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Lecture 05 Friday, January 18, 2013 Chittu Tripathy
Rules of Inference
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Lecture 05 Friday, January 18, 2013 Chittu Tripathy
Today’s Menu
• Quantifiers: Universal and Existential
• Nesting of Quantifiers
• Applications
Rules of Inference
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Lecture 05 Friday, January 18, 2013 Chittu Tripathy
• Suppose we have:
“All human beings are mortal.”
“Sachin is a human being.”
• Does it follow that “Sachin is mortal?”
Our Old Example:
Solution:
• Let H(x): “x is a human being.”
• Let M(x): “x is mortal.”
• The domain of discourse U is all human beings.
• “All human beings are mortal.” translates to x (H(x) M(x))
“Sachin is a human being.” translates to H(Sachin)
• Therefore, for H(Sachin) M(Sachin) to be true it must be the case that M(Sachin).
Old Example Re-Revisited
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Lecture 05 Friday, January 18, 2013 Chittu Tripathy
Arguments in Propositional Logic
• A argument in propositional logic is a sequence of propositions. • All but the final proposition are called premises. The last
statement is the conclusion. • The argument is valid if the premises imply the conclusion. • An argument form is an argument that is valid no matter what
propositions are substituted into its propositional variables. • If the premises are p1 ,p2, …,pn and the conclusion is q then (p1 ∧ p2 ∧ … ∧ pn ) → q is a tautology. • Inference rules are all argument simple argument forms that will
be used to construct more complex argument forms.
Next, we will discover some useful inference rules!
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Lecture 05 Friday, January 18, 2013 Chittu Tripathy
Modus Ponens or Law of Detachment
Example: Let p be “It is snowing.” Let q be “I will study discrete math.” “If it is snowing, then I will study discrete math.” “It is snowing.” “Therefore , I will study discrete math.”
Corresponding Tautology: (p ∧ (p →q)) → q
(Modus Ponens = mode that affirms)
p p q
∴ q p q p →q
T T T
T F F
F T T
F F T
Proof using Truth Table:
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Lecture 05 Friday, January 18, 2013 Chittu Tripathy
Modus Tollens
Example: Let p be “it is snowing.” Let q be “I will study discrete math.” “If it is snowing, then I will study discrete math.” “I will not study discrete math.” “Therefore , it is not snowing.”
Corresponding Tautology: (¬q ∧(p →q))→¬p
aka Denying the Consequent
¬q p q
∴ ¬p p q p →q
T T T
T F F
F T T
F F T
Proof using Truth Table:
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Lecture 05 Friday, January 18, 2013 Chittu Tripathy
Hypothetical Syllogism aka Transitivity of Implication or Chain Argument
Example: Let p be “it snows.” Let q be “I will study discrete math.” Let r be “I will get an A.” “If it snows, then I will study discrete math.” “If I study discrete math, I will get an A.” “Therefore , If it snows, I will get an A.”
Corresponding Tautology: ((p →q) ∧ (q→r))→(p→r)
p q q r
∴ p r
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Lecture 05 Friday, January 18, 2013 Chittu Tripathy
Disjunctive Syllogism aka Disjunction Elimination or OR Elimination
Example: Let p be “I will study discrete math.” Let q be “I will study English literature.” “I will study discrete math or I will study English literature.” “I will not study discrete math.” “Therefore , I will study English literature.”
Corresponding Tautology: ((p ∨q) ∧ ¬p) → q
p ∨q ¬p
∴ q
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Lecture 05 Friday, January 18, 2013 Chittu Tripathy
Addition
Example: Let p be “I will study discrete math.” Let q be “I will visit Las Vegas.” “I will study discrete math.” “Therefore, I will study discrete math or I will visit Las Vegas.”
Corresponding Tautology: p →(p ∨q)
aka Disjunction Introduction
p
∴ (p ∨q)
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Lecture 05 Friday, January 18, 2013 Chittu Tripathy
Simplification
Example: Let p be “I will study discrete math.” Let q be “I will study English literature.” “I will study discrete math and English literature” “Therefore, I will study discrete math.”
Corresponding Tautology: (p∧q) →p
aka Conjunction Elimination
p ∧q
∴ p
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Lecture 05 Friday, January 18, 2013 Chittu Tripathy
Conjunction
Example: Let p be “I will study discrete math.” Let q be “I will study English literature.” “I will study discrete math.” “I will study English literature.” “Therefore, I will study discrete math and I will study English literature.”
Corresponding Tautology: ((p) ∧ (q)) →(p ∧ q)
aka Conjunction Introduction
p q
∴ p ∧ q
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Lecture 05 Friday, January 18, 2013 Chittu Tripathy
Resolution
Example: Let p be “I will study discrete math.” Let q be “I will study databases.” Let r be “I will study English literature.” “I will study discrete math or I will study databases.” “I will not study discrete math or I will study English literature.” “Therefore, I will study databases or I will English literature.”
Corresponding Tautology: ((p ∨ q) ∧ (¬p ∨ r )) →(q ∨ r)
Resolution plays an important role in Artificial Intelligence and is used in the programming language Prolog.
p ∨ q ¬p ∨ r
∴ q ∨ r
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Lecture 05 Friday, January 18, 2013 Chittu Tripathy
Proof by Cases p q r q p r
∴ q
aka Disjunction Elimination
Corresponding Tautology: ((p q) ∧ (r q) ∧ (p r )) q
Example: Let p be “I will study discrete math.” Let q be “I will study Computer Science.” Let r be “I will study databases.” “If I will study discrete math, then I will study Computer Science.” “If I will study databases, then I will study Computer Science.” “I will study discrete math or I will study databases.” “Therefore, I will study Computer Science.”
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Lecture 05 Friday, January 18, 2013 Chittu Tripathy
Constructive Dilemma Disjunction of modus ponens
p q r s p r
∴ q s Corresponding Tautology: ((p q) ∧ (r s) ∧ (p r )) (q s )
Example: Let p be “I will study discrete math.” Let q be “I will study computer science.” Let r be “I will study protein structures.” Let s be “I will study biochemistry.” “If I will study discrete math, then I will study computer science.” “If I will study protein structures, then I will study biochemistry.” “I will study discrete math or I will study protein structures.” “Therefore, I will study computer science or biochemistry.”
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Lecture 05 Friday, January 18, 2013 Chittu Tripathy
Destructive Dilemma Disjunction of modus tollens
p q r s
¬q ¬s
∴ ¬p ¬r
Corresponding Tautology: (p q) ∧ (r s) ∧ (¬q ¬s ) (¬p ¬r )
Example: Let p be “I will study discrete math.” Let q be “I will study computer science.” Let r be “I will study protein structures.” Let s be “I will study biochemistry.” “If I will study discrete math, then I will study computer science.” “If I will study protein structures, then I will study biochemistry.” “I will not study computer science or I will not study biochemistry.”
“Therefore, I will not study discrete math or I will not study protein structures.”
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Lecture 05 Friday, January 18, 2013 Chittu Tripathy
Absorption
q is absorbed by p in the conclusion! p q
∴ p (p ∧q) Corresponding Tautology: (p q) (p (p ∧q))
Example: Let p be “I will study discrete math.” Let q be “I will study computer science.” “If I will study discrete math, then I will study computer science.”
“Therefore, if I will study discrete math, then I will study discrete mathematics and I will study computer science.”
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Lecture 05 Friday, January 18, 2013 Chittu Tripathy
Building Valid Arguments
• A valid argument is a sequence of statements where each statement is either a premise or follows from previous statements (called premises) by rules of inference. The last statement is called conclusion.
• A valid argument takes the following form:
Premise 1
Premise 2
Conclusion
Premise n
∴
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Lecture 05 Friday, January 18, 2013 Chittu Tripathy
Valid Arguments
Example: From the single proposition Show that q is a conclusion.
Solution:
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Lecture 05 Friday, January 18, 2013 Chittu Tripathy
Valid Arguments Example: • With these hypotheses:
“It is not sunny this afternoon and it is colder than yesterday.” “We will go swimming only if it is sunny.” “If we do not go swimming, then we will take a canoe trip.” “If we take a canoe trip, then we will be home by sunset.”
• Using the inference rules, construct a valid argument for the conclusion: “We will be home by sunset.”
Solution: 1. Choose propositional variables:
p : “It is sunny this afternoon.” q : “It is colder than yesterday.” r : “We will go swimming.” s : “We will take a canoe trip.” t : “We will be home by sunset.”
2. Translation into propositional logic:
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Lecture 05 Friday, January 18, 2013 Chittu Tripathy
Valid Arguments
Remember you can also use truth table to show this albeit with 32 = 25 rows!
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Lecture 05 Friday, January 18, 2013 Chittu Tripathy
How do we use quantifiers with rules of inference?
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Lecture 05 Friday, January 18, 2013 Chittu Tripathy
Universal Instantiation (UI)
Example: Our domain consists of all students and Sachin is a student. “All students are smart” “Therefore, Sachin is smart.”
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Lecture 05 Friday, January 18, 2013 Chittu Tripathy
Universal Generalization (UG)
Used often implicitly in Mathematical Proofs.
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Lecture 05 Friday, January 18, 2013 Chittu Tripathy
Existential Instantiation (EI)
Example: “There is someone who got an A in COMPSCI 230.” “Let’s call her Amelie and say that Amelie got an A”
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Lecture 05 Friday, January 18, 2013 Chittu Tripathy
Existential Generalization (EG)
Example: “Amelie got an A in the class.” “Therefore, someone got an A in the class.”
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Lecture 05 Friday, January 18, 2013 Chittu Tripathy
• Suppose we have:
“All human beings are mortal.”
“Sachin is a human being.”
• Does it follow that “Sachin is mortal?”
Our Old Example:
Solution:
• Let H(x): “x is a human being.”
• Let M(x): “x is mortal.”
• The domain of discourse U is all human beings.
• “All human beings are mortal.” translates to x H(x) M(x)
“Sachin is a human being.” translates to H(Sachin)
Old Example Re-Revisited
To show: x (H(x) M(x)) H(Sachin)
∴ M(Sachin)
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Lecture 05 Friday, January 18, 2013 Chittu Tripathy
Old Example Re-Revisited
To show: x (H(x) M(x)) H(Sachin)
∴ M(Sachin)
Valid Argument Reason Step
x (H(x) M(x))
H(Sachin) M(Sachin) H(Sachin)
M(Sachin)
Premise
Universal instantiation from (1) Premise
Modus ponens from (2) and (3)
(1)
(2) (3)
(4)
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Lecture 05 Friday, January 18, 2013 Chittu Tripathy
Universal Modus Ponens
Universal modus ponens combines universal instantiation and modus ponens into one rule.
This is what our previous example used!
x (P(x)→ Q(x)) P(a), where a is a particular element in the domain
∴ Q(a)
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Lecture 05 Friday, January 18, 2013 Chittu Tripathy
The Lewis Carroll Example Revisited • Premises:
1. “All lions are fierce.”
2. “Some lions do not drink coffee.”
Conclusion: Can we conclude the following?
3. “Some fierce creatures do not drink coffee.”
• Let L(x): “x is a lion.” F(x): “x is fierce.” and C(x): “x drinks coffee.” Then the above three propositions can be written as:
1. x (L(x)→ F(x))
2. x (L(x) ∧ ¬C(x))
3. x (F(x) ∧ ¬C(x))
• How to conclude 3 from 1 and 2?
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Lecture 05 Friday, January 18, 2013 Chittu Tripathy
The Lewis Carroll Example Revisited 1. x (L(x)→ F(x)) 2. x (L(x) ∧ ¬C(x)) 3. x (F(x) ∧ ¬C(x))
1. x (L(x) ∧ ¬C(x)) Premise 2. L(Foo) ∧ ¬C(Foo) Existential Instantiation from (1) 3. L(Foo) Simplification from (2) 4. ¬C(Foo) Simplification from (2) 5. x (L(x)→ F(x)) Premise 6. L(Foo) → F(Foo) Universal instantiation from (5) 7. F(Foo) Modus ponens from (3) and (6) 8. F(Foo) ∧ ¬C(Foo) Conjunction from (4) and (7) 9. x (F(x) ∧ ¬C(x)) Existential generalization from (8)
How to conclude 3 from 1 and 2?
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Lecture 05 Friday, January 18, 2013 Chittu Tripathy
Example: Is Moo Carnivorous?
• Premises:
1. “If x is a lion, then x is carnivorous.”
2. “Moo is not carnivorous.”
Conclusion: Can we conclude the following?
3. “Moo is not a lion.”
• Let L(x): “x is a lion.” C(x): “x is carnivorous.”
• Then the above three propositions can be written as:
1. x (L(x)→ C(x))
2. ¬C(Moo)
3. ¬L(Moo)
• How to conclude 3 from 1 and 2?
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Lecture 05 Friday, January 18, 2013 Chittu Tripathy
Example: Is Moo Carnivorous? 1. x (L(x)→ C(x))
2. ¬C(Moo)
3. ¬L(Moo)
How to conclude 3 from 1 and 2?
1. x (L(x)→ C(x)) Premise 2. L(Moo) → C(Moo) Universal instantiation from (1) 3. ¬C(Moo) Premise 4. ¬L(Moo) Modus tollens from (1) and (2)
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Lecture 05 Friday, January 18, 2013 Chittu Tripathy
Universal Modus Tollens
Universal modus tollens combines universal instantiation and modus ponens into one rule.
This is what our previous example used!
x (P(x)→ Q(x)) ¬Q(a), where a is a particular element in the domain
∴ ¬P(a)