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## Logic What is it?

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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

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

P1: All geniuses are illogical.

P2: Some politicians are illogical.

Conclusion: Some politicians are geniuses.

Valid or not?

Not valid

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.

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

- 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.

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

- 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.

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

- Propositional calculus
- Quantification theory (predicate logic)

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.

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.

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

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.

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.

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

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

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

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

Only M is W If W then M

De Morgan’s laws:

Tautology is a proposition that is always true.

Contradiction is a proposition that is always

false:

Falsum: symbolizes a contradiction

if is a tautology, then is a contradiction.

if is a contradiction, then is a tautology.

Proof or Deduction

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

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

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

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)

(if x is a teacher then x is friendly)

Some teachers are unfriendly

(x is a teacher and x is unfriendly)

Some teachers are unfriendly

Some teachers are unfriendly

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.

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

“___ loves ___”

There is something with a particular property.

Every man is mortal.

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.

Some teachers are unfriendly

1. Interpretation

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”.

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.

Dog is a quadruped.

Only drunk drivers under eighteen cause bad accident.

Driving is risky, if you are drunk.

Everything with F has G Nothing with F has G

Something with F has G Something with F has not G

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.,

‘there is no x with the property P’

i.e., all x have the property notP

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.

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.

Let R stands for raven and B for black.

Then the sentence expresses that all ravens are black.

Black animals

Ravens

Animals

‘There is a dog that is toothless’.

Domain: All animals

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

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