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CSE 240Logic and Discrete Math

Lecture notes 3

Weixiong ZhangWashington University in St. Louis

http://www.cse.wustl.edu/~zhang/teaching/cse240/Spring10/index.html

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Today

Refresher: Chapter 1.2

Chapter 1.3 : Arguments

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Interpretation

In propositional logic interpretation is a mapping from variables in your formulae to {true, false}

Example:Formula: A v BInterpretation 1: A = true, B = falseInterpretation 2: A = false, B = false

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Interpretations

How many interpretations do the following formulae allow?A B

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(AB & A) B4

Why not 8 or 16?

The number of interpretations is 2N where Nis the number of independent variables

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Questions?

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ConditionsSuppose we care about statement X

X = “this assignment is copied”We want to evaluate X (true/false?)

Suppose we know A such that AXA is a sufficient conditionA=“the cheater is caught in the act”

Suppose we know B such that XBB is a necessary conditionB=“there was an original assignment to copy from”

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Criteria

Suppose we know C such that CXC is a criterionC=“someone has copied this assignment”

Graphically:

X

A

B

C

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Wanted : Criteria

Medical testsSoftware/hardware correctnessFraud/cheatingFinancial marketPsychology (e.g., in sales)Science : mathematics, physics, chemistry, etc.Logic :

If C is a criterion for X then C ≡ X !

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PracticeIt is frequently non-trivial to derive a criterion for a real-life property X

Then we have to settle for:Sufficient conditions:

“If this quality test passes then the product is fine”Necessary conditions:

“If the patient breaks a leg they will be in pain”

Statistical validity : the condition works most of the timeIn logic : the condition works all the time!

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Derivation of Criteria

Logic/Mathematics/Theoretical sciences:Equivalent transformationsProofs by contradiction

Empirical sciences:Statistical testsFunction approximation

Artificial Intelligence:Machine learning

These methods are

notguaranteed to produce true

criteria…

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Questions?

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Logic EquivalencePropositions/statements/formulae A and B are logically equivalent when:

A holds if and only if B holdsNotation: A ≡ BExamples:

A v A is equivalent to:A

A v ~A is equivalent to:true

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Challenge

Theorem 1.1.1 : Boolean AlgebraDerive the rest (e.g., #8) from the first 5 equivalences…

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Use of Equivalences

Deriving equivalent formulae!Of course, but why do we care?

Simplification of formula

Simplification of code

Simplification of hardware (e.g., circuits)

Derivation of criteria!

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Limitations

Not all statements are equivalent!Of course not, but what else is there?

Some formulae are stronger than others

They imply or entail other formula but not the other way around…

Equivalences cannot directly help us proving such entailments…

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EntailmentA collection of statements P1,…,Pn (premises) entails statement Q (conclusion) if and only if:

Whenever all premises hold the conclusion holds

For every interpretation I that makes all Pj hold, I also makes Q hold

Example:Premises:

P1 = “If Socrates is human then Socrates is mortal”P2 = “Socrates is human”

Conclusion:Q = “Socrates is mortal”

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Valid/Invalid Arguments

Suppose someone makes an argument:P1,..,PN therefore Q

The argument is called valid iff:P1,…,PN logically entail Q

That is:Q must hold if all Pi hold

Otherwise the argument is called invalid

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Example

Sample argument:P1 = “If Socrates is human then Socrates is mortal”P2 = “Socrates is human”Therefore:Q = “Socrates is mortal”

Valid / invalid?

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Entailment

Then is the following argument valid?P1

P2

entailsQ

Yes?Very well, but what if my interpretation I sets P1 and P2 to true but Q to false?

Then by definition Q is not entailed by P1 and P2

So do P1,P2 entail Q or do they not?

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What Happened…We considered P1, P2, and Q under a particular(common sense) interpretation:

P1 = “If Socrates is human then Socrates is mortal” trueP2 = “Socrates is human” trueQ = “Socrates is mortal” true

Thus, they were merely logical constants to us:P1=trueP2=trueQ=true

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Generality…Thus our argument was:

TrueTrueentailsTrue

Well, this is not very useful because it doesn’t tell us anything about validity of other arguments. For example:

P1=“If J.B. broke his leg then J.B. is in pain”P2=“J.B. broke his leg”entailsQ=“J.B. is in pain”

Is this argument valid?

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Extracting the Essence

How do we know it is valid?Because regardless of who J.B. is and what happened to him/her, we somehow know that:

If P1 and P2 holdThen Q will hold

But how do we know that?How can we extract the essence of the “dead Socrates” and “J.B. in pain” arguments?

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General Structure!Recall both arguments:

P1 “If Socrates is human then Socrates is mortal” “If J.B. broke his leg then J.B. is in pain”P2 “Socrates is human” “J.B. broke his leg”entailsQ “Socrates is mortal” “J.B. is in pain”

Note that P1, P2, and Q are related!Both arguments share the same structure:

P1 If X then YP2 XentailsQ Y

Then for any interpretation I as long as I satisfies P1and P2, interpretation I must satisfy Q

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Modus PonensThe “generalized” argument

P1 = X YP2 = XentailsQ = Y

…is much more useful!Why?

Because it captures the essence of both arguments and can be used for infinitely many more

“method of affirming” (Lat.)

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Valid Arguments (Revisited)Suppose someone makes an argument:

P1,..,PN therefore QThe argument is called valid iff:

P1,…,PN logically entail QThat is:

For any interpretation I that satisfies all Pj, interpretation I must necessarily satisfy Q

Usually: Pj and Q are somehow related formulae and P1 & … & PN can be true or false depending on the interpretation I

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Logical Form

Since:we consider all possible interpretationsthe conjunction of premises:

P1 & … & PNis not always true or false

The conclusion Q must follow from / be entailed by the premises… …by logical form of Pj and Q alone(p. 29 in the text)

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Questions?

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How Do We:Tell between a valid argument and an invalid argument:

People are mortal. Socrates is a man. Socrates is mortal.Ducks fly. F-16 flies. F-16 is a duck.

Prove that something logically follows from something else:

1: Everybody likes Buddha2: Everybody likes someone

Prove that something is logically equivalent to something else:

1: Everybody likes cream and sugar2: Everybody likes cream and everybody likes sugar

Prove that there is a contradiction?

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Propositional Logic

Method #1:Go through all possible interpretations and check the definition of valid argument

Method #2:Use derivation rules to get from the premises to the conclusion in a logically sound way

“derive the conclusions from premises”

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Method #1

Section 1.3 in the text proves many arguments/inference rules using truth tables

Suppose the argument is:P1,…,PN therefore Q

Create a truth table for formulaF=(P1 & … & PN Q)

Check if F is a tautology

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But Why? Recall:

Formula A entails formula Biff (A B) is a tautology

In general:premises P1,…,PN entail Qiffformula F=(P1 & … & PN Q)is a tautology

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Example #1

P v Q v R~R entails P v Q

valid/invalid?

(example 1.3.1 in the book, p. 30)

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Example #2

P v Q v R~R entails Q

valid/invalid?

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Example #3

P QP entails Q

valid/invalid?Modus ponens

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Example #4

P QQ entails P

valid/invalid?

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Example #5

P Q~Q entails ~P

valid/invalid?Modus tollens

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Example #6

P Qentails ~Q ~P

valid/invalid?

In fact, we proved last time that: (P Q) ≡ (~Q ~P)

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Example #7

P v Q~P & ~Qentails P & Q

valid/invalid?

Any argument with a contradiction in its premises is valid by default…

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Pros & Cons

Method #1:Pro: straight-forward, not much creativity machines can do

Con: the number of interpretations grows exponentially with the number of variables cannot do for many variables

Con: in predicate and some other logics even a small formula may have an infinitenumber of interpretations

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Questions?

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Method #2 : Derivations

To prove that an argument is valid:

Begin with the premises

Use valid/sound inference rules

Arrive at the conclusion

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Inference Rules

But what are these “inference rules”?They are simply…

…valid arguments!

Example:X & Y X & Y Z & WthereforeZ & W by modus ponens

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Example #1

(X&Y Z&W) & KX&YthereforeZ&WHow?(X&Y Z&W) & KX&Y Z&W by conjunctive simplificationX&YZ&W by modus ponens

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Derivations

The chain of inference rules that starts with the premises and ends with the conclusion…is called a derivation:

The conclusion is derived from the premises

Such a derivation makes a proof of argument’s validity

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Example #1

(X&Y Z&W) & KX&YthereforeZ&WHow?(X&Y Z&W) & KX&Y Z&W by conjunctive simplificationX&YZ&W by modus ponens

derivation

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Pros & Cons

Method #2:Pro: often can get a dramatic speed-up over truth tables.

Con: requires creativity and intuition harder to do by machines

Con: semi-decidable : there is no algorithm that can prove any first-order predicate logic argument to be valid or invalid

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Questions?

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Fallacies

An error in derivation leading to an invalid argumentVague formulations of premises/conclusionMissing stepsUsing non-sound inference rules, e.g.:

Converse errorInverse error

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Converse Error

If John is smart then John makes a lot of moneyJohn makes a lot of moneyTherefore:John is smart

Tries to use this non-sound “inference rule”:AB, BThus: A

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Inverse Error

If John is smart then John makes a lot of moneyJohn is not smartTherefore:John doesn’t make a lot of money

Tries to use this non-sound “inference rule”:AB, ~AThus: ~B

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Questions?

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Truth of facts vs. Validity of Arguments

The premises are assumed to be true ONLY in the context of the argument

The following argument is valid:If John Lennon was a rock star then he was a womanJohn Lennon was a rock starThus:John Lennon was a woman

But the 1st premise doesn’t hold under the common sense interpretation

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Inference Rules

Table 1.3.1 on page 39

If practice with the rules then will be more fluent using them

If are more fluent using them then will be more likely to get a better mark on exams

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Summary

Equivalence:A ≡ BA holds iff B holdsA is a criterion for BB is a criterion for AA entails BB entails AA and B are “equivalently strong”Formula F=(AB) is a tautology

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Summary

Entailment:A entails BB follows from AA∴B is a valid argumentA is a sufficient condition for BB is a necessary condition for AIf A holds then B holdsA may be “stronger than” BFormula F=(AB) is a tautology

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The Big Picture

Logic is being used to verify validity of arguments

An argument is valid iff its conclusion logically follows from the premises

Derivations are used to prove validity

Inference rules are used as part of derivations

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Questions?