Rules of Inference and Proof Methods in Propositional Logic, Study notes of Mathematics

Download Rules of Inference and Proof Methods in Propositional Logic and more Study notes Mathematics in PDF only on Docsity!

Inference Rules and Proof Methods

Lucia Moura

Winter 2010

Introduction

Rules of Inference and Formal Proofs

Proofs in mathematics are valid arguments that establish the truth of

mathematical statements.

An argument is a sequence of statements that end with a conclusion.

The argument is valid if the conclusion (final statement) follows from

the truth of the preceding statements (premises).

Rules of inference are templates for building valid arguments.

We will study rules of inferences for compound propositions, for quantified

statements, and then see how to combine them.

These will be the main ingredients needed in formal proofs.

Rules of Inference for Propositional Logic

Valid Arguments using Propositional Logic

Consider the following argument (sequence of propositions):

If the prof offers chocolate for an answer, you answer the prof’s question.

The prof offers chocolate for an answer.

Therefore, you answer the prof’s question.

Let p be “the prof offers chocolate for an answer”

and q be “you answer the prof’s question”.

The form of the above argument is:

p → q

p ∴ q

The argument is valid since ((p → q) ∧ p) → q is a tautology.

Rules of Inference for Propositional Logic

Arguments, argument forms and their validity

Definition

An argument in propositional logic is sequence of propositions. All but the

final proposition are called premises and the final proposition is called the conclusion. An argument is valid if the truth of all its premises implies

that the conclusion is true. An argument form in propositional logic is a sequence of compound

propositions involving propositional variables. An argument form is valid if no matter which propositions are substituted for the propositional variables

in its premises, if the premises are all true, then the conclusion is true.

In other words, an argument form with premises p 1 , p 2 ,... , pn and

conclusion q is valid if and only if

(p 1 ∧ p 2 ∧ · · · ∧ pn) → q

is a tautology.

Rules of Inference for Propositional Logic

Rules of Inference for Propositional Logic II

p

∴ p ∨ q

p → (p ∨ q) addition

p ∧ q ∴ p

(p ∧ q) → p simplification

p

q ∴ p ∧ q

((p) ∧ (q)) → (p ∧ q) conjunction

p ∨ q

¬p ∨ r ∴ q ∨ r

((p ∨ q) ∧ (¬p ∨ r)) → (q ∨ r) resolution

Rules of Inference for Propositional Logic

Which rule of inference is used in each argument below?

Alice is a Math major. Therefore, Alice is either a Math major or a CSI major.

Jerry is a Math major and a CSI major. Therefore, Jerry is a Math major.

If it is rainy, then the pool will be closed. It is rainy. Therefore, the pool is closed.

If it snows today, the university will close. The university is not closed today. Therefore, it did not snow today.

If I go swimming, then I will stay in the sun too long. If I stay in the sun too long, then I will sunburn. Therefore, if I go swimming, then I

will sunburn.

l go swimming or eat an ice cream. I did not go swimming.

Therefore, I eat an ice cream.

Rules of Inference for Propositional Logic

Determine whether the argument is valid and whether the

conclusion must be true

If

2 > 32 then (

2)^2 > ( 32 )^2. We know that

2 > 32. Therefore,

(

2)^2 = 2 > ( 32 )^2 = 94.

Is the argument valid?

Rules of Inference for Propositional Logic

Determine whether the argument is valid and whether the

conclusion must be true

If

2 > 32 then (

2)^2 > ( 32 )^2. We know that

2 > 32. Therefore,

(

2)^2 = 2 > ( 32 )^2 = 94.

Is the argument valid?

Does the conclusion must be true?

Rules of Inference for Propositional Logic

Determine whether the argument is valid and whether the

conclusion must be true

If

2 > 32 then (

2)^2 > ( 32 )^2. We know that

2 > 32. Therefore,

(

2)^2 = 2 > ( 32 )^2 = 94.

Is the argument valid?

Does the conclusion must be true?

What is wrong?

The argument is valid: modus ponens inference rule.

Rules of Inference for Propositional Logic

Determine whether the argument is valid and whether the

conclusion must be true

If

2 > 32 then (

2)^2 > ( 32 )^2. We know that

2 > 32. Therefore,

(

2)^2 = 2 > ( 32 )^2 = 94.

Is the argument valid?

Does the conclusion must be true?

What is wrong?

The argument is valid: modus ponens inference rule.

We cannot conclude that the conclusion is true, since one of its premises,

2 > 32 , is false.

Rules of Inference for Propositional Logic

Formal Proofs: using rules of inference to build arguments

Definition

A formal proof of a conclusion q given hypotheses p 1 , p 2 ,... , pn is a sequence of steps, each of which applies some inference rule to hypotheses

or previously proven statements (antecedents) to yield a new true statement (the consequent).

A formal proof demonstrates that if the premises are true, then the conclusion is true.

Note that the word formal here is not a synomym of rigorous.

A formal proof is based simply on symbol manipulation (no need of thinking, just apply rules).

A formal proof is rigorous but so can be a proof that does not rely on symbols!

Rules of Inference for Propositional Logic

Formal proof example

Show that the 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.

lead to the conclusion:

We will be home by the sunset.

Main steps:

(^1) Translate the statements into proposional logic.

(^2) Write a formal proof, a sequence of steps that state hypotheses or

apply inference rules to previous steps.

Rules of Inference for Propositional Logic

Resolution and Automated Theorem Proving

We can build programs that automate the task of reasoning and proving

theorems.

Recall that the rule of inference called resolution is based on the

tautology: ((p ∨ q) ∧ (¬p ∨ r)) → (q ∨ r)

If we express the hypotheses and the conclusion as clauses (possible by

CNF, a conjunction of clauses), we can use resolution as the only inference rule to build proofs!

This is used in programming languages like Prolog. It can be used in automated theorem proving systems.

Rules of Inference for Propositional Logic

Proofs that use exclusively resolution as inference rule

Step 1: Convert hypotheses and conclusion into clauses:

Original hypothesis equivalent CNF Hypothesis as list of clauses

(p ∧ q) ∨ r (p ∨ r) ∧ (q ∨ r) (p ∨ r), (q ∨ r) r → s (¬r ∨ s) (¬r ∨ s)

Conclusion equivalent CNF Conclusion as list of clauses

p ∨ s (p ∨ s) (p ∨ s)

Step 2: Write a proof based on resolution:

Step Reason

1. p ∨ r hypothesis
2. ¬r ∨ s hypothesis
3. p ∨ s resolution of 1 and 2