Math Problem Statement

Use a truth table to determine if the following symbolic form of an argument is valid or invalid: 1. p → q 2. ~p ∴ q Is the symbolic argument valid or invalid?

Solution

To determine if the argument is valid or invalid using a truth table, let's analyze the given argument step by step:

The argument is as follows:

  1. pqp \to q (If pp, then qq)
  2. p\sim p (Not pp)
  3. q\therefore q (Conclusion: qq)

We need to create a truth table to verify whether the premises logically lead to the conclusion in all cases.


Step 1: Structure the Truth Table

We include columns for:

  • pp: Truth values of pp
  • qq: Truth values of qq
  • pqp \to q: Implication (pp implies qq)
  • p\sim p: Negation of pp
  • Premises and Conclusion: Verify if the conclusion qq holds when premises are true.

ppqqpqp \to qp\sim pPremises TrueConclusion qq
TTTF?T
TFFF?F
FTTT?T
FFTT?F

Step 2: Analyze the Truth Table

  1. Implication (pqp \to q):

    • pqp \to q is false only when pp is true and qq is false; otherwise, it is true.
  2. Negation (p\sim p):

    • p\sim p is the opposite of pp.
  3. Premises:

    • Both premises, pqp \to q and p\sim p, must be true for the conclusion qq to be valid.

Step 3: Identify Rows Where Premises Are True

From the table:

  • Row 3: p=Fp = F, q=Tq = T

    • pq=Tp \to q = T, p=T\sim p = T, and q=Tq = T.
    • Both premises are true, and the conclusion holds (q=Tq = T).
  • Row 4: p=Fp = F, q=Fq = F

    • pq=Tp \to q = T, p=T\sim p = T, but q=Fq = F.
    • Here, the premises are true, but the conclusion does not hold.

Step 4: Conclusion

Since there is at least one row (Row 4) where the premises are true and the conclusion is false, the argument is invalid.


Final Answer:

The symbolic argument is Invalid.


Do you need a further breakdown or explanation of the truth table? Let me know!


Related Questions:

  1. What is the definition of a valid argument in propositional logic?
  2. How do we construct a truth table for multiple premises?
  3. Why does a single counterexample invalidate an argument?
  4. What are the common forms of invalid arguments in logic?
  5. How is modus tollens different from this argument structure?

Tip: In propositional logic, an argument is valid if there is no situation (truth table row) where the premises are true and the conclusion is false.

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Math Problem Analysis

Mathematical Concepts

Propositional Logic
Truth Tables

Formulas

Implication (p → q): False only when p is True and q is False
Negation (~p): Opposite of p

Theorems

Logical Validity in Propositional Logic

Suitable Grade Level

Grades 10-12