1. Propositional Logic Computational Logic Lecture 2
Michael Genesereth Autumn 2009

2. Complexity
The cherry blossoms in the spring … sank.
2/24/2019

3. Ambiguity Leland Stanford Junior University
2/24/2019

4. Ambiguity
There’s a girl in the room with a telescope.
2/24/2019

5. Ambiguity Lettuce won’t turn brown if you put your head
in a plastic bag before placing it in the refrigerator.
The manager of a nudist colony complains that
a hole was cut in the wall surrounding the camp.
Police are looking into it.
2/24/2019

6. Ambiguity - Headlines Crowds Rushing to See Pope Trample 6 to Death
Tips to Help You Prevent Headaches After You Die
Bundy Beats Date With Chair
Police Oversight Group Likes San Jose Model
Japanese Scientists Grow Frog Eyes and Ears
Food Stamp Recipients Turn to Plastic
Indian Ocean Talks
2/24/2019

7. Meaning not Syntax Champagne is better than soda.
Soda is better than swill.
Therefore, champagne is better than swill.
Bad sex is better than nothing.
Nothing is better than good sex.
Therefore, bad sex is better than good sex.
2/24/2019

8. Propositional Languages
A propositional signature is a set/sequence of primitive symbols, called proposition constants.
Given a propositional signature, a propositional sentence is either (1) a member of the signature or (2) a compound expression formed from members of the signature. (Details to follow.)
A propositional language is the set of all propositional sentences that can be formed from a propositional signature.
2/24/2019

9. Proposition Constants
By convention (in this course), proposition constants are written as strings of alphanumeric characters beginning with a lower case letter.
Examples:
raining
r32aining
rAiNiNg
rainingorsnowing
Non-Examples:
324567
raining.or.snowing
2/24/2019

10. Compound Sentences Negations: raining
The argument of a negation is called the target.
Conjunctions:
(raining  snowing)
The arguments of a conjunction are called conjuncts.
Disjunctions:
(raining  snowing)
The arguments of a disjunction are called disjuncts.
2/24/2019

11. Compound Sentences (concluded)
Implications:
(raining  cloudy)
The left argument of an implication is the antecedent.
The right argument is the consequent.
Reductions:
(cloudy  raining)
The left argument of a reduction is the consequent.
The right argument of a reduction is the antecedent.
Equivalences:
(cloudy  raining)
2/24/2019

12. Parenthesis Removal Dropping Parentheses is good: (p  q)  p  q
But it can lead to ambiguities:
((p  q)  r)  p  q  r
(p  (q  r))  p  q  r
2/24/2019

13. Precedence
Parentheses can be dropped when the structure of an expression can be determined by precedence.   
  
NB: An operand associates with operator of higher precedence. If surrounded by operators of equal precedence, the operand associates with the operator to the right.
p  q  r p  q  r p  q
p  q  r p  q  r
2/24/2019

14. Example o  (p  q)  (p  q) a  r  o b  p  q
s  (o  r)  (o  r)
c  a  b p q r s c o a b
2/24/2019

15. (r  ((p  q)  (p  q)))  (p  q)
Example
(r  ((p  q)  (p  q)))  (p  q) p q r
2/24/2019

16. Propositional Interpretation
A propositional interpretation is an association between the propositional constants in a propositional language and the truth values T or F.
We sometimes view an interpretation as a Boolean vector of values for the items in the signature of the language (when the signature is ordered).
i = TFT
2/24/2019

17. Sentential Interpretation
A sentential interpretation is an association between the sentences in a propositional language and the truth values T or F.
pi = T (p  q)i = T
qi = F (q  r)i = T
ri = T ((p  q)  (q  r))i = T
A propositional interpretation defines a sentential interpretation by application of operator semantics.
2/24/2019

18. Operator Semantics Negation:
For example, if the interpretation of p is F, then the interpretation of p is T.
For example, if the interpretation of (pq) is T, then the interpretation of (pq) is F.
2/24/2019

19. Operator Semantics (continued)
Conjunction: Disjunction:
NB: The type of disjunction here is called inclusive or, which says that a disjunction is true if and only if at least one of its disjuncts is true. This contrasts with exclusive or, which says that a disjunction is true if and only if an odd number of its disjuncts is true.
2/24/2019

20. Operator Semantics (continued)
Implication: Reduction:
NB: The semantics of implication here is called material implication. Any implication is true if the antecedent is false, whether or not there is a connection to the consequent.
If George Washington is alive, I am a billionaire.
2/24/2019

21. Operator Semantics (concluded)
Equivalence:
2/24/2019

22. Evaluation Interpretation i: Compound Sentence (p  q)  (q  r)
2/24/2019

23. (r  ((p  q)  (p  q)))  (p  q)
Example
pi = T
qi = T
ri = T
(r  ((p  q)  (p  q)))  (p  q) p q r
2/24/2019

24. Multiple Interpretations
Logic does not prescribe which interpretation is “correct”. In the absence of additional information, one interpretation is as good as another.
Interpretation i Interpretation j
Examples:
Different days of the week
Different locations
Beliefs of different people
2/24/2019

25. Truth Tables A truth table is a table of all possible interpretations
for the propositional constants in a language.
One column per constant.
One row per interpretation.
For a language with n constants,
there are 2n interpretations.
2/24/2019

26. Properties of Sentences
Valid
Contingent
Unsatisfiable
A sentence is valid if and only if
every interpretation satisfies it.
A sentence is contingent if and only if
some interpretation satisfies it and
some interpretation falsifies it.
A sentence is unsatisfiable if and
only if no interpretation satisfies it.
2/24/2019

27. Properties of Sentences
Valid
Contingent
Unsatisfiable 
A sentences is satisfiable if and only
if it is either valid or contingent.
A sentences is falsifiable if and only
if it is contingent or unsatisfiable. 
2/24/2019

28. Example of Validity
2/24/2019

29. (p  (q  r))  ((p  q)  (p  r))
More Validities
Double Negation:
p  p
deMorgan's Laws:
(pq)  (pq)
(pq)  (pq)
Implication Introduction:
p  (q  p)
Implication Distribution
(p  (q  r))  ((p  q)  (p  r))
2/24/2019

30. Evaluation Versus Satisfaction
2/24/2019

31. ((r  ((p  q)  (p  q)))  (p  q))i = T
Example
pi = ?
qi = ?
ri = ?
((r  ((p  q)  (p  q)))  (p  q))i = T p q r
2/24/2019

32. Satisfaction Method to find all propositional interpretations that
satisfy a given set of sentences:
Form a truth table for the propositional constants.
(2) For each sentence in the set and each row in the truth table, check whether the row satisfies the sentence. If not, cross out the row.
(3) Any row remaining satisfies all sentences in the set. (Note that there might be more than one.)
2/24/2019

33. Satisfaction Example
qr
2/24/2019

34. Satisfaction Example (continued)
qr
p qr
2/24/2019

35. Satisfaction Example (concluded)
qr
p qr r
What if we were to add p to this set?
2/24/2019

36. Axiomatizability
A set of boolean vectors of length n is axiomatizable in propositional logic if and only if there is a signature of size n and a set of sentences from the corresponding language such that the vectors in the set correspond to the set of interpretations satisfying the sentences.
A set of sentences defining a set of vectors is called the axiomatization of the set of vectors.
2/24/2019

37. Example Set of bit vectors: {TFF, FTF, FTT} Signature: {p, q, r}
Axiomatization:
(p  q  r)  (p  q)
2/24/2019

38. The Big Game
Stanford people always tell the truth, and Berkeley people always lie. Unfortunately, by looking at a person, you cannot tell whether he is from Stanford or Berkeley.
You come to a fork in the road and want to get to the football stadium down one fork. However, you do not know which to take. There is a person standing there. What single question can you ask him to help you decide which fork to take?
2/24/2019

39. Basic Idea
2/24/2019

40. The Big Game Solved
Question: The left road the way to the stadium if and only if you are from Stanford. Is that correct?
2/24/2019