STEP-BY-STEP LESSON · §2.3
Valid and Invalid Arguments
Distinguish valid inference from an accidentally true conclusion.
Before you begin
Conditional Statements
In p→q, p is the hypothesis and q the conclusion. The implication fails only when p is true and q false.
p→q≡¬p∨q
The equivalent contrapositive reverses and negates both parts: if not q, then not p. The converse q→p is not equivalent in general.
p→q≡¬q→¬p
Symbols
- ∴
- therefore
Definitions and notation for this topic
Therefore / hence / thus
∴
Signals that the following conclusion follows from the preceding reasoning.
The word therefore does not itself prove the conclusion; the step needs justification.
This is a conclusion marker, not a logical connective and not a guarantee that the preceding reasoning is valid.
Separate glossary example
n=2k and k∈ℤ. Therefore n is even.
Step by step
Step 1 / 6
Premises and conclusion
An argument is valid when its premises cannot all be true while its conclusion is false.
We assess the connection between the statements.
Worked example
From p→q and ¬q, justify ¬p one step at a time.
- The second premise makes q false. We must show that p cannot be true when both premises hold.
- Temporarily suppose p is true. The first premise p→q then requires q to be true.
- This conflicts with ¬q. Therefore p is false and ¬p is true. This valid pattern is modus tollens.
Optional self-check
Which truth-table row refutes an argument?
Show answer
All premises T and conclusion F.