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

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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

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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!

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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!

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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

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Propositional logic

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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.

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Implication can be non-intuitive

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Implication

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Tautology

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Equivalence

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Some important equivalences De Morgan’s laws

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Some important equivalences

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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.

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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.

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More manipulation examples

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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

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

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Mathematical Structures A collection of objects with operations defined on them and the accompanying properties form a mathematical structure or system.

Mathematical Structures A collection of objects with operations defined on them and the accompanying properties form a mathematical structure or system.

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