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Introduction to Logic Introduction Michael Genesereth Computer Science Department Stanford University Lecture will begin at ~1:35 PDT.

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Page 1: Introduction to Logic Introduction

Introduction to LogicIntroduction

Michael GeneserethComputer Science Department

Stanford University

Lecture will begin at ~1:35 PDT.

Page 2: Introduction to Logic Introduction

Greek Trivium

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History

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History of Logic

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Uses of Logic

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Mathematics

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Hardware Engineering

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Software Engineering

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Rules and Regulations

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Games

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Debate

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Communication

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Logic and Computer Science

Logic is the mathematics of Computer Science as

Calculus is the mathematics of Physics.

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Elements of Logic

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Friends

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Friends

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Friends

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Friends

... ... ... ... ... ... ... ...

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Logical Language

Dana likes Cody.Abby does not like Dana.

Dana does not like Abby.Bess likes Cody or Dana.

Abby likes everyone that Bess likes.

Cody likes everyone who likes her.

Everyone likes herself.

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Reasoning as Information Processing

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Premises:Dana likes Cody.Abby does not like Dana.Everybody likes somebody.Bess likes Cody or Dana. Abby likes everyone that Bess likes.Cody likes everyone who likes her.Everyone likes herself.

True: False: Unknown:Bess likes Cody. Bess likes Dana. Dana likes Bess.Bess does not like Dana. Everybody likes everybody.Everybody likes someone.

Logical Reasoning

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Model Checking

... ... ... ... ... ... ... ...

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All of Abby's friends are Bess's friends.All of Bess's friends are Cody's friends.Therefore, all of Abby's friends are Cody's friends.

Symbolic Manipulation

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All Accords are Hondas.All Hondas are Japanese.Therefore, all Accords are Japanese.

Sample Rule of Inference

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Sample Rule of Inference

All borogoves are slithy toves.All slithy toves are mimsy.Therefore, all borogoves are mimsy.

Sample Rule of Inference

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All x are y.All y are z.Therefore, all x are z.

General Rule of Inference

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Bertrand Russell

Logic "may be defined as the subject in which we never know what we are talking about nor whether what we are saying is true.''

- Bertrand Russell

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All x are y.Some y are z.Therefore, some x are z.

No! No!! No!!!

Unsound Rule of Inference

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All Toyotas are Japanese cars.Some Japanese cars are made in America.Therefore, some Toyotas are made in America.

Sometimes produces a result that happens to be true.

Using Unsound Rule of Inference

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All Toyotas are cars.Some cars are Porsches.Therefore, some Toyotas are Porsches.

Sometimes produces a result that happens to be false.

Using Unsound Rule of Inference

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A rule of inference is sound if and only if the conclusion is true whenever the premises are true.

The application of sound rules of inference is called deduction.

Deduction

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Incomplete Induction

Induction is reasoning from the specific to the general.

I have seen 1000 black ravens.I have never seen a raven that is not black.Therefore, every raven is black.

Induction is not necessarily sound (but it can be useful).Deduction is necessarily sound.

Induction is the basis for Science (and machine learning)Deduction is the subject matter of Logic.

Science aspires to discover new knowledge.Logic aspires to derive conclusions implied by premises.

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Niels Bohr to Albert Einstein

“You are not thinking; you are just being logical.”

https://www.youtube.com/watch?v=Oseqh7SMIvo

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Logical Language Definitions: A triangle is a polygon with three sides.  Constraints: Parents are older than their children. Partial Information: Abby likes one of Cody or Dana.

Logical Reasoning Model Checking - Truth Tables, Logic Grids Equivalence Transformations - Symbolic Manipulation Proofs

"Metalevel" Concepts and Analysis Validity, Contingency, Unsatisfiability Equivalence, Entailment, Consistency Soundness, Completeness, Decidability

Elements of Logic

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Language Probabilities Metaknowledge - knowledge about knowledge Paradoxes, e.g. This sentence is false.

Reasoning Negation as Failure - knowing not versus not knowing Induction, Abduction, Analogical Reasoning Paraconsistent Reasoning (reasoning with inconsistency)

Logical Extensions

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Symbolic Logic

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Logical Sentences

Dana likes Cody.Abby does not like Dana.

Dana does not like Abby.Bess likes Cody or Dana.

Abby likes everyone that Bess likes.

Cody likes everyone who likes her.

Everyone likes herself.

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One grammatically correct sentence:

The cherry blossoms in the spring.

Another grammatically correct sentence:

The cherry blossoms in the spring

Complexity of Natural Language

sank.

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There’s a girl in the room with a telescope.

Grammatical Ambiguity

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Crowds Rushing to See Pope Trample 6 to Death

Newseum Headlines

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Crowds Rushing to See Pope Trample 6 to Death

Scientists Grow Frog Eyes and Ears

Newseum Headlines

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Crowds Rushing to See Pope Trample 6 to Death

Scientists Grow Frog Eyes and Ears

Fried Chicken Cooked in Microwave Wins Trip

Newseum Headlines

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Crowds Rushing to See Pope Trample 6 to Death

Scientists Grow Frog Eyes and Ears

Fried Chicken Cooked in Microwave Wins Trip

British Left Waffles on Falkland Islands

Newseum Headlines

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Residents report that a hole was cut in the fence surrounding a nudist colony.

Misteaks in Print

Police are looking into it.

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Champagne is better than beer.Beer is better than soda. Therefore, champagne is better than soda.

Doug Lenat's Logic

X is better than Y.Y is better than Z. Therefore, X is better than Z.

Bad sex is better than nothing.Nothing is better than good sex. Therefore, bad sex is better than good sex. Really?

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Xavier is three times as old as Yolanda. Xavier's age and Yolanda's age add up to twelve. How old are Xavier and Yolanda?

x − 3y = 0x + y = 12−4y = −12

Symbolic Algebra

y = 3x = 9

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If Mary loves Pat, then Mary loves Quincy. If it is Monday and raining, then Mary loves Pat or Quincy. If it is Monday and raining, does Mary love Quincy?

p ⇒ q m ∧ r ⇒ p ∨ q

m ∧ r ⇒ q ∨ q m ∧ r ⇒ q

Answer: Yes

Symbolic Logic

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Logistics

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Propositional Logic (logical operators)

If it is raining and cold, then the ground is wet.

Relational Logic (variables and quantifiers)

If x is a parent of y, then x is older than y.

Functional Logic (compound terms)

{a, b} is a subset of {a, b, c}.

Multiple Logics

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Schedule

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Numerical Grade 30% - quiz on Propositional Logic (Week 4) 30% - quiz on Relational Logic (Week 7) 30% - quiz on Herbrand Logic (Week 10) 10% - class attendance, discussion forum, puzzles, ...

Letter Grade Based on numerical grade (see above) *No* curve - i.e. independent of number of students A, B, C distributed uniformly over 70% - 100%

NB: We may / MAY offer an optional final exam for those wishing to improve their grades. This MAY not happen.

Grades

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http://cs157.stanford.edu

Course Website

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Materials of the Course Lectures Textbook / Lessons Exercises Puzzles Tools

Ed Discussion Read discussion Post questions Answer questions

Hints on How to Take the Course

Working in groups is okay /

recommended!!

Read the notes.Do the exercises.Do the exercises!Do the exercises!!

Learn actively.

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Value of Practice

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Value of Theory

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We will frequently write sentences about sentences.

Sentence: When it rains, it pours.Metasentence: That sentence contains two verbs.

We will frequently prove things about proofs.

Proof: If it is Monday and raining, Mary loves Quincy.Metaproof: Our proof system is sound and complete.

Meta

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Mike took it twice!

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