Problem 23 - Thinking Mathematically

Question

Let \(p\) and \(q\) represent the following simple statements: \(p: Y o u\) are human. q: You have feathers. Write each compound statement in symbolic form. Being human is sufficient for not having feathers.

Step-by-Step Solution

Answer

The symbolic form of the compound statement 'Being human is sufficient for not having feathers' is \(p -> ~q\).

Step 1: Understanding the Terms

We are given two simple statements represented by \(p\) and \(q\). \(p\) represents 'You are human' and \(q\) represents 'You have feathers'. The compound statement that needs to be written in symbolic form is 'Being human is sufficient for not having feathers'.

Step 2: Identify the Logical Connective

'Sufficient for' is a logical connective that represents a conditional statement in logic. It can be symbolized by an arrow (->). 'Not' negates the truth value of the statement it's applied to and in symbolic form it's represented by '~' sign.

Step 3: Write the Compound Statement in Symbolic Form

The statement can be written symbolically as 'If \(p\) then not \(q\)'. Symbolically, this is represented as \(p -> ~q\).

Key Concepts

Compound Statement

In symbolic logic, a compound statement is a combination of two or more simple statements. These can be linked using logical connectives. For example, let's consider the statements:

• Statement 1: "You are human." (Symbolized by \(p\))
• Statement 2: "You have feathers." (Symbolized by \(q\))

When these simple statements are combined, they form a compound statement. In our exercise, the compounded form becomes, "Being human is sufficient for not having feathers." Each simple statement plays a role in forming the overall meaning of the compound statement. Understanding the components is crucial for linking them suitably with logical connectives.

Logical Connective

Logical connectives are symbols or words used to connect simple statements to form a compound statement. They are essential in the field of symbolic logic for building complex logical expressions. In the given exercise, the logical connective is 'sufficient for'.

• This type of connective is used to indicate a conditional relationship between the two statements.
• The word 'sufficient for' is translated into a conditional symbol \(\rightarrow\) in logic.

By using connectives like \(\rightarrow\), \(\land\) (and), \(\lor\) (or), and \(\lnot\) (not), individuals can construct various logical statements. Connectives help in understanding complex logical relationships and arguments.

Conditional Statement

A conditional statement is a type of compound statement that usually takes the form "If...then...". In logic, it represents a relationship between two statements where the truth of one statement depends on the other.

• The standard form is "If \(p\), then \(q\)", represented symbolically as \(p \rightarrow q\).
• In our exercise, this translates to "If you are human (\(p\)), then you do not have feathers (not \(q\))."

This statement is vital since it clarifies the dependency or expectation linked between two logical conditions. Conditional statements are foundational in mathematical proofs and logical deductions.

Negation

Negation is a fundamental concept in symbolic logic used to alter the truth value of a statement.

• The negation of a statement expresses the opposite of the original statement.
• In symbolic logic, negation is usually represented by the symbol \(\lnot\) or a tilde \(\sim\).

For example, the negation of the statement "You have feathers" (represented by \(q\)) is "You do not have feathers," and is written symbolically as \(\sim q\). In logical reasoning, negations are crucial for forming more nuanced statements and understanding the inverse of given conditions. Practically, they are often used to express limitations or exceptions in rules or statements.