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INTRODUCTION TO ARTIFICIAL
INTELLIGENCE
Massimo Poesio
LECTURE 2: Logic, propositional
calculus
WHAT KIND OF KNOWLEDGE HUMANS
HAVE?
[Bach]’s air followed. Mr. Stolzman tied the
composer in by proclaiming him the great
improviser of the 18th century (ARRAU corpus)
p: Bach is a composer
[Swallows] are also mentioned in the Quran as
attacking Abraha, the Christian king of Yemen
who was attacking Mecca. The tiny birds hurled
small stones and forced Abraha's mighty army
to retreat. (Wikipedia)
q: Swallows are birds
REPRESENTING KNOWLEDGE: LOGIC
• Logic is the branch of mathematics /
philosophy concerned with knowledge and
reasoning
• Classical logic (as developed by Frege and
colleagues) has limitations but it is still the
starting point for any study of knowledge and
inference
– It also introduces a mathematical rigor that is
essential
LOGIC
• Aristotle distinguished between three types of
arguments:
– LOGIC arguments: that produce true conclusions
from true premisses
– DIALECTIC arguments: that produce true
conclusions from plausible premisses
– SOPHISTIC arguments: incorrect
VALID ARGUMENTS: CONNECTIVES
If students like AI, Massimo is happy
Students like AI
Massimo is happy
VALID ARGUMENTS: QUANTIFIERS
Birds fly
Swallows are birds
Swallows fly
INVALID ARGUMENTS (FALLACIES)
If students like AI, Massimo is happy
Massimo is happy
Students like AI
TWO MAIN FORMS OF LOGIC
• PROPOSITIONAL CALCULUS
– Valid arguments involving CONNECTIVES
– Propositions remain unanalyzed
• PREDICATE CALCULUS
– Analyze propositions into PREDICATES and
ARGUMENTS
– This makes it possible to study valid arguments
involving QUANTIFIERS, as well (FIRST ORDER LOGIC)
– (A generalization of Syllogism logic)
CHARACTERISTICS OF A MODERN
LOGIC
• VOCABULARY: the set of SYMBOLS in the
language
• SYNTAX: a set of rules to combine symbols
into phrases
• SEMANTICS: the interpretation of the symbols
and the phrases
• A PROOF THEORY: a system of formal rules to
derive formulas from other formulas
– In classical logic, preserving VALIDITY
PROPOSITIONAL CALCULUS
• The logic of CONNECTIVES: and, or, not, if …
then
• Originally formulated by the Stoics (Crisippo)
REPRESENTING KNOWLEDGE IN
LOGIC: PROPOSITIONAL CALCULUS
• p: students like AI
• q: Massimo is happy
• p  q?
• r: Every bicycle has 2 wheels
THE CONNECTIVES: CONJUNCTION
• p: Trento is in Northern Italy
• q: Trento has 100,000 inhabitants
• p & q: Trento is in Northern Italy
and
(Trento) has 100,000 inhabitants
• Trento and Bolzano are in Northern Italy
• The Facolta’ di Lettere has offices in Palazzo
Verdi and Santa Chiara
THE CONNECTIVES: NEGATION
• p: The University of Trento has a Faculty of
Medicine
• ~p: It is not the case that the University of
Trento has a Faculty of Medicine
• ~p: The University of Trento does not have a
Faculty of Medicine
• ~p: There is no Faculty of Medicine at the
University of Trento
THE CONNECTIVES: DISJUNCTION
• p: Trento has 100,000 inhabitants
• q: Trento has 110,000 inhabitants
• p \/ q: Trento has 100,000
or
(Trento has) 110,000 inhabitants
• Massimo studied in Torino or Rochester
THE CONNECTIVES: IMPLICATION
• If students like AI, Massimo is happy
– Two clear cases:
• If the students like AI and Massimo is happy, the implication
is true
• If the students like AI, but Massimo is not happy, the
implication is false
– There are also difficult cases:
• If the students don’t like AI, is the implication true or false?
• Convention ever since Filone of Megara: yes!
• Argument: implication does not make any claim at all about
these cases)
• Unpleasant consequence: If 2+2=5, I am the pope: true!!
THE CONNECTIVES: BICONDITIONAL
• I will go walking if I get my car fixed
– IMPLICATION: only false if I get my car fixed but then I
don’t go walking. OK if I go walking even if I don’t get
my car fixed
– Often more intuitive if reverse: If I get my car fixed, I
will go walking.
• I will go walking if, and only if, I get my car fixed
– BICONDITIONAL: if I don’t get my car fixed I don’t go
walking
– Getting the car fixed is a NECESSARY and SUFFICIENT
condition to go walking
PROPOSITIONAL CALCULUS: SYNTAX
• Vocabulary:
– PROPOSITIONAL SYMBOLS: p, q, r …
– CONNECTIVES: ~, &, ∨, , <-->
• Combination rules:
– If p is a propositional symbol, then p is an (atomic)
formula
– If α is a formula, then ~α is a formula
– If α and β are formulas, then α & β is a formula
– If α and β are formulas, then α ∨ β is a formula
– If α and β are formulas, then α  β is a formula
– If α and β are formulas, then α <--> β is a formula
PRACTICE WITH THE SYNTAX OF THE
PROPOSITIONAL CALCULUS
•
•
•
•
•
Today is October 2nd
Trento is in Italy
Sunday is a holiday and there are no lectures
There is no mensa in Palazzo Verdi
If we don’t finish the lecture in time we will not get
lunch
• Students in Filosofia e Linguaggi della Modernita’
follow either the percorso in Filosofia, logica e
informatica, or the percorso in Etica, politica e scienze
delle religioni, or the percorso Storia della filosofia e
scienze umane
THE OTHER WAY AROUND
•
•
•
•
p&q
p \/ q
~p \/ q
p  ~q
PROPOSITIONAL CALCULUS:
SEMANTICS
• TERTIUM NON DATUR: every proposition is
either TRUE (T, 1) or FALSE (F, 0)
– p \/ ~p
– Aristotle, De Interpretatione, Metaphysics: “it will
not be possible to be and not to be the same
thing”
PROPOSITIONAL CALCULUS: COMPUTING
THE TRUTH VALUE OF FORMULAE
• Atomic formulae are either TRUE (T, 1) or
FALSE (F, 0)
• The truth of more complex formulae can be
computed using TRUTH TABLES
PROPOSITIONAL CALCULUS: TRUTH
TABLE FOR CONJUNCTION
PROPOSITIONAL CALCULUS:
TT FOR NEGATION
THE TRUTH TABLES FOR OR
• Two types of OR in natural language:
– INCLUSIVE OR (Latin: vel)
– EXCLUSIVE OR (Latin: aut .. aut)
• Inclusive or usually adopted
TRUTH TABLE FOR INCLUSIVE OR
IMPLICATION
• Remember Filone’s proposal:
– Antecedent true, consequent true (true)
– Antecedent true, consequent false (false)
– Antecedent false, consequent true: TRUE
– Antecedent false, consequent false: TRUE
– I.e., p  q = ~p \/ q
TRUTH TABLE FOR IMPLICATION
TRUTH TABLE FOR BI-IMPLICATION
p
q
p <--->q
1
1
1
1
0
0
0
1
0
0
0
1
USING TRUTH TABLES TO COMPUTE
THE TRUTH OF COMPLEX SENTENCES
p
q
r
(p & q)
p \/ r
~ (p \/ r )
((p&q)~(p\/r))
1
1
1
1
1
0
0
1
1
0
1
1
0
0
1
0
1
0
1
0
1
1
0
0
0
1
0
1
0
1
1
0
1
0
1
0
0
1
0
1
0
1
0
0
0
0
0
1
1
TAUTOLOGIES, CONTRADICTIONS, AND
CONTINGENCIES
• TAUTOLOGY: a statement that is always true
(the last column in the TT is always 1)
• CONTRADICTION: a statement that is always
false
• CONTINGENCY: a statement that is true or
false depending on the values of the
propositional variables
EXAMPLES
• TAUTOLOGY: p \/ ~p, (p  (q  p))
• CONTRADICTION: p & ~ p
• CONTINGENCIES: p, p \/ p
LOGICAL EQUIVALENCES
• If a statement with a biconditional is a
tautology, the component statements are
LOGICALLY EQUIVALENT
• Example: De Morgan’s laws
– ~ (p \/ q) <--> ~p & ~ q
– ~ (p & q) <--> ~p \/ ~ q
EXERCISES
• Using truth tables, show that:
– IDEMPOTENCE: (p \/ p) <--> p
– ASSOCIATIVITY: ( (p \/ q) \/ r) <--> (p \/ (q \/r ))
– DISTRIBUTIVITY:
(p \/ (q & r)) <--> ((p \/ q) & (p \/ r))
NATURAL DEDUCTION
• Now we can go back to the original motivation
for developing a logic: having a way of
checking which ARGUMENTS are VALID
ARGUMENTS
• An argument consists of
– A number of statements called PREMISSES,
assumed to be true (for the sake of the argument)
– A statement called the CONCLUSION
– Claim: the truth of the conclusion follows from the
truth of the premisses
• An argument is VALID iff in any situation in
which the premisses are true, the conclusion
is also true
AN EXAMPLE OF ARGUMENT
If students like AI, Massimo is happy
Students like AI
Massimo is happy
SEMANTIC PROOF OF VALIDITY
If P  Q is a tautology, then
P
Q
Is a VALID ARGUMENT, and
Q is said to be a LOGICAL
CONSEQUENCE of P
PROVING VALIDITY
Demonstrating the validity
of the argument:
If students like AI, Massimo is happy
Students like AI
Massimo is happy
Amounts to showing that
((p  q) & p)  q)
Is a tautology
NON-VALIDITY
By contrast, the argument:
If students like AI, Massimo is happy
Massimo is happy
Students like AI
Is not a tautology
MODUS PONENS
The valid arguments of the
form
If P, then Q
P
Q
Are called MODUS PONENS
DOING EVERYTHING SEMANTICALLY IS
A PAIN
• Once we go beyond the simpler cases, proving
logical equivalence and logical consequence
using truth tables is very space- and timeconsuming
– The number of rows requested for a formula with
n propositional atoms is 2n
• The answer: use rules for deriving formulas
from other formulas that are known to
preserve validity
THE SOLUTION: PROOF THEORY
• A set of valid arguments, or INFERENCE
RULES, for the logic that make it possible to
derive all and only the consequences of an
initial set of statements
VALIDITY VS. PROVABILITY
• Every modern logic has TWO notions of valid
argument:
– SEMANTIC: an argument is demonstrated to be
true on SEMANTIC grounds (in propositional logic)
• P |= Q
– PROOF-THEORETICAL: argument is valid because
the conclusion can be derived from the premisses
applying valid patterns of inference
• P |- Q
AXIOMS OF THE STOICS
• MODUS PONENS:
– P  Q, P |- Q
• MODUS TOLLENS:
– P  Q, ~Q |- ~P
– If it’s day time, there is light. There is no light. Therefore,
it’s not daytime.
• DISJUNCTIVE SYLLOGISM:
– A \/ B, ~A, |- B
– It is either daytime or nighttime. It’s not daytime.
Therefore, it’s night time.
• TWO MORE LESS USED THESE DAYS
– E.g., ~ (A & B), A, |- ~B
INFERENCE RULES FOR THE
PROPOSITIONAL CALCULUS
•
•
•
•
MODUS PONENS
MODUS TOLLENS
DISJUNCTIVE SYLLOGISM
HYPOTHETICAL SYLLOGISM
– P  Q, Q  R |- P  R
– If Massimo lives in Trento, he lives in Italy. If he lives in Italy, he
lives in Europe. Therefore, if Massimo lives in Trento, he lives in
Europe.
• CONJUNCTION ELIMINATION
– P & Q |- P
• CONJUNCTION INTRODUCTION
– P, Q |- P&Q
• DISJUNCTION INTRODUCTION
– P |- P \/ Q
AUTOMATED REASONING
• What we have discussed so far is ‘pure’ Logic
• The contribution of AI to our understanding of
reasoning is the development of AUTOMATIC
methods of reasoning
• Two methods are particularly well known:
– TABLEAUX
– RESOLUTION
• We will concentrate on the first
(SEMANTIC) TABLEAUS
• A SYNTACTIC method for proving a formula
(only use rules that work on the basis of the
syntactic structure of the formula)
• By REFUTATION:
– Show that all attempts to falsify the formula fail
(i.e., lead to a CONTRADICTION)
TABLEAUS: THE METHOD
• Start by adding to the tableaus the negation of
A
• Then apply TABLEAU EXPANSION RULES
• Stop when all paths contain a contradiction, or
no more rules apply
• If all paths contain a contradiction, A is valid
TABLEAU EXPANSION RULES
TABLEAUS: AN EXAMPLE
Suppose you want to prove that the following is valid:
(~p  ~q) \/ q
Then you try to prove that the negation leads to
contradiction:
(~ p & ~~ q) & ~q
You make that the first line in your tableau:
1.
(~ p & ~~ q) & ~q
TABLEAUS EXAMPLE
PROPOSITIONAL LOGIC ON THE WEB
• There are several demonstrators available for
both propositional calculus and first order
logic
• Some can be downloaded, some can be
accessed directly on the web
• Example of the second:
www.thelogiccourse.com/bluestorm/builder.html
PREDICATE CALCULUS
• In predicate calculus, o predicate logic, atomic
statements are decomposed into TERMS and
PREDICATES
– Massimo is happy: HAPPY(m)
– Students like AI: LIKE(students,AI)
• In this way it is possible to state general
properties about predicates: for instance,
every professor at the University of Trento is
happy, etc.
READINGS
• B. Partee, A. ter Meulen, R. Wall,
Mathematical Methods in Linguistics,
Springer, ch. 5, 6, 7
• P. Odifreddi, Le menzogne di Ulisse, Tea, ch. 17
• P. Blackburn, J. Bos, Representation and
Inference for Natural Language, CSLI
• Logic on the Web:
– http://www.thelogiccourse.com/