Validity & Proof Construction
∴Why This Matters
An argument can have true premises and a true conclusion but still be logically invalid. Validity is about the structure of reasoning, not the truth of individual claims.
A valid argument is one where if the premises are true, the conclusion must be true. We'll learn the fundamental inference rules that let us construct valid proofs.
Key Inference Rules
Modus Ponens
P → Q
P
∴ Q
If P implies Q, and P is true, then Q is true.
Modus Tollens
P → Q
¬Q
∴ ¬P
If P implies Q, and Q is false, then P is false.
Hypothetical Syllogism
P → Q
Q → R
∴ P → R
Chain implications together.
Disjunctive Syllogism
P ∨ Q
¬P
∴ Q
If one disjunct is false, the other must be true.
Common Fallacies
!Affirming the Consequent
Invalid: P → Q, Q, therefore P.
The conclusion doesn't follow! Q could be true for other reasons.
The conclusion doesn't follow! Q could be true for other reasons.
!Denying the Antecedent
Invalid: P → Q, ¬P, therefore ¬Q.
Just because P didn't happen doesn't mean Q can't happen another way.
Just because P didn't happen doesn't mean Q can't happen another way.
?Quick Check
Is this valid? 'If it rains, the ground is wet. The ground is wet. Therefore, it rained.'
Constructing Proofs
A proof is a sequence of statements, each justified by an inference rule, leading from premises to conclusion.