Rules of Inference, Study notes of Computer Science

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Rules of Inference

Today’s Menu

• Quantifiers: Universal and Existential
• Nesting of Quantifiers
• Applications

Rules of Inference

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 p 1 ,p 2 , …, pn and the conclusion is q then (p 1 ∧ p 2 ∧ … ∧ 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!

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:

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

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

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

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

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.”

Constructive Dilemma

p  q Disjunction of modus ponens 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.”

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.”

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

∴

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:

Valid Arguments

Remember you can also use truth table to show this albeit with 32 = 2^5 rows!