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LogicWhat 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
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A valid argument is one whose conclusion is true in every case in which all its premises are true.
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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
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P1: All geniuses are illogical.P2: Some politicians are illogical.
Conclusion: Some politicians are geniuses.
Valid or not?
Not valid
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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.
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P1
P2
.
.
.Pn
C
IF P1 and P2 and … Pn THEN C
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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.
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PostulatesAxioms
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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.
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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.
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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.
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Euclidian Elliptic Hyperbolic
Sum of the angles:
180 180 180
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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.
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Logic
• Propositional calculus
• Quantification theory (predicate logic)
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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.
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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 snowingIt is raining and it is snowing.
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If A then Bnot AA or BA and B
If it is raining then it is it is snowing.It is not raining.It is raining or it is snowingIt 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.
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Compare with mathematics.
))(( )1( 22 yxyxyx
)715)(715(715 )2( 22
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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.
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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.
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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 V q
F F FF T TT F TT T T
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‘Socrates is alive V Plato is alive’ is false.
‘Socrates is alive V Plato is dead’ is true.
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p q p V q
0 0 00 1 11 0 11 1 1
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Not:
raining.not isIt
or raining, isit that)case not the is(it :
raining. isIt :
A
A
pp
0 1 1 0
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AND: Socrates is alive Plato is alive
qpqp
0 0 00 1 01 0 01 1 1
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)( qpqpqqppqp
T T F T F F T T F F F T T F
. . .
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An ‘if … then ___’ is called a conditional. The proposition replacing ‘…’ is called the antecedent, and that replacing the ‘___’ is called the consequent.
How is ‘if … then ___’ represented?
‘if it isn’t raining, then he is at game’
antecedent consequent
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Truth value for conditionals.
qthenpif
Things that are q
Things that are p
qp
Things that are neither p nor q
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Truth Table
qpqp
0 0 10 1 11 0 01 1 1
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qp
if p, then q
p only if q
q
p
You have malaria only if you have fever.
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Only men are whisky-drinkers.
Only M is W If W then M
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Equivalence
qpqp
De Morgan’s laws:
qpqp
qpqp
qp
)(
)(
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Tautology is a proposition that is always true.
)( pqp
Contradiction is a proposition that is always false:
Falsum: symbolizes a contradiction
any ,
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if is a tautology, then is a contradiction.
if is a contradiction, then is a tautology.
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Inference rules
ponens) (modus ,
n eliminatio
negation Double
and on introducti
,,
n eliminatio
,on introducti
and n eliminatio
p
qpp
p
p
qp
q
qp
p
q
qrqprp
qp
qp
q
qp
p
qp
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introduction
qp
q
qp
p
and .1
qpqp
.2
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I I
E E
pqqppq
pqp
qqp
1 1
1
qpqp
Compare:
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I])[(
I )(
E
ppp
pp1
1
2
2
qpqp
Compare:
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I )(
I
pqppq
p
qp
q
1
1
Compare
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Proof or Deduction
CP,PP n ,...,21
stands for a proof: there is one or more inferences that together lead to C.
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I ][)]([
I
E
)( E
E
rqprqprqp
rrq
rqppqp
qqp1
1
12
2
A Proof
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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.
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Quantification Theory
For all member in a set …
There exist a member in a set …
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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)
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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)
x
x
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All teachers are friendly.
Some teachers are unfriendly
friendly) is teacher a is ( xxx
)unfriendly is teacher a is ( xxx
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All teachers are friendly.
Some teachers are unfriendly
friendly) is teacher a is ( xxx
friendly)not is teacher a is ( xxx
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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.
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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
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Another example:
“___ is less than ___”.
“2 is less than 3”.
In mathematical language:
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)(SocratesMan
),( JohnJaneLoves
ohnJaneLovesJalternatively
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“___ is a man”
“___ loves ___”
)(xMan
),( yxLoves
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)(xxUnicorn
Existential quantifier
There is something with a particular property.
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Universal quantifier
))()(( xMortalxManx
Every man is mortal.
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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.
))()(( xangledrightxtrianglex
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All teachers are friendly.
Some teachers are unfriendly
))( )(( xFxTx
))( )(( xFxTx
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)))((0( ySxyxx
Combinations of universal and existential quantifiers.
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Problems
1. Interpretation
yxySxS )()(
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yxySxS )()(
Three different interpretations:
1. S is the “security number of a person”.2. S is the “successor function in arithmetic.3. S is the “DNA sequence of a person”.
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2. Difficulties in expressing natural language sentences.
a. All men are mortal.b. Dog is a quadruped.c. Only drunk drivers under eighteen cause bad accident.a. Driving is risky, if you are drunk.
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All have the same form:
))()(( xBxAx
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All men are mortal.
Dog is a quadruped.
Only drunk drivers under eighteen cause bad accident.
Driving is risky, if you are drunk.
mortal is )(man a is )( xxBxxA
quadruped a is )( dog a is )( xxBxxA
18under driver drunk a is )(
accident bad causes )(
xxB
xxA
risky is drive tofor )( drunk is )( xxBxxA
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Al men are not good.
There are no good men.
Not all men are good.
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Everything with F has G Nothing with F has G
))()(( xGxFx ))()(( xGxFx
Something with F has G Something with F has not G
))()(( xGxFx ))()(( xGxFx
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Negation
)(xxP 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., )(xPx
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)()( xPxxxP
‘there is no x with the property P’i.e., all x have the property notP
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Interpretation
What does it mean for a quantified expression to be true or false?
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)(xxPsays 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.
)(xxPsays that in the domain, decided by the interpretation of P, there is at least one element with the property P.
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),( yxD
The meaning of D: ‘x has the dog y’Domain: Växjö.
The interpretation of D is the all pairs such that p is the name of a person in Växjö that has a dog with the name d.
dp,
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))()(( xBxRx
Let R stands for raven and B for black.
Then the sentence expresses that all ravens are black.
Black animals
Ravens
Animals
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))()(( xTxDx
‘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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END
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