Math Problem Statement
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:
- (If , then )
- (Not )
- (Conclusion: )
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:
- : Truth values of
- : Truth values of
- : Implication ( implies )
- : Negation of
- Premises and Conclusion: Verify if the conclusion holds when premises are true.
Premises True | Conclusion | ||||
---|---|---|---|---|---|
T | T | T | F | ? | T |
T | F | F | F | ? | F |
F | T | T | T | ? | T |
F | F | T | T | ? | F |
Step 2: Analyze the Truth Table
-
Implication ():
- is false only when is true and is false; otherwise, it is true.
-
Negation ():
- is the opposite of .
-
Premises:
- Both premises, and , must be true for the conclusion to be valid.
Step 3: Identify Rows Where Premises Are True
From the table:
-
Row 3: ,
- , , and .
- Both premises are true, and the conclusion holds ().
-
Row 4: ,
- , , but .
- 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:
- What is the definition of a valid argument in propositional logic?
- How do we construct a truth table for multiple premises?
- Why does a single counterexample invalidate an argument?
- What are the common forms of invalid arguments in logic?
- 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