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Introduction to Logic

Overview:What is Logic?

Propositional Logic

Predicate Logic

Literature:

Handouts: (J. Kelly)The Essence of Logic:

chapter 1 (not 1.6)

chapter 6

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What is Logic?

(logik) = relative to logos (reason)

Definitions of logic The art of reasoning

the branch of philosophy that analyzes inference

Reasoned and reasonable judgment; "it made a certain kind oflogic"

the principles that guide reasoning within a given field orsituation: "economic logic requires it"; "by the logic of war"

a system of reasoning

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Socrates is mortal

All man are mortalSocrates is a man

Aristotoles (384322 B.C.)

The instrument (the "organon") by means ofwhich we come to know anything.

formal rules for correct reasoning

Syllogism:

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Gottfried Leibniz (1646-1716)

Human reasoning can be reduced to calculation

characteristica universalis universal language to describe all scientific concepts

calculus ratiocinator method to reason with that language

salva veritate two expressions can be interchanged without changing the

truth-value of the statements in which they occur

when there are disputes among persons, we can simplysay: Let us calculate [calculemus]!

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Criticism to logic

logic is not a model of reasoning at all but is more ameans of constraining the reasoning process; we [people]dont solve problems using logic, we just use it to explainour solutions.

[Marvin Minsky]

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The Importance of Logic

High-level language for expressing knowledge

High expressive power

Well-understood formal semantics

Precise notion of logical consequence

Proof systems that can automatically derive statementssyntactically from a set of premises

Logic can provide explanations for answers By tracing a proof that leads to a logical consequence

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Uses of Logic

Argumentation

Inference (systems)

Cognitive psychology & AI descriptive use of logic

Normative systems prescriptive use of logic

Knowledge representation

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Warning!

Meta-language Different levels

Abstraction

Difference to everyday use of concepts Temporal, causal, relations

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Propositional Logic and Predicate Logic

Propositional Logic

The study of statements and their connectivity structure.

Predicate Logic

The study of individuals and their properties.

Propositional logic more abstract and hence less detailedthan predicate logic.

Propositional/predicate logic are unique in the sense thatsound and complete proof systems do exist.

Not for more expressive logics (higher-order logics)

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Reasoning = Reach valid conclusions

Reasoning is valid (deduction, syllogism)

Nothing is said about the actual truth of premisses!

The concern of logic is that

never a false conclusion can be reached from truepremises!

Socrates is mortal

All man are mortal

Socrates is a man

Socrates has blue skin

All man have blue skin

Socrates is a man

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Propositional Logic

Truth Tables

Logical equivalence

Tautology

Contradiction

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Basic elements

Logical connectives and

or

not

implication, if then

bi-implication, iff (if and only if)

Connectives are used to link atoms

atoms = facts = statements

Propositions can be constructed based on atoms andconnectives

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Truth values

A proposition has a truth value:

true (T) or false (F)

v(A) represents the truth value of A

T and F are often represented as 1 resp. 0

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Truth tables

Truth tables give the operational definition of logicalconnectives List ALL possible cases

Example, for negation :

F

T

AA

F

T

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V( ) =

Truth table for AND ()

TT

FT

TF

FF

ABBA

F

F

F

T

(1+1 = 2) (the sun is a star) T

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V( ) =

Truth table for OR ()

TT

FT

TF

FF

ABBA

F

T

T

T

(1+1 = 2) (the sun is a planet) T

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V( ) =

Truth table for implication ()

TT

FT

TF

FF

ABBA

T

T

F

T

(the sun is a planet) (1+1 = 3)

(dog is animal) (dog breaths)

T

V( ) = T

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Truth table for iff ()

TT

FT

TF

FF

ABBA

T

F

F

T

(the sun is a star) (1+1 = 2)V( ) = T

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Remarks on implication

Implication in propositional logic is different from its use innatural language Temporal issues

Causal relations

Implication in propositional logic refers to the truth valuesof the atoms!

E.g. compare: Ifthe moon is made of cheese then it is tasty

Ifthe moon is made of cheese then 2 x 2 = 5

Both are true, but the first sounds more logical!!

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Tautology and contradiction

Tautology A logical expression that has truth value T is all cases

Contradiction A logical expression that has truth value F is all cases

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Example: tautology

TTTFFTTTT

TTTTFTFTT

TTFFTFTFT

TTTTTFFFT

TTTFFFTTF

TTTTFFFTF

TTFFTFTFF

TTTTTFFFF

(AB) (C(BC))C(BC)BCCBABCBA

(AB) (C(BC))

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Example: tautology

TTTFFTTTT

TTTTFTFTT

TTFFTFTFT

TTTTTFFFT

TTTFFFTTF

TTTTFFFTF

TTFFTFTFF

TTTTTFFFF

(AB) (C(BC))C(BC)BCCBABCBA

(AB) (C(BC))

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Example: Contradiction

FFT

FTF

AAAA

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Logical equivalence

Two logical expressions are logically equivalent ifthey have the same truth table That is, each truth assignment of the atoms results in

the same truth value for the expression

Notation: A B A and B are logically equivalent

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Example

TTT

FFT

FTF

FFF

ABBA

TFFFTT

FTTFFT

FTFTTF

FTTTFF

(AB)ABBABA

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Example

TTT

FFT

FTF

FFF

ABBA

TFFFTT

FTTFFT

FTFTTF

FTTTFF

(AB)ABBABA

Conclusion:

A B (A B)

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Equivalence laws

A 0 0

A 1 A

A 0 A

A 1 1

A A A

A A A

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Equivalence laws

A 0 0

A 1 A

A 0 A

A 1 1

A A A

A A A

A A 0

A A 1

A A

A B B A

A B B A

A (A B) A

A (A B) A

A (A B) A B

A (A B) A B

(A B) (A B) A

A B A B

A B (A B)

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Equivalence laws (general)

Distributivity A (B C) (A B) (A C)

A (B C) (A B) (A C)

De Morgan Laws (A B) A B

(A B) A B

Logical equivalence satisfies the laws of a Boolean algebra.

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Inference

How to reason (deduce new facts) with propositional logic

Modus Ponens

A B

A

B

IfA B and A hold (are true) then it can be safely

concluded that B also is true E.g. if the rule is (smoke fire) and you see smoke

then you can conclude that there is fire

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Predicate logic

Objects, predicates, functions

Quantifiers

1st order language

Substitutions, interpretations

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Syllogistic reasoning as

cannot be expressed in propositional logic

Socrates is mortal

Syllogistic reasoning

All man are mortal

Socrates is a man

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Objects, predicates, quantifiers

Needed: Objects, such as Socrates

Predicates, such as mortal

Quantifiers, such as all

Formally: x : M(x) S(x)

M(s)

S(s)

or:

x : man(x) mortal(x)man(socrates)mortal(socrates)

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Quantifiers

Universal quantifier Notation:

for all

Ex. (x)M(x) or x : M(x)

Existential quantifier Notation:

exists, there is

Ex. (x)M(x) of x : M(x)

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Often used formulas

All A are B:

For all x: ifx is A then x is B

(x)(A(x) B(x))

Some A are B:

There is x: x is A and x is B

(x)(A(x) B(x))

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Warning

Order of quantifiers is essential!

Big difference: (x)(y)A(x,y)

For all x there is a y such that A(x,y)

For all men x there is a woman y such that mother(x,y)

(y)(x)A(x,y) There is a y such that for all x, A(x,y)

There is a woman y such that for all man x, mother(x,y)

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Quantifiers in finite domains

When the domain is finite,say {a1, , an}, the universal and existential quantifiersare a abbreviation for the finite conjunction resp.disjunction:

(x)A(x) = A(a1) A(an)

(x)A(x) = A(a1) A(an)

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Relation between and

(x)A(x) (x)A(x)

(x)A(x) (x)A(x)

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Inference

How to reason (deduce new facts) with predicate logic

Modus ponens

Universal Elimination (Syllogism)

(x) P(x) Q(x)

P(a)

Q(a)where x X, a X

E.g (all kids like ice-cream) and (Bob is a kid) hold, then

deduce (Bob likes ice-cream) Exercise: translate into predicate logic!

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Next module:

Rule Based Systems

(27 november 2007)

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