Verify Logical Statement Using Truth Table (Tautology, Contradiction, Contingency) | Query Point Official
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Using a Truth Table to Determine Whether an Argument Form Is Valid or Invalid
Problem Statement
Using the truth table, determine whether the following argument form is valid or invalid:
- `p → q`
- `p ∨ r`
- ∴ `r`
We will evaluate whether the conclusion logically follows from the premises.
Concept: Validity Using Truth Tables
In propositional logic, an argument form is valid if in every row of the truth table where all premises are true, the conclusion is also true. If there is at least one row where the premises are true but the conclusion is false, then the argument form is invalid.
Constructing the Truth Table
| p | q | r | `p → q` | `p ∨ r` | Conclusion: `r` |
|---|---|---|---|---|---|
| T | T | T | T | T | T |
| T | T | F | T | T | F |
| T | F | T | F | T | T |
| T | F | F | F | T | F |
| F | T | T | T | T | T |
| F | T | F | T | F | F |
| F | F | T | T | T | T |
| F | F | F | T | F | F |
Validity Check
To test validity, we examine only those rows where all premises (`p → q` and `p ∨ r`) are true.
- In the row where `p → q = T` and `p ∨ r = T`, but `r = F`, the conclusion is false.
Conclusion
Because there is at least one row in the truth table where the premises are true but the conclusion is false, the argument form:
`p → q`, `p ∨ r` ∴ `r`
is invalid. This means the conclusion does not logically follow from the given premises in all cases.
Explanation
Using a truth table is a systematic way to check logical arguments. You list all possible combinations of truth values for the atomic propositions (here p, q, and r) and evaluate both premises and conclusion. If every time the premises are all true, the conclusion is also true, the argument form is valid. Otherwise, it’s invalid.
FAQ
What is a truth table?
A truth table is a table that shows all possible truth values of logical expressions based on the truth values of their atomic propositions.
What does it mean for an argument to be valid?
An argument is valid if the conclusion must be true whenever all the premises are true. Otherwise, it is invalid.
Where is this method used?
Truth tables are widely used in discrete mathematics, logic, computer science, and digital circuit design to test propositions, tautologies, valid arguments, and equivalences.
Related Topics
See Mathematics Notes & MCQs for more problems on logic, truth tables, and validity tests.
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