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Logic What is it?. Formal logic is the science of deduction . It aims to provide systematic means for telling whether or not given conclusions follow from given premises, i.e., whether arguments are valid or not. A valid argument is one whose conclusion is true in every case

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slide2
Formal logic is the science of deduction.
  • It aims to provide systematic means for telling whether or not given conclusions follow from given premises, i.e., whether arguments are valid or not
slide3

A valid argument is one whose

conclusion is true in every case

in which all its premises are true.

slide4

Premise 1: Some cave dwellers use fire.

Premise 2: All who use fire have intelligence.

Conclusion: Some cave dwellers have

intelligence.

Valid or not?

Valid

slide5

P1: All geniuses are illogical.

P2: Some politicians are illogical.

Conclusion: Some politicians are geniuses.

Valid or not?

Not valid

slide6

P1: If you overslept, you will be late.

P2: You aren’t late.

Conclusion: You didn’t oversleep.

IF you oversleep, you will be late

AND you are not late

THEN you didn’t oversleep.

slide7

P1

P2

.

.

.

Pn

C

IF P1 and P2 and … Pn THEN C

slide8

A valid argument does not say that C is true but that C is true if all the premises are true.

That is, there are NO counterexamples.

P1: Bertil is a professional musician.

P2: All professional musicians have pony-tail.

Therefore: Bertil has pony-tail.

einstein s postulates for the special theory of relativity
Einstein's Postulates for the Special Theory of Relativity
  • The laws of physics are the same in all reference frames.
  • The speed of light through a vacuum (300,000,000 m/s) is constant as observed by any observer, moving or stationary.
slide11

Euclid's fifth axiom (parallel axiom):

For each point P and each line l, there exists at most one line through P parallel to l.

different geometries
Different Geometries
  • Euclidean: Given a line L and a point P not on L, there is exactly one line passing through P, parallel to L.
  • Hyperbolic: Given a line L and a point P not on L, there are at least two lines passing through P, parallel to L.
  • Elliptic: Given a line L and a point P not on L, there are no lines passing through P, parallel to L.
slide13

Hyperbolic

Euclidian

Elliptic

Sum of the angles:

slide14

Premises/Postulates/Axioms Conclusion

Logic is about how to deduce, on mere form, a valid argument.

Valid is a semantic concept.

Deduction is a syntactic concept.

logic
Logic
  • Propositional calculus
  • Quantification theory (predicate logic)
slide16

Proposition

Propositions are expressed, in natural language, in sentences.

It is raining.

It is snowing.

Where is Jack? (NOT a proposition)

Propositions are declarative sentences: saying something that is true or false.

slide17

More examples:

Napoleon was German.

All men are mortal.

Tweety is a robin.

Oxygen is an element.

Jenkins is a bachelor.

No bachelor are married.

If it is raining then it is it is snowing.

It is not raining.

It is raining or it is snowing

It is raining and it is snowing.

slide18

Propositional CALCULUS uses variables for propositions and study the form, not the content (semantics), in order to deduce valid conclusions.

If it is raining then it is it is snowing.

It is not raining.

It is raining or it is snowing

It is raining and it is snowing.

If A then B

not A

A or B

A and B

slide20

We shall treat propositions as unanalyzed, thus making no attempt to discern their logical structure.

We shall be concerned only with the relations between propositions, and then only insofar as those relations concern truth or falsehood.

slide21

OR

Either you wash the car or you cut the grass.

The symbol V (from Latin vel) is used to indicate ‘or’

as in ‘Either A or B’.

V is inclusive.

slide22

Exact definition of V.

Lower-case letters ‘p’, ‘q’, ‘r’, etc., are used for propositional variables, just as ‘x’, ‘y’ are numerical variables.

p q p Vq

F F F

F T T

T F T

T T T

slide23

‘Socrates is alive V Plato is alive’ is false.

‘Socrates is alive V Plato is dead’ is true.

slide24

p q p V q

0 0 0

0 1 1

1 0 1

1 1 1

slide25

Not:

0 1

1 0

slide26

AND:

Socrates is alive Plato is alive

  • 0 0 0
  • 0 1 0
  • 0 0
  • 1 1 1
slide27

T T F T F F T

T F F F T T F .

.

.

slide28

How is ‘if … then ___’ represented?

An ‘if … then ___’ is called a conditional.

The proposition replacing ‘…’ is called the antecedent,and that replacing the ‘___’ is called the consequent.

‘if it isn’t raining, then he is at game’

antecedent

consequent

slide29

Truth value for conditionals.

Things that are q

Things that are p

Things that are

neither p nor q

slide30

Truth Table

0 0 1

0 1 1

1 0 0

1 1 1

slide31

if p, then q

p only if q

q

p

You have malaria only if you have fever.

slide32

Only men are whisky-drinkers.

Only M is W If W then M

slide33

Equivalence

De Morgan’s laws:

slide34

Tautology is a proposition that is always true.

Contradiction is a proposition that is always

false:

Falsum: symbolizes a contradiction

slide35

if is a tautology, then is a contradiction.

if is a contradiction, then is a tautology.

slide38

1

1

1

Compare:

slide39

1

2

1

2

Compare:

slide40

1

1

Compare

proof or deduction
Proof or Deduction

stands for a proof: there is one or more inferences that together lead to C.

slide42

A Proof

1

2

1

1

2

slide43

Soundness and completeness

A logic is sound if a deduction yields a valid argument.

It is complete if there is a deduction for a valid argument.

quantification theory
Quantification Theory

For all member in a set …

There exist a member in a set …

slide45

All teachers are friendly.

For all x (if x is a teacher then x is friendly)

Some teachers are unfriendly

There is (exists) a teacher such that

(x is a teacher and x is unfriendly)

slide46

All teachers are friendly.

(if x is a teacher then x is friendly)

Some teachers are unfriendly

(x is a teacher and x is unfriendly)

slide47

All teachers are friendly.

Some teachers are unfriendly

slide48

All teachers are friendly.

Some teachers are unfriendly

slide49

A predicate is expressed by an incomplete sentence or sentence skeleton containing an open place.

“___ is a man” expresses a predicate.

When we fill the open place with the name of a subject, such as Socrates, the sentence “Socrates is a man” is obtained.

slide50

Consider the skeleton “___ loves ___”.

In grammatical terminology, this consists of a transitive verb and two open places, one to be filled by the name of a subject, such as “Jane”, the other of an object, such as “John”.

Binary relation

slide51

Another example:

“___ is less than ___”.

“2 is less than 3”.

In mathematical language:

slide53

“___ is a man”

“___ loves ___”

slide54

Existential quantifier

There is something with a particular property.

slide55

Universal quantifier

Every man is mortal.

slide56

there are right-angled triangles

OR

there is a triangle that is right-angled

OR

there is a triangle with the property of being right-angled.

slide57

All teachers are friendly.

Some teachers are unfriendly

slide59

Problems

1. Interpretation

slide60

Three different interpretations:

  • S is the “security number of a person”.
  • S is the “successor function in arithmetic.
  • S is the “DNA sequence of a person”.
slide61

2. Difficulties in expressing natural language sentences.

  • All men are mortal.
  • Dog is a quadruped.
  • Only drunk drivers under eighteen cause
  • bad accident.
  • Driving is risky, if you are drunk.
slide63

All men are mortal.

Dog is a quadruped.

Only drunk drivers under eighteen cause bad accident.

Driving is risky, if you are drunk.

slide64

Al men are not good.

There are no good men.

Not all men are good.

slide65

Everything with F has G Nothing with F has G

Something with F has G Something with F has not G

slide66

Negation

can be read

‘it is not the case that all x have the property P’

i.e., ‘some x has not property P’

i.e., there exist some x which not has property P

i.e.,

slide67

‘there is no x with the property P’

i.e., all x have the property notP

interpretation
Interpretation

What does it mean for a quantified expression

to be true or false?

slide69

says that all element in a particular domain (a nonempty set) have the property P,

i.e., all x belongs to the set that is decided by the interpretation of P.

says that in the domain, decided by the interpretation of P,

there is at least one element with the property P.

slide70

The meaning of D: ‘x has the dog y’

Domain: Växjö.

The interpretation of D isthe all pairs such that p is the name of a person in Växjö that has a dog with the name d.

slide71

Let R stands for raven and B for black.

Then the sentence expresses that all ravens are black.

Black animals

Ravens

Animals

slide72

‘There is a dog that is toothless’.

Domain: All animals

If there is a dog that is toothless the sentence is true.