Lecture 5, DSCI 220, 2026W1
Treat a valid argument like a legal move in a game.
Today we define the moves: eight named rules, and two rules?
A valid argument is a legal inference.
If the lab is open, I’ll go. The lab is open. So I’ll go.
If the exam is today, the forum is quiet. The forum isn’t quiet. So the exam isn’t today.
Your friend _______ .
Do they love kittens?
From p→q and p, infer ________.
Ex:
If the bus is full, I’ll walk. The bus is full. ________
____________________ is a tautology
From p→q and ¬q, infer ________.
Ex:
If office hours moved, Slack has an announcement. No announcement. ________
____________________ is a tautology
From p→q and q→r, infer ________.
Ex:
If tidy then joins are simpler; if simpler joins then faster viz; ________
____________________ is a tautology
From p∨q and ¬q, infer ________.
Ex:
Dinner is sushi or tacos. Not tacos. ________
____________________ is a tautology
CONJ: from p, q infer ________
SPEC: from p∧q infer ________
__________________ and __________________ are tautologies
From p infer ________.
Ex: I rode my scooter today, ________
________________ is a tautology
From p∨q and ¬p∨r infer ________.
Ex:
Either study or soccer; if study then library. ________
________________ is a tautology
From p→r and q→r infer ________.
Ex:
If it’s a weekday I take the bus; if it’s raining I take the bus; ________
________________ is a tautology
From p→F infer ________.
Ex:
If I’m free Friday, then 1 = 2. ________
________________ is a tautology
Affirming the consequent ❌
from p→q, q infer p
Denying the antecedent ❌
from p→q, ¬p infer ¬q
SPEC: \((p\wedge q)\rightarrow p\) is this always true?
Affirming Consequent: \(((p\rightarrow q)\wedge q)\rightarrow p\) is this always true?
| \(p\) | \(q\) | \(p\rightarrow q\) | \((p\rightarrow q)\wedge q\) | \(((p\rightarrow q)\wedge q)\rightarrow p\) |
|---|---|---|---|---|
| F | F | |||
| F | T | |||
| T | F | |||
| T | T |
Your friend _______ .
Do they love kittens?
These are the ten on your Formula Sheet, in the Sheet’s own names. Keep it next to you. You will need them Monday.
p→q, p infer qp→q, ¬q infer ¬pp→q, q→r infer p→rp∨q, ¬q infer pp, q infer p∧qp∧q infer p (or q)p infer p∨q (any q)p∨q, ¬p∨r infer q∨rp→r, q→r infer (p∨q)→rp→F infer ¬pCommon traps:
p→q, q ∴ p ❌p→q, ¬p ∴ ¬q ❌Rules of Inference
Modus Ponens, Modus Tollens
Transitivity, Elimination
Conjunction, Specialization, Generalization
Resolution, Proof by Cases, Contradiction
Fallacies