lecture 03© vadim bulitko : cmput 272, winter 2004, uofa1 cmput 272 formal systems & logic in...

Lecture 03 © Vadim Bulitko : CMPUT 272, Winter 2004, UofA 1

CMPUT 272CMPUT 272Formal Systems & Logic in Formal Systems & Logic in

CSCS

CMPUT 272CMPUT 272Formal Systems & Logic in Formal Systems & Logic in

CSCS

Vadim BulitkoUniversity of Alberta

http://www.cs.ualberta.ca/~bulitko/W04


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Refresher: Chapter 1.2

Chapter 1.3 : Arguments

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InterpretationInterpretationInterpretationInterpretation

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


InterpretationsInterpretationsInterpretationsInterpretations

How many interpretations do the following formulae allow?A B

4

(AB & A) B4

Why not 8 or 16?

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


Questions?Questions?Questions?Questions?


ConditionsConditionsConditionsConditions

Suppose we care about statement XX = “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”


CriteriaCriteriaCriteriaCriteria

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

Graphically:

XXXX

AAAA

BBBB

CCCC


Wanted : CriteriaWanted : CriteriaWanted : CriteriaWanted : 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 !


PracticePracticePracticePractice

It 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!


Derivation of CriteriaDerivation of CriteriaDerivation of CriteriaDerivation of Criteria

Logic/Mathematics/Theoretical sciences:Equivalent transformationsProofs by contradiction

Empirical sciences:Statistical testsFunction approximation

Artificial Intelligence:Machine learning

These methods are

not guaranteed to produce

true criteria…


Logic EquivalenceLogic EquivalenceLogic EquivalenceLogic Equivalence

Propositions/statements/formulae A and B are logically equivalent when:

A holds if and only if B holds

Notation: A BExamples:

A v A is equivalent to:A

A v ~A is equivalent to:true


ChallengeChallengeChallengeChallenge

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


Use of EquivalencesUse of EquivalencesUse of EquivalencesUse 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!


LimitationsLimitationsLimitationsLimitations

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…


EntailmentEntailmentEntailmentEntailment

A 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”


Valid/Invalid Valid/Invalid ArgumentsArguments

Valid/Invalid Valid/Invalid ArgumentsArguments

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


ExampleExampleExampleExample

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

Therefore:Q = “Socrates is mortal”

Valid / invalid?


EntailmentEntailmentEntailmentEntailment

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?


What Happened…What Happened…What Happened…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=true

P2=trueQ=true


Generality…Generality…Generality…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 Is this argument argument

valid?valid?

Is this Is this argument argument

valid?valid?


Extracting the EssenceExtracting the EssenceExtracting the EssenceExtracting 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 hold

Then 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?


General Structure!General Structure!General Structure!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 Y

P2 XentailsQ Y

Then for any interpretation I as long as I satisfies P1 and P2, interpretation I must satisfy Q


Modus PonensModus PonensModus PonensModus Ponens

The “generalized” argumentP1 = X Y

P2 = 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 method of affirming” affirming”

(Lat.)(Lat.)

““method of method of affirming” affirming”

(Lat.)(Lat.)


Valid Arguments Valid Arguments (Revisited)(Revisited)

Valid Arguments Valid Arguments (Revisited)(Revisited)

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

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

That 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


Logical FormLogical FormLogical FormLogical Form

Since:we consider all possible interpretationsthe conjunction of premises:

P1 & … & PN

is 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)


How Do We:How Do We:How Do We: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?


Propositional LogicPropositional LogicPropositional LogicPropositional 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”


Method #1Method #1Method #1Method #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


But Why? Recall:But Why? Recall:But Why? Recall:But Why? Recall:

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

In general:premises P1,…,PN entail Q

iffformula F=(P1 & … & PN Q)

is a tautology


Example #1Example #1Example #1Example #1

P v Q v R~R entails P v Q

valid/invalid?

(example 1.3.1 in the book, p. 30)



P v Q v R~R entails Q

valid/invalid?



P QP entails Q

valid/invalid?Modus ponens



P QQ entails P

valid/invalid?



P Q~Q entails ~P

valid/invalid?Modus tollens



P Qentails ~Q ~P

valid/invalid?

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



P v Q~P & ~Qentails P & Q

valid/invalid?

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


Pros & ConsPros & ConsPros & ConsPros & 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 infinite number of interpretations


Method #2 : Method #2 : DerivationsDerivationsMethod #2 : Method #2 : DerivationsDerivations

To prove that an argument is valid:

Begin with the premises

Use valid/sound inference rules

Arrive at the conclusion


Inference RulesInference RulesInference RulesInference Rules

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

…valid arguments!

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



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


DerivationsDerivationsDerivationsDerivations

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



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

derivationderivation


Pros & ConsPros & ConsPros & ConsPros & 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


FallaciesFallaciesFallaciesFallacies

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

Converse errorInverse error


Converse ErrorConverse ErrorConverse ErrorConverse 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


Inverse ErrorInverse ErrorInverse ErrorInverse 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


Truth of facts vs. Validity of Truth of facts vs. Validity of ArgumentsArguments

Truth of facts vs. Validity of Truth of facts vs. Validity of ArgumentsArguments

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


Inference RulesInference RulesInference RulesInference 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


SummarySummarySummarySummary

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


SummarySummarySummarySummary

Entailment:A entails BB follows from AAB 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


The Big PictureThe Big PictureThe Big PictureThe 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

lecture 03© vadim bulitko : cmput 272, winter 2004, uofa1 cmput 272 formal systems & logic in...

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