Math Problem Statement

Let p, q, and r be the following statements: p: It is Saturday. q: Reuben is practicing basketball. r: Reuben is out of breath. Argument (a): If it is Saturday, then Reuben is practicing basketball. If Reuben is practicing basketball, then Reuben is out of breath. Therefore, if it is Saturday, then Reuben is out of breath. Argument (b): If it is Saturday, then Reuben is practicing basketball. Reuben is practicing basketball. Therefore, it is Saturday.

Solution

The image contains a logic problem involving three statements:

  • pp: It is Saturday.
  • qq: Reuben is practicing basketball.
  • rr: Reuben is out of breath.

There are two arguments given in the table, where the goal is to express each argument in symbolic form and determine whether the argument is valid.

Argument (a)

Argument:

  • If it is Saturday, then Reuben is practicing basketball.
  • If Reuben is practicing basketball, then Reuben is out of breath.
  • Therefore, if it is Saturday, then Reuben is out of breath.

We need to represent this in symbolic form:

  1. "If it is Saturday, then Reuben is practicing basketball" is pqp \rightarrow q.
  2. "If Reuben is practicing basketball, then Reuben is out of breath" is qrq \rightarrow r.
  3. Conclusion: "If it is Saturday, then Reuben is out of breath" is prp \rightarrow r.

This forms a classic hypothetical syllogism: (pq)(qr)(pr)\left( p \rightarrow q \right) \land \left( q \rightarrow r \right) \rightarrow \left( p \rightarrow r \right)

So the symbolic form for argument (a) should be:

pq,qrprp \rightarrow q, \quad q \rightarrow r \quad \therefore \quad p \rightarrow r

In the choices, this corresponds to the form found in the third option for argument (a).

Validity:
This argument is valid, as it follows the hypothetical syllogism rule.

Argument (b)

Argument:

  • If it is Saturday, then Reuben is practicing basketball.
  • Reuben is practicing basketball.
  • Therefore, it is Saturday.

Symbolically:

  1. "If it is Saturday, then Reuben is practicing basketball" is pqp \rightarrow q.
  2. "Reuben is practicing basketball" is qq.
  3. Conclusion: "Therefore, it is Saturday" is pp.

This commits the fallacy of affirming the consequent. Just because qq (Reuben practicing basketball) is true does not necessarily imply that pp (it being Saturday) is true.

So the symbolic form is:

pq,qpp \rightarrow q, \quad q \quad \therefore \quad p

This corresponds to the second option for argument (b).

Validity:
This argument is not valid because it commits the affirming the consequent fallacy.

Summary:

  • Argument (a): Valid, symbolic form is pqp \rightarrow q, qrq \rightarrow r, therefore prp \rightarrow r.
  • Argument (b): Invalid, symbolic form is pqp \rightarrow q, qq, therefore pp.

Would you like more details or have any further questions?

Here are 5 related questions to extend understanding:

  1. How does the hypothetical syllogism work in logic?
  2. What are common logical fallacies, like affirming the consequent?
  3. Can you explain the difference between necessary and sufficient conditions?
  4. What is the Modus Ponens rule in logic?
  5. How do logical arguments relate to mathematical proofs?

Tip: Always identify the form of logical argument before determining validity to avoid confusion.

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Math Problem Analysis

Mathematical Concepts

Propositional Logic
Logical Reasoning
Symbolic Logic

Formulas

If p → q and q → r, then p → r (Hypothetical Syllogism)

Theorems

Hypothetical Syllogism
Fallacy of Affirming the Consequent

Suitable Grade Level

College-level or High School (Advanced Logic)