Problem 2 - A Gentle Introduction to the Art of Mathematics

Question

Below is a rule of inference that we call extended elimination. $$ \begin{aligned} &(A \vee B) \vee C \\ & \neg A \\ & \neg B \\ \hline \therefore C & \end{aligned} $$ Use a truth table to verify that this rule is valid.

Step-by-Step Solution

Answer

The rule is valid because whenever \((A \vee B) \vee C\), \(!A\), and \(!B\) are true, C is always true in the truth table.

Step 1 - List Possible Truth Values

For the propositions A, B, and C, list all possible combinations of truth values. There are 8 possible combinations: (A=true, B=true, C=true), (A=true, B=true, C=false), (A=true, B=false, C=true), (A=true, B=false, C=false), (A=false, B=true, C=true), (A=false, B=true, C=false), (A=false, B=false, C=true), (A=false, B=false, C=false).

Step 2 - Evaluate Disjunctions

Determine the truth value of \((A \vee B) \vee C\) for each combination. For a disjunction (OR), the result is true if at least one operand is true, otherwise it is false.

Step 3 - Evaluate Negations

Calculate the truth values of \(\eg A\) and \(!B\) for each combination of A and B. A negation is true if the operand is false, and false if the operand is true.

Step 4 - Analyze the Conclusion

Check whether C is true whenever \((A \vee B) \vee C\) is true, \(!A\) and \(!B\) are true for each row in the truth table. Track the rows where all premises are true.

Step 5 - Construct the Truth Table

Construct the truth table including columns for A, B, C, \((A \vee B) \vee C\), \(!A\), and \(!B\). Verify that in all the rows where \((A \vee B) \vee C\), \(!A\), and \(!B\) are true, C must also be true.

Key Concepts

logical propositions

Logical propositions are statements that can either be true or false. In logic, these propositions are often represented by variables like A, B, or C.
For example, the statement 'A = It is raining' can either be true or false depending on the weather.
Logical propositions are the building blocks for more complex logical expressions and rules.
Important points to remember:

• A proposition has a clear truth value, either true or false.
• Each variable (like A, B, or C) represents an individual proposition.

Understanding these basic components is crucial for building and analyzing more complex logical structures, as seen in the extended elimination rule.

disjunctions

Disjunctions are logical operations that represent 'or' statements. In formal logic, the disjunction of two propositions A and B is written as (A ∨ B).
This statement is true if at least one of the propositions is true.
For example, if A is 'I will study' and B is 'I will play games,' then the disjunction statement 'I will study or I will play games' is true if either A, B, or both are true.
Key points about disjunctions:

• A disjunction is true if either one (or both) of the components are true.
• If both components are false, then the disjunction is false.

In the extended elimination rule, disjunctions are used to combine multiple conditions, making it easier to evaluate complex logical propositions.

negations

A negation in logic is used to flip the truth value of a proposition. It is denoted by the symbol ¬ before the proposition.
For example, if A represents 'It is sunny,' then ¬A represents 'It is not sunny.'
The truth value of ¬A is opposite to the truth value of A. If A is true, then ¬A is false, and vice versa.
Important concepts about negations:

• Negation changes true to false and false to true.
• It is a unary operation, meaning it only affects one proposition at a time.

In the given exercise, understanding negation is crucial for analyzing the combined logical statement and proving that the rule is valid using a truth table.