Transcript Chapter 7 Propositional and Predicate Logic 1

Chapter 7
Propositional and Predicate Logic
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Chapter 7 Contents (1)
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What is Logic?
Logical Operators
Translating between English and Logic
Truth Tables
Complex Truth Tables
Tautology
Equivalence
Propositional Logic
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Chapter 7 Contents (2)
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Deduction
Predicate Calculus
Quantifiers  and 
Properties of logical systems
Abduction and inductive reasoning
Modal logic
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What is Logic?
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Reasoning about the validity of arguments.
An argument is valid if its conclusions
follow logically from its premises – even if
the argument doesn’t actually reflect the
real world:
 All lemons are blue
 Mary is a lemon
 Therefore, Mary is blue.
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Logical Operators
And
 Or
 Not
 Implies
 Iff
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Λ
V
¬
→
↔
(if… then…)
(if and only if)
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What is a Logic?
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What is a Logic?
_ A logic consists of three components:
1. Syntax: A language for stating
propositions/sentences.
2. Semantics: A way of determining whether a
given proposition/sentence is true or false.
(Model theory)
3. Inference system: Rules for
inferring/deducing theorems from other
theorems.
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Translating between English and Logic
Facts and rules need to be translated
into logical notation.
 For example:
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It is Raining and it is Thursday:
R Λ T
R means “It is Raining”, T means “it is
Thursday”.
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Translating between English and Logic
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More complex sentences need
predicates. E.g.:
It is raining in New York:
R(N)
Could also be written N(R), or even just R.
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It is important to select the correct
level of detail for the concepts you
want to reason about.
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Truth Tables
Tables that show truth values for all
possible inputs to a logical operator.
 For example:
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A truth table shows the semantics of
a logical operator.
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Complex Truth Tables
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We can produce
truth tables for
complex logical
expressions, which
show the overall
value of the
expression for all
possible
combinations of
variables:
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Tautology
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The expression A v ¬A is a tautology.
This means it is always true, regardless of the value of
A.
A is a tautology: this is written
╞ A
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A tautology is true under any interpretation.
Example: A
A
A V ¬A
An expression which is false under any interpretation is
contradictory.
Example: A Λ ¬ A
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Equivalence
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Two expressions are equivalent if
they always have the same logical
value under any interpretation:
A Λ B  B Λ A
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Equivalences can be proven by
examining truth tables.
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Some Useful Equivalences
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A
A
A
A
A
A
A
vAA
ΛAA
Λ (B Λ C)  (A Λ B) Λ C
v (B v C)  (A v B) v C
Λ (B v C)  (A Λ B) v (A Λ C)
Λ (A v B)  A
v (A Λ B)  A
A Λ true  A
A v true  true
A Λ false  false
A v false  A
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Propositional Logic
Propositional logic is a logical system.
 It deals with propositions.
 Propositional Calculus is the language
we use to reason about propositional
logic.
 A sentence in propositional logic is
called a well-formed formula (wff).
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Propositional Logic
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The following are wff’s:
P, Q, R…
true, false
(A)
¬A
AΛB
AvB
A→B
A↔B
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Deduction
The process of deriving a conclusion from a set of
assumptions.
 Use a set of rules, such as:
A
A→B
B
If A is true, and A implies B is true, then we
know B is true.
 (Modus Ponens)
 If we deduce a conclusion C from a set of
assumptions, we write:
 {A1, A2, …, An} ├ C
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Deduction - Example
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Predicate Logic
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The first of these, predicate logic,
involves using standard forms of
logical symbolism which have been
familiar to philosophers and
mathematicians for many decades.
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Most simple sentences,
 for example, ``Peter is generous'' or
``Jane gives a painting to Sam,''
 can be represented in terms of logical
formulae in which a predicate is
applied to one or more arguments
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Predicate Calculus
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Predicate Calculus extends the
syntax of propositional calculus with
predicates and quantifiers:
P(X) – P is a predicate.
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First Order Predicate Calculus
(FOPC) allows predicates to apply to
objects or terms, but not functions or
predicates.
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Quantifiers  and 
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 - For all:
 xP(x) is read “For all x’es, P (x) is true”.
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 - There Exists:
 x P(x) is read “there exists an x such that P(x)
is true”.
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Relationship between the quantifiers:
 xP(x)  ¬(x)¬P(x)
 “If There exists an x for which P holds, then it is not
true that for all x P does not hold”.
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Existential Quantifier
 -”there exists”
There are times when, rather than
claim that something is true about all
things, we only want to claim that it is
true about at least one thing.
 For example, we might want to make
the claim that "some politicians are
honest," but we would probably not
want to claim this universally.
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A way that mathematicians often phrase
this is "there exists a politician who is
honest."
Our abbreviation for "there exists" is " ",
which is called the existential quantifier
because it claims the existence of
something.
If we use P for the predicate "is a
politician" and H for the predicate "is
honest," we can write "some politicians
are honest" as:
x[Px Hx].
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Properties of Logical Systems
Soundness: Is every theorem valid?
 Completeness: Is every tautology a
theorem?
 Decidability: Does an algorithm exist
that will determine if a wff is valid?
 Monotonicity: Can a valid logical
proof be made invalid by adding
additional premises or assumptions?
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Abduction and Inductive Reasoning
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Abduction:
B
A→B
A
Not logically valid, BUT can still be useful.
In fact, it models the way humans reason all
the time:
 Every non-flying bird I’ve seen before has been a
penguin; hence that non-flying bird must be a
penguin.
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Not valid reasoning, but likely to work in many
situations.
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Inductive Reasoning
Inductive Reasoning enable us to
make predictions based on what has
happened in the past.
 Example: “The Sun came up
yesterday and the day before, and
everyday I know before that, so it will
come up again tomorrow.”
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Three Kinds of Reasoning
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Broadly speaking there are 3 kinds of
reasoning:
deductive – Based on the use of modus
ponens and other deductive rules and
reasoning.
abductive – Based on common fallacy.
inductive – Based on history (what has
happened in the past)
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Examples
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A deductive argument consists of n
premisses and a conclusion.
If the argument is valid, then if the
premisses are true the conclusion must be
true:
Premiss 1: If it's raining then the streets
are wet
Premiss 2: It's raining
----------------Therefore the streets are wet
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All horses have brains
Herman is a horse
-------------Therefore Herman has a brain
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When Conclusion Does Not Follow
From the Premisses
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The following are invalid:
If it's raining then the streets are wet
The streets are wet
--------------Therefore it's raining
All horses have brains
Herman has a brain
--------------Therefore Herman is a horse
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Examples of Invalid Arguments
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The following two arguments are invalid:
If it's raining then the streets are wet
The streets are wet
-------------Therefore it's raining
All horses have brains
Herman has a brain
-------------Therefore Herman is a horse
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More on Deductive Reasoning
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An argument can have any number of
premisses:
If p then q
If q then r
If r then s
If s then t
p
------Therefore t
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Abductive reasoning
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Abduction is "reasoning backwards". We
start with some facts and reason back to a
hypothesis. E.g.
If someone has measles they have spots
and a sore throat
Jimmy has spots and a sore throat
-----------------------Therefore Jimmy has measles
This isn't formally valid, of course. In fact it
is a famous fallacy, called "confirming the
consequent".
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An Earlier Example
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If it's raining then the streets are wet
The streets are wet
-------------Therefore it's raining
Nevertheless this does seem to be how
doctors work.
They use abduction to generate
hypotheses, which they then test (for
instance, by doing a blood test).
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Inductive reasoning
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Inductive reasoning is reasoning from
particular cases or facts to a general
conclusion:
raven 1 is black
raven 2 is black
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raven n is black
----------Therefore all ravens are black
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More Examples
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horse 1 has a brain
horse 2 has a brain
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horse n has a brain
------------Therefore all horses have brains
These go from SOME to ALL:
All observed (i.e. some) Xs have property P
------------------------------Therefore all Xs have P
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Limitations
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This isn't formally valid.
The conclusion does not formally follow
from the observed facts.
At one time people believed that all
observed swans are white, therefore all
swans are white.
This is false, of course, because there are
black swans in Western Australia!
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Modal logic
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Modal logic is a higher order logic.
Allows us to reason about certainties, and
possible worlds.
If a statement A is contingent then we say that
A is possibly true, which is written:
◊A
If A is non-contingent, then it is necessarily
true, which is written:
A
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Reasoning in Modus Logic
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The following rules are examples of the
axioms that can be used to reason in
modus logic:
A ◊A
 ¬A ¬◊A
◊A
¬A
We cannot draw truth tables to prove them;
however, you can reason by your
understanding of the meaning of the
operators.
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Class Exercise
Draw a truth table for the following
expressions:
 1. ¬AΛ(AVB)Λ(BVC)
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2. ¬AΛ(AVB)Λ(BVC)Λ¬D
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