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Logical Arguments and Counterexamples: Understanding Forms and Validity, Study notes of Reasoning

Texas A&M University (A&M)Reasoning

The concept of logical arguments and counterexamples through various examples. It introduces the idea of argument forms, substitution instances, and counterexamples, and demonstrates how to identify and evaluate the validity of arguments using these concepts. The document also discusses the limitations of the method of counterexamples.

Typology: Study notes

Pre 2010

Uploaded on 02/10/2009

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1.2 Forms and Counterexamples
Consider the following arguments:
1. All oaks are trees.
2. All trees are plants.
So, all oaks are plants.
1. All monauli are flageolets.
2. All flageolets are fipple-flutes.
So, all monauli are fipple-flutes.
These arguments have the same form, namely:
Form 1
1. All A are B.
2. All B are C.
So, all A are C.
‘A’, ‘B’, and ‘C’ here stand for terms:
Definition: Aterm is a word or phrase that stands for a
class, i.e., a collection or set of things.
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1.2 Forms and Counterexamples

Consider the following arguments:

  1. All oaks are trees.
  2. All trees are plants. So, all oaks are plants.
  3. All monauli are flageolets.
  4. All flageolets are fipple-flutes. So, all monauli are fipple-flutes.

These arguments have the same form , namely:

Form 1

  1. All A are B.
  2. All B are C. So, all A are C.

‘A’, ‘B’, and ‘C’ here stand for terms :

Definition: A term is a word or phrase that stands for a

class , i.e., a collection or set of things.

General diagram of the logic of this argument form:

C (Plants)

B (Trees)

A (Oaks)

Another valid form:

  1. All emeralds are gems.
  2. Some rocks are not gems. So, some rocks are not emeralds.
  3. All collies are dogs.
  4. Some animals are not dogs. So, some animals are not collies.

Definition (sorta): An argument form is a pattern of

reasoning.

Definition: An argument that results from uniformly re-

placing letters in an argument form with terms (or state-

ments, when appropriate) is called a substitution in-

stance of that form.

Comment: The two arguments preceding the form in each case above are substitution instances of the respective forms.

Definition: A counterexample to an argument form is

a substitution instance whose premises are well-known

truths and whose conclusion is a well-known falsehood.

Form 3

  1. All A are B.
  2. All C are B. So, all A are C.

A Counterexample to Form 3

  1. All birds are animals.
  2. All dogs are animals. So, all birds are dogs.

Here is a diagram for the counterexample:

B

C

A

Showing invalidity by counterexample

1. Identify the form of the argument

2. If the validity of the argument is suspect, attempt to produce

a substitution instance of the argument form in which the

premises are obviously true and the conclusion obviously false.

3. Conclude that the argument is invalid.

Form of the argument:

  1. No A are B.
  2. All B are C. So, no A are C.

Counterexample (2 steps): Begin with an obviously false conclu-

sion and work backwards — use simple, well understood concepts,

e.g., biological kinds:

  1. No dogs are B.
  2. All B are animals. So, no dogs are animals.

Notice we start with our obviously false conclusion and fill in for the

terms A and C. Now all we need to do is find an appropriate term

for B to complete our counterexample:

  1. No dogs are cats.
  2. All cats are animals. So, no dogs are animals.

Limitations of the method of Counterexamples

1. The method cannot show that a valid form is valid.

Finding a substitution instance with true premises

and a true conclusion does not show validity!

2. Inability to find a counterexample of itself does not

establish validity.

Perhaps we are just not being clever enough.

A final complication

Arguments can, and typically do, have more than one

form.

Example

  1. All determinists are fatalists.
  2. All fatalists are unhappy. So, all determinists are unhappy.

In addition to a valid form, this has the invalid form “A; B;

so C”. But note: an argument is valid if any of its forms

is valid!