using the counter-example method.

Eg. there are instances of

Some F are G

n is F

so n is G

that are valid, such as

some logicians are Belgians, Lorenz is a logician, so Lorenz is a logician.

When searching for an invalid instantiation of argument A

find an argument form F such that

• A is an instance of F

• F needs to be valid in order for A to be valid (It cannot be the case that A is valid yet F is invalid).

Find an argument C such that

• C is an instance of F (A and C have the same form, F).

• C has true premises and a false conclusion (otherwise it is not a counterexample).

Suppose argument A:

some logicians are Belgians, Lorenz is a logician, so Lorenz is a logician.

We find argument form F:

Some F are G

n is F

so n is G

and so we take on the argument:

Some animals are zebras, Garfield is an animal, so Garfield is a zebra

but it can go wrong because you have to show whether the initial argument is valid or invalid, in order to know the validity of the form of the argument. Thus, whatever you are showing is circular, because you deduce that it is invalid because of a ”leap of faith” regarding the general argument form.

Logical Validity

Generally what we will be studying in this course.

Logic is about truth-preservation, not about truth.

No dogs are chickens, so no chickens are dogs

No logicians are irrational, so no irrational people are logicians

• Both of these arguments are deductively valid. (However, on purely formal grounds)

Argument form: no F are G, so no G are F

Ann is a widow, therefore Ann is a woman

Sophie is a widow, so Sophie is a woman

• Both of these are deductively valid

But really, the fact that these arguments are valid should have nothing to do with the content of the arguments.

Argument form 1

n is F

so n is G

Invalid: Lorenz is a logician, so Lorenz is silly

Argument form 2

n is a widow, so n is a woman

• Valid

• All its instances are valid.

• But not on formal grounds

• The necessary truth preservation is based on the meaning of ’widow’

• Has nothing to do with what we replace the variables with however.

Deductive validity = necessary truth-preservation (no further questions asked)

So in deductive validity, we can form logical validity:

Necessary truth-preservation on purely formal grounds

ie. it doesn not rely on the meaning of words.

An argument is logically valid if and only if, it is necessarily truth-preserving in virtue of the topic-neutral words that occur in its premises and conclusion.

If an argument is logically valid, it is also deductivaly valid. However, an argument does not have to be logically valid if it is deductively valid.

Logical necessity: true in all logically possible situations on purely formal grounds.

Deductive necessity (not standard terminology): true in all logically possible situations.

(It is what we earlier called logical necessity, but we have made logical necesity more strictly defined.)

Every proposition that is logically necessary, is also deductively necessary.

Eg. It is raining or it is not raining

P or -P

logically necessary

All P are P

Logically necessary

2 +2 = 4

it is controversial whether this is actually logically necessary.