Propositional Logic

1
Propositional Logic
2
Grammatical Complexity

• A natural language is complex. E.g.
• The dog chased the cat.
• The dog that ate the rat chased the cat.
• The cherry blossoms in the spring sank.

3
Ambiguity

• A natural language has ambiguity, e.g.
• Theres a girl in the room with a telescope.
• Which one of the figures the sentence intends to
  capture?

• In short we need to define a syntax.

4
Syntax
5
Propositional Constants

• Examples
• raining
• r32aining
• rAiNiNg
• rainingorsnowing
• Non-Examples
• 324567
• raining.or.snowing

6
Compound Sentences

• Negations
• Øraining
• Conjunctions
• (rainingÙsnowing)
• The arguments of a conjunction are called
  conjuncts.
• Disjunctions
• (rainingÚsnowing)
• The arguments of a disjunction are called
  disjuncts.

7
Compound Sentences

• Implications
• (raining Þ cloudy)
• The left argument of an implication is the
  antecedent.
• The right argument of an implication is the
  consequent.
• Reductions
• (cloudy Ü raining)
• The left argument of a reduction is the
  consequent.
• The right argument of a reduction is the
  antecedent.
• Equivalences
• (cloudy Û raining)
• Notice that the constituent sentences within any
  compound sentence can be either simple sentences
  or compound sentences or a mixture of two.

8
Parenthesis Removal

• One disadvantage of our notation, as written, is
  that the parentheses tend to build up and need to
  be matched correctly. It would be nice if we
  could dispense with parentheses, e.g. simplifying
  
• ((pÚq) Þ Ør) into
• pÚq Þ Ør
• Unfortunately, we cannot do without parentheses
  entirely, since then we would be unable to render
  certain sentences unambiguously.
• For example, the sentence shown above could have
  resulted from dropping parentheses from either of
  the following sentences.
• ((p Úq)Þ Ør )
• (p Ú(q Þ Ør ))

9
Operator Precedence

• The solution to this problem is the use of
  operator precedence. The following table gives a
  hierarchy of precedences for our operators.
• The Ø operator has higher precedence than Ù, Ù
  has higher precedence than Ú,and Ú has higher
  precedence than Þ,
• Ü, and Û.
• Ø
• Ù
• Ú
• ÜÛÞ

10
Parenthesis

• Note that just because precedence allows us to
  delete parentheses in some cases, this doesnt
  mean that we can dispense with parentheses
  entirely.
• E.g.
• (p Þ q) Ú (sÞ r)
• Here, we cannot remove the parenthesis

11
Semantics
12
Algebra and Logic

• The semantics of logic is similar to the
  semantics of algebra.
• Algebra is unconcerned with the real-world
  meaning of variables like x and y.
• What is interesting is the relationship between
  the variables expressed in the equations we
  write and
• algebraic methods are designed to respect these
  relationships, no matter what meanings or values
  are assigned to the constituent variables.
• In a similar way, logic itself is unconcerned
  with what sentences say about the world being
  described.
• What is interesting is the relationship between
  the truth of simple sentences and the truth of
  compound sentences within which the simple
  sentences are contained.
• Also, logical reasoning methods are designed to
  work no matter what meanings or values are
  assigned to the logical variables used in
  sentences.

13
Interpretation

• Although the values assigned to variables are not
  crucial in the sense just described, in talking
  about logic itself, it is sometimes useful to
  make these assignments explicit and to consider
  various assignments or all assignments and so
  forth. Such an assignment is called an
  interpretation.
• A propositional logic interpretation (i) is an
  association between the propositional constants
  in a propositional language and the truth values
  T or F.
• p ¾i T pi T
• q ¾i F can be written also qi F
• r ¾i T ri T

14
Operator Semantics

• The notion of interpretation can be extended to
  all sentences by application of operator
  semantics.
• Negation
• f Øf
• T F
• F T
• For example, if the interpretation of p is F,
  then the interpretation of Øp is T.
• For example, if the interpretation of Øp is T,
  then the interpretation of ØØp is F.

15
Operator Semantics (continued)

• Conjunction
• f y f Ùy
• T T T
• T F F
• F T F
• F F F
• Disjunction
• f y f Úy
• T T T
• T F T
• F T T
• F F F

16
Operator Semantics (continued)

• Implication
• f y f Þy
• T T T
• T F F
• F T T
• F F T
• Reduction
• f y f Üy
• T T T
• T F T
• F T F
• F F T

17
Evaluation

• We can easily determine for any given
  interpretation whether or not any sentence is
  true or false under that interpretation.
• The technique is simple. We substitute true and
  false values for the propositional constants and
  replace complex expressions with the
  corresponding values, working from the inside
  out.
• We say that an interpretation i satisfies a
  sentence if and only if it is true under that
  interpretation.
• As an example, consider the interpretation i show
  below.
• pi true
• qi false
• ri true
• We can see that i satisfies (pÚq)Ù(ØqÚr).
• (true Ú false) Ù (Øfalse Ú true)
• true Ù(Øfalse Ú true)
• true Ù(true Ú true)
• true Ù true
• true

18
Evaluation (continued)

• Now consider interpretation j defined as follows.
• pj true
• qj true
• rj false
• In this case, j does not satisfy (pÚq)Ù(ØqÚr).
• (true Ú true) Ù(Øtrue Ú false)
• trueÙ (Øtrue Ú false)
• true Ù( false Ú false)
• trueÙ false
• false
• Using this technique, we can evaluate the truth
  of arbitrary sentences in our language.
• The cost is proportional to the size of the
  sentence.

19
Reverse Evaluation

• Reverse evaluation is the opposite of evaluation.
  We begin with one or more compound sentences and
  try to figure out which interpretations satisfy
  those sentences.
• One way to do this is using a truth table for the
  language.
• A truth table for a propositional language is a
  table showing all of the possible interpretations
  for the propositional constants in the language.
• The interpretations i and j correspond to the
  third and seventh rows of this table,
  respectively.
• Note that, for a propositional language with n
  logical constants, there are n columns in the
  truth tables 2n rows.

p q r T T T T T F T F T T F
F F T T F T F F F T F F F
20
Validity, Satisfiability, Unsatisfiability

• A sentence is valid if and only if it is
  satisfied by every interpretation.
• p Ú Øp is valid.
• A sentence is satisfiable if and only if it is
  satisfied by at least one interpretation.
• We have already seen several examples of
  satisfiable sentences.
• A sentence is unsatisfiable if and only if it is
  not satisfied by any interpretation.
• p Û Øp is unsatisfiable. No matter what
  interpretation we take, the sentence is always
  false.
• In one sense, valid sentences and unsatisfiable
  sentences are useless. Valid sentences do not
  rule out any possible interpretations
  unsatisfiable sentences rule out all
  interpretations thus they say nothing about the
  world.
• On the other hand, from a logical perspective,
  they are extremely useful in that, as we shall
  see, they serve as the basis for legal
  transformations that we can perform on other
  logical sentences.
• Note that we can easily check the validity,
  satisfiability, or unsatisfiability of a sentence
  can easily by looking at the truth table for the
  propositional constants in the sentence.

• In one sense, valid sentences and unsatisfiable
  sentences are useless.
• Valid sentences do not rule out any possible
  interpretations
• Unsatisfiable sentences rule out all
  interpretations thus they say nothing about the
  world.
• On the other hand, they are very useful in that,
  they serve as the basis for legal transformations
  that we can perform on other logical sentences.
• Note that we can easily check the validity,
  satisfiability, or unsatisfiability of a sentence
  by looking at the truth table for the
  propositional constants in the sentence.

21
Entailment

• We would like to know, given some sentences,
  whether other sentences are or are not logical
  conclusions.
• This relative property is known as logical
  entailment.
• A set of sentences D logically entails a sentence
  j
• D j
• iff every interpretation that satisfies D also
  satisfies j.
• E.g.
• The sentence p logically entails the sentence (p
  Ú q).
• Since a disjunction is true whenever one of its
  disjuncts is true, then (p Ú q) must be true
  whenever p is true.
• On the other hand, the sentence p does not
  logically entail (p Ù q).
• A conjunction is true if and only if both of its
  conjuncts are true, and q may be false.

22
Semantic Reasoning Methods

• Several basic methods for determining whether a
  given set of premises propositionally entails a
  given conclusion.

23
Truth Table Method

• One way of determining whether or not a set of
  premises logically entails a possible conclusion
  is to check the truth table for the logical
  constants of the language.
• This is called the truth table method and can be
  formalized as follows.
• Starting with a complete truth table for the
  propositional constants, iterate through all the
  premises of the problem, for each premise
  eliminating any row that does not satisfy the
  premise.
• Do the same for the conclusion.
• Finally, compare the two tables. If every row
  that remains in the premise table, i.e. is not
  eliminated, also remains in the conclusion table,
  i.e. is not eliminated, then the premises
  logically entail the conclusion.

24
Amys
Simple sentences Amy loves Pat. loves(amy,
pat) Since we dont have vars but only
constants, it the same as saying
lovesAmyPat. Amy loves Quincy. loves(amy,
quincy) Same as lovesAmyQuincy It is
Monday. ismonday
Premises If Amy loves Pat, Amy loves
Quincy. lovesAmyPat Þ lovesAmyQuincy If it
Monday, Amy loves Pat or Quincy. ismonday Þ
lovesAmyPat Ú lovesAmyQuincy Question If it
is Monday, does Amy love Quincy? I.e. is
ismonday ÞlovesAmyQuincy entailed by the premises?
25
Truth table for the premises
lovesAmyPat lovesAmyQuincy ismonday lovesAmyPat Þ lovesAmyQuincy ismonday Þ lovesAmyPat Ú lovesAmyQuincy
T T T T T
T T F T T
T F T F T
T F F F T
F T T T T
F T F T T
F F T T F
F F F T T
26
Crossing out non-sat interpretations
lovesAmyPat lovesAmyQuincy ismonday lovesAmyPat Þ lovesAmyQuincy ismonday Þ lovesAmyPat Ú lovesAmyQuincy
T T T T T
T T F T T
T F T F T
T F F F T
F T T T T
F T F T T
F F T T F
F F F T T
27
Truth table for the conclusion
lovesAmyPat lovesAmyQuincy ismonday ismonday Þ lovesAmyQuincy
T T T T
T T F T
T F T F
T F F T
F T T T
F T F T
F F T F
F F F T
28
Crossing out non-sat interpretations
lovesAmyPat lovesAmyQuincy ismonday ismonday Þ lovesAmyQuincy
T T T T
T T F T
T F T F
T F F T
F T T T
F T F T
F F T F
F F F T
29
Comparing tables

• Finally, in order to make the determination of
  logical entailment, we compare the two rightmost
  tables and notice that every row remaining in the
  premise table also remains in the conclusion
  table.
• In other words, the premises logically entail the
  conclusion.
• The truth table method has the merit that it is
  easy to understand. It is a direct implementation
  of the definition of logical entailment.
• In practice, it is awkward to manage two tables,
  especially since there are simpler approaches in
  which only one table needs to be manipulated.

30
Validity checking

• One of these approaches is the validity checking
  method. It goes as follows.
• To determine whether a set of sentences
• j1,,jn
• logically entails a sentence j, we form the
  sentence
• (j1 Ù Ù jn Þ j)
• and check that it is valid.
• To see how this method works, consider the
  problem of Amy and Pat and Quincy once again. In
  this case, we write the tentative conclusion as
  shown below.
• (lovesAmyPat Þ lovesAmyQuincy) Ù (ismonday Þ
  lovesAmyPat Ú lovesAmyQuincy) Þ (ismonday Þ
  lovesAmyQuincy)
• We then form a truth table for our language with
  an added column for this sentence and check its
  satisfaction under each of the possible
  interpretations for our logical constants.

31
Unsatisfability Checking

• The satisfiability checking method is another
  single table approach. It is almost exactly the
  same as the validity checking method, except that
  it works negatively instead of positively.
• To determine whether a finite set of sentences
  j1,,jn logically entails a sentence j, we form
  the sentence
• (j1 Ù Ù jn Ù Øj)
• and check that it is unsatisfiable.
• Both the validity checking method and the
  satisfiability checking method require about the
  same amount of work as the truth table method,
  but they have the merit of manipulating only one
  table.

32
A truth table
p q r p Þ q p Þ r p Þ r ? q (p Þ q) ? (p Þ r) Þ (p Þ r ? q) p Þ r ? q (p Þ q) ? (p Þ r) Þ (p Þ r ? q)
T T T T T T T T T
T T F T F T T F T
T F T F T T T F T
T F F F F F T F T
F T T T T T T T T
F T F T T T T T T
F F T T T T T T T
F F F T T T T T T
33
Proofs
34
Intro

• Semantic methods for checking logical entailment
  have the merit of being conceptually simple they
  directly manipulate interpretations of sentences.
• Unfortunately, the number of interpretations of a
  language grows exponentially with the number of
  logical constants.
• When the number of logical constants in a
  propositional language is large, the number of
  interpretations may be impossible to manipulate.
• Proof methods provide an alternative way of
  checking and communicating logical entailment
  that addresses this problem.
• In many cases, it is possible to create a proof
  of a conclusion from a set of premises that is
  much smaller than the truth table for the
  language
• moreover, it is often possible to find such
  proofs with less work than is necessary to check
  the entire truth table.

35
Schemata

• An important component in the treatment of proofs
  is the notion of a schema. A schema is an
  expression satisfying the grammatical rules of
  our language except for the occurrence of
  metavariables in place of various subparts of the
  expression.
• For example, the following expression is a
  pattern with metavariables j and y.
• j Þ (y Þj)
• An instance of a sentence schema is the
  expression obtained by substituting expressions
  for the metavariables.
• For example, the following is an instance of the
  preceding schema.
• p Þ(qÞ p)

36
Rules of Inference

• The basis for proof methods is the use of correct
  rules of inference that can be applied directly
  to sentences to derive conclusions that are
  guaranteed to be correct under all
  interpretations.
• Since the interpretations are not enumerated,
  time and space can often be saved.
• A rule of inference is a pattern of reasoning
  consisting of
• one set of sentence schemata, called premises,
  and
• a second set of sentence schemata, called
  conclusions.
• A rule of inference is sound if and only if, for
  every instance, the premises logically entail the
  conclusions.

37
E.g. Modus Ponens

• j Þy
• j
• y
• rainingÞ wet
• raining
• wet
• wet Þ slippery
• wet
• slippery

• p Þ(qÞ r)
• p
• q Þ r
• (p Þ q)Þ r
• p Þ q
• r
• I.e. we can substitute for the metavariables
  complex sentences
• Note that, by stringing together applications of
  rules of inference, it is possible to derive
  conclusions that cannot be derived in a single
  step. This idea of stringing together rule
  applications leads to the notion of a proof.

38
Axiom schemata

• The implication introduction schema (II),
  together with Modus Ponens, allows us to infer
  implications.
• j Þ (y Þj)
• The implication distribution schema (ID) allows
  us to distribute one implication over another.
• (j Þ (y Þ c)) Þ((jÞy) Þ(jÞ c))
• The contradiction realization schemata (CR)
  permit us to infer a sentence if the negation of
  that sentence implies some sentence and its
  negation.
• (y Þ Øj) Þ((y Þj) ÞØy)
• (Øy ÞØj) Þ((Øy Þj)Þy)
• The equivalence schemata (EQ) capture the meaning
  of the Û operator.
• (j Ûy)Þ (j Þy)
• (j Ûy)Þ (y Þj)
• (j Þy)Þ ((y Þj)Þ (y Ûj))
• The meaning of the other operators in
  propositional logic is captured in the following
  axiom schemata.
• (j Üy)Û(y Þj)
• (j Úy )Û(Øj Þy)
• (j Ùy)Û Ø(Øj Ú Øy)
• The axiom schemata in this section are jointly
  called the standard axiom schemata for
  Propositional Logic. They all are valid.

39
Proofs

• A proof of a conclusion from a set of premises is
  a sequence of sentences terminating in the
  conclusion in which each item is either (1) a
  premise, (2) an instance of an axiom schema, or
  (3) the result of applying a rule of inference to
  earlier items in sequence.
• 1. p Þ q Premise
• 2. q Þ r Premise
• 3. (q Þ r)Þ (p Þ(q Þ r)) II
• 4. p Þ(qÞ r) MP 3,2
• 5. (p Þ(qÞ r)) Þ(( pÞ q) Þ(p Þ r)) ID
• 6. (p Þ q)Þ (p Þr ) MP 5,4
• 7. p Þ r MP 6,1

40
Proofs (continued)

• If there exists a proof of a sentence j from a
  set D of premises and the standard axiom schemata
  using Modus Ponens, then j is said to be provable
  from D (written as D - j) and is called a
  theorem of D.
• Earlier, it was suggested that there is a close
  connection between provability and logical
  entailment. In fact, they are equivalent. A set
  of sentences D logically entails a sentence j if
  and only if j is provable from D.
• Soundness Theorem If j is provable from D, then
  D logically entails j.
• Completeness Theorem If D logically entails j,
  then j is provable from D.
• The concept of provability is important because
  it suggests how we can automate the determination
  of logical entailment.
• Starting from a set of premises D, we enumerate
  conclusions from this set.
• If a sentence j appears, then it is provable from
  D and is, therefore, a logical consequence.
• If the negation of j appears, then Øj is a
  logical consequence of D and j is not logically
  entailed (unless D is inconsistent).
• Note that it is possible that neither j nor Øj
  will appear.

41
Resolution
42
Intro

• Propositional resolution is an extremely powerful
  rule of inference for Propositional Logic. Using
  propositional resolution (without axiom schemata
  or other rules of inference), it is possible to
  build a theorem prover that is sound and complete
  for all of Propositional Logic.
• What's more, the search space using propositional
  resolution is much smaller than for standard
  propositional logic.

43
Clausal Form

• Propositional resolution works only on
  expressions in clausal form. Before the rule can
  be applied, the premises and conclusions must be
  converted to this form.
• A literal is either an atomic sentence or a
  negation of an atomic sentence. For example, if p
  is a logical constant, the following sentences
  are both literals p, Øp
• A clause expression is either a literal or a
  disjunction of literals. If p and q are logical
  constants, then the following are clause
  expressions
• p, Øp, p Úq
• A clause is the set of literals in a clause
  expression. For example, the following sets are
  the corresponding clauses p, Øp, p,q
• Note that the empty set is also a clause. It
  is equivalent to an empty disjunction and,
  therefore, is unsatisfiable.

44
Converting to clausal form

• 1. Implications (I)
• j1 Þ j2 Øj1 Ú j2
• j1 Ü j2 j1 Ú Øj2
• j1 Ûj2 (Øj1 Ú j2 ) Ù(j1 Ú Øj2 )
• 2. Negations (N)
• ØØj j
• Ø(j1 Ù j2 ) Øj1 ÚØj2
• Ø(j1 Új2 ) Øj1 ÙØj2
• 3. Distribution (D)
• j1 Ú (j2 Ù j3 ) (j1 Ú j2) Ù(j1 Ú j3 )
• (j1 Ùj2 ) Ú j3 (j1 Ú j3 ) Ù (j2 Ú j3)
• (j1 Új2 ) Ú j3 j1 Ú (j2 Ú j3 )
• (j1 Ùj2 ) Ù j3 j1 Ù (j2 Ù j3)
• 4. Operators (O)
• j1 Ú... Újn j1,...,jn
• j1 Ù ...Ùjn j1...jn

45
Examples

• g Ù (r Þ f )
• I g Ù (Ør Ú f )
• N g Ù (Ør Ú f )
• D g Ù (Ør Ú f )
• O g
• Ør, f

Ø(g Ù (rÞ f )) I Ø(g Ù (Ør Ú f )) N Øg ÚØ(Ør Ú
f )) Øg Ú(ØØr Ù Øf ) Øg Ú(r ÙØf ) D (Øg Úr)
Ù(Øg Ú Øf ) O Øg,r Øg,Øf
46
Propositional Resolution

• j1,..., c,...,jm
• y1,...,Øc,...,yn
• j1,...,jm,y1,...,yn
• E.g.
• p,q
• Øp,r
• q,r

• The idea of resolution is simple.
• Suppose we know that p is true or q is true, and
  suppose we also know that p is false or r is
  true.
• One clause contains p, and the other contains Øp.
  
• If p is false, then by the first clause q must be
  true.
• If p is true, then, by the second clause, r must
  be true.
• Since p must be either true or false, then it
  must be the case that q is true or r is true.
• In other words, we can cancel the p literals.

47
Example

• 1. p,q Premise
• 2. Øp, q Premise
• 3. p,Øq Premise
• 4. Øp,Øq Premise
• 5. q 1,2
• 6. Øq 3,4
• 7. 5,6

48
Two finger (or pointer) method

• We keep two pointers (the slow and the fast)
• On each step we compare the sentences under the
  pointers. If we can resolve, we add the new
  derived sentence at the end of the list.
• At the start of the inference we initialize slow
  and fast at the top of the list.
• As long as the two pointers point to different
  positions, we leave the slow where it is and
  advance the fast.
• When they meet, we move the fast at the top of
  the list and we move the slow one position down
  the list.

49
Example (two finger method)

• 1. p,q Premise
• 2. Øp,r Premise
• 3. Øq, r Premise
• 4. Ør Premise
• 5. q,r 1,2
• 6. p,r 1,3
• 7. Øp 2,4
• 8. Øq 3,4

9. r 3,5 10. q 4,5 11. r 2,6 12. p
4,6 13. q 1,7 14. r 6,7 15. p 1,8 16.
r 5,8 17. 4,9
50
TFM With Identical Clause Elimination

• 1. p,q Premise
• 2. Øp,r Premise
• 3. Øq, r Premise
• 4. Ør Premise
• 5. q,r 1,2
• 6. p,r 1,3
• 7. Øp 2,4
• 8. Øq 3,4
• 9. r 3,5
• 10. q 4,5
• 11. p 4,6
• 12. 4,9

51
Another TFM example

• 1. p,q
• 2. p,Øq
• 3. Øp,q
• 4. Øp,Øq
• 5. p 1,2
• 6. q 1,3
• 7. Øq, q 2,3
• 8. p,Øp 2,3
• 9. q,Øq 1,4
• 9.5 p,Øp 1,4
• 10. Øq 2,4

11. Øp 3, 4 12. q 3,5 13. Øq 4,5 14. p
2,6 15. Øp 4,6 16. p,q 1,7 17. Øq, p
2,7 18. Øp,q 3,7 19. Øq,Øp 4,7 20. q 6,7
52
continued

• 21. Øq, q 7, 7
• 22. Øq, q 7, 7
• 23. q, p 1,8
• 24. Øq, p 2,8
• 25. Øp,q 3,8
• 26. Øp,Øq 4,8
• 27. p 5,8
• 28. Øp, p 8,8
• 29. Øp, p 8,8
• 30. p,q 1,9

31. Øq, p 2,9 32. Øp, q 3,9 33. Øq,Øp 4,
9 34. q 6,9 35. Øq,q 7,9 36. q,Øq
9,9 37. q,Øq 9,9 38. p 1,10 39. Øp
3,10 40. 6,10
53
Proof With Identical Clause Elimination

• 1. p,q
• 2. p,Øq
• 3. Øp,q
• 4. Øp,Øq
• 5. p 1,2
• 6. q 1,3
• 7. Øq, q 2,3
• 8. p,Øp 2,3
• 9. q,Øq 1,4
• 10. Øq 2,4
• 11. Øp 3,4
• 12. 6,10

54
Motivation for Tautology Elimination

• A tautology is a clause with a complementary pair
  of literals.
• 1. p,q Premise
• 2. p,Øp Premise
• 3. p,q 1,2

55
Proof with TE and ICE

• 1. p, q
• 2. p,Øq
• 3. Øp, q
• 4. Øp,Øq
• 5. p 1,2
• 6. q 1,3
• 7. Øq 2, 4
• 8. Øp 3,4
• 9. 6,7