Problem 2 - A Gentle Introduction to the Art of Mathematics

Question

In constructive dilemma, the antecedent of the conditional sentences are usually chosen to represent opposite alternatives. This allows us to introduce their disjunction as a tautology. Consider the following proof that there is never any reason to worry (found on the walls of an Irish pub). Either you are sick or you are well. If you are well there's nothing to worry about. If you are sick there are just two possibilities: Either you will get better or you will die. If you are going to get better there's nothing to worry about. If you are going to die there are just two possibilities: Either you will go to Heaven or to Hell. If you go to Heaven there is nothing to worry about. If you go to Hell, you'll be so busy shaking hands with all your friends there won't be time to worry \(\ldots\) Identify the three tautologies that are introduced in this "proof."

Step-by-Step Solution

Answer

Three tautologies are: 'Either you are sick or you are well,' 'Either you will get better or you will die,' and 'Either you will go to Heaven or to Hell.'

Step 1: Understand the Constructive Dilemma

A constructive dilemma involves two conditional statements and a statement of their disjunction. The conclusion follows from the disjunction.

Step 2: Identify the Conditional Statements

The first set of conditional statements are: 'If you are well there's nothing to worry about,' and 'If you are sick there are just two possibilities.'

Step 3: Form the First Tautology

Disjunction of the antecedents: 'Either you are sick or you are well.' This is the first tautology.

Step 4: Identify Conditional Statements for Sickness

The second set of conditional statements if you are sick are: 'If you are going to get better there's nothing to worry about,' and 'If you are going to die there are just two possibilities.'

Step 5: Form the Second Tautology

Disjunction of the antecedents: 'Either you will get better or you will die.' This is the second tautology.

Step 6: Identify Conditional Statements for Death

The third set of conditional statements if you die are: 'If you go to Heaven there is nothing to worry about,' and 'If you go to Hell, you will be busy shaking hands with friends.'

Step 7: Form the Third Tautology

Disjunction of the antecedents: 'Either you will go to Heaven or to Hell.' This is the third tautology.

Key Concepts

Tautologies in Logic

A tautology in logic is a statement that is always true, no matter what the truth values of its components are. In simple terms, it’s like saying, 'It will either rain tomorrow, or it will not rain tomorrow,' which is invariably true. When we deal with constructive dilemmas, introducing tautologies helps simplify and guarantee the truthfulness of the arguments.

In the given problem, we have three main tautologies:

• 'Either you are sick or you are well.'
• 'Either you will get better or you will die.'
• 'Either you will go to Heaven or to Hell.'

These statements are tautologies because they cover all possibilities. There’s no situation where these statements would be false. This property is what makes them powerful tools in logical arguments, especially for constructing valid proofs.

Conditional Statements

Conditional statements, often written in the 'if-then' format, are foundational in logic. They link an antecedent (initial condition) to a consequent (result). For example, 'If you are well, there's nothing to worry about.' Here, the well-being is the antecedent, and the lack of worry is the consequent.

Understanding these statements in the context of the dilemma:

• If you are well (antecedent), then there's nothing to worry about (consequent).
• If you are sick (antecedent), then you will either get better or die (consequent).
• If you get better (antecedent), then there's nothing to worry about (consequent).
• If you die (antecedent), then you will either go to Heaven or Hell (consequent).
• If you go to Heaven (antecedent), then there's nothing to worry about (consequent).
• If you go to Hell (antecedent), then you will be busy shaking hands with friends (consequent).

Understanding how these conditions follow one another helps in constructing the overall logical dilemma, ensuring that the chain of reasoning remains intact and valid.

Disjunction in Logic

Disjunction in logic refers to a situation where at least one of the propositions in a compound statement is true. It's represented by the keyword 'or.' For instance, 'Either you are sick or you are well.' This means that at least one of these conditions must be true.

In our example, disjunction helps in combining different outcomes to form a comprehensive logical statement. Here are the disjunctions used:

• 'Either you are sick or you are well.'
• 'Either you will get better or you will die.'
• 'Either you will go to Heaven or to Hell.'

These disjunctions ensure that all possible scenarios are covered, leading to a valid logical outcome.

The use of disjunctions simplifies complex logical arguments by ensuring that all cases are accounted for. This helps in drawing conclusions without ambiguity and strengthens the logical structure of the proof.