1. Propositional Logic Discrete Structures (CS 173) Madhusudan Parthasarathy, University of Illinois School of Athens Fresco by Raphael Wikimedia Commons

2. Mathematical logic (symbolic logic) Study of inference using abstract rules that does not assume any particular knowledge of things or of properties. E.g.: All men are mortal Socrates is a man Inference: Socrates is mortal. E.g. All pigs are boisterous Alfred is a pig. Inference: Alfred is boisterous

3. All snarks are frabjous Yeti is a snark. Inference: Yeti is frabjous Key idea: Inference is independent of the subjects (men, pigs, snarks) and properties (mortality, boisterousness, frabjousness). Inference follows simply from language!

5. But inference rules needn’t hold in natural language! … quirks of English Sam and Sally are programmers. Inference: Sam is a programmer Sam and Sally are together. Inference: Sam is together! So we need a formal language…. logic!

6. Propositional logic A proposition is a statement that is either true or false. Examples: Socrates is a man This car is purple 43 is prime Non-examples: Trucks Hello Trkjkjugirtu

7. Propositional logic

10. Propositional logic: Semantics A formula f, in general, doesn’t have a “truth” value associated to it. Model: M - Assigns truth/falsehood to each proposition Any formula f evaluates to true/false in such a model.

12. Implication can be non-intuitive

13. Implication

14. Tautology

15. Equivalence

17. Some important equivalences De Morgan’s laws

19. Some important equivalences

20. Contrapositive, converse, negation Proposition: “If the sky is green, then I’m a monkey’s uncle.” Converse – If I’m a monkey’s uncle, then the sky is green. Contrapositive – If I’m not a monkey’s uncle, then the sky is not green. Negation – The sky is green, but I am not a monkey’s uncle.

21. Contrapositive, converse, negation Proposition: “If the sky is green, then I’m a monkey’s uncle.” Converse – If I’m a monkey’s uncle, then the sky is green. Contrapositive – If I’m not a monkey’s uncle, then the sky is not green. Negation – The sky is green, but I am not a monkey’s uncle.

22. More manipulation examples

24. Logistics If you’re not registered yet and – Sign sheet at end of class (again) – Sign up for moodle and piazza – Keep on top of homeworks only mini-homework for next week will be released by Friday No discussion sections this week

25. See you next week! Tuesday – More logic Predicate logic Quantifiers Binding and scope – Direct proofs Thursday – More proof practice and strategies