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Mustafa Jarrar: Lecture Notes in Discrete Mathematics.Birzeit University, Palestine, 2020

mjarrar©2020

Propositional Logic

2.1. Introduction and Basics2.2 Conditional Statements

2.3 Inferencing

Keywords:Logic, Propositional Logic, Reasoning, inference, Inference rules, Numeration method, validity of argument, Modus Ponens, Modus Tollens, Generalization, Specialization, Conjunction, Elimination, Transitivity, Division into Cases, Contradiction

جاتنتسالا دعاوق ،يقطنم جاتنتسا ،جاتنتسا ،ایاضقلا قطنم ،قطنم

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Watch this lecture and download the slides

Acknowledgement: This lecture is based on (but not limited to) to chapter 5 in “Discrete Mathematics with Applications by Susanna S. Epp (3rd Edition)”.

More Online Courses at: http://www.jarrar.infoCourse Page: http://www.jarrar.info/courses/DMath/

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2.3 Logical Inferencing

In this lecture:

qPart 1: Numeration MethodqPart 2: Rules of Inference

qPart 3: Example

Keywords:

Propositional Logic

Mustafa Jarrar: Lecture Notes in Discrete Mathematics.Birzeit University, Palestine, 2015

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p → qp

∴q?

Numeration MethodExample 1

If today is Friday then today is holiday

Today is Friday

∴today is holiday?

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Numeration MethodExample 2

p → q ∨ ∼r q→p∧r

∴ p→r ?

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Numeration Method

What is wrong with this Numeration Method?

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2.3 Logical Inferencing

Keywords:

Propositional Logic

Mustafa Jarrar: Lecture Notes in Discrete Mathematics.Birzeit University, Palestine, 2015

In this lecture:

qPart 1: Numeration Method

qPart 2: Rules of InferenceqPart 3: Example

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Rules of Inference

p→q p

If today is Friday then today is holiday

Today is Friday

∴ q ∴ Today is holiday

1. Modus Ponens:

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Rules of Inference

If today is Friday then today is holiday

Today is not holiday

∴ ∼p ∴ Today is not Friday

2. Modus Tollens:

p→q ~q

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Rules of Inference

p Today is Saturday

∴p∨q ∴ Today is Saturday or today is Sunday

3. Generalization:

q

∴p∨q

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Rules of Inference

p∧q Today is Friday and today is holiday

∴p ∴ Today is Friday

4. Specialization:

p∧q

∴q ∴ Today is Holiday

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Rules of Inference

pq

Today is Friday

Today is Holiday

∴ p∧q ∴ Today is Friday and today is holiday

5. Conjunction:

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Rules of Inference

p∨q∼p

p∨q∼q

Today is Saturday or today is Sunday

Today is not Saturday

∴ p

∴ Today is Sunday

6. Elimination:

∴ q

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Rules of Inference

p→q q→r

If today is Friday then Today is holiday

If today is holiday then I am happy

∴ p→r ∴ If today is Friday then I am happy

7. Transitivity:

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Rules of Inference

p∨q p→r q→r

Today is Friday or today is Sunday

If today is Friday then I am happy

If today is Sunday then I am happy

∴r ∴ I am happy

8. Division into Cases:

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Rules of Inference

∴ p

If “Today is not Friday” is false

9. Contradiction Rule:

∼p → c

∴ Today is Friday

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Rules of InferenceSummary

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2.3 Logical Inferencing

Keywords:

Propositional Logic

Mustafa Jarrar: Lecture Notes in Discrete Mathematics.Birzeit University, Palestine, 2015

In this lecture:

qPart 1: Numeration MethodqPart 2: Rules of Inference

qPart 3: Example

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If I was reading the newspaper in the kitchen, then my

glasses are on the kitchen table.

If my glasses are on the kitchen table, then I saw them at

breakfast.

I did not see my glasses at breakfast.

I was reading the newspaper in the living room or I was

reading the newspaper in the kitchen.

If I was reading the newspaper in the living room then

my glasses are on the coffee table.

Where are the glasses?

Inferencing ExampleFormalize the following text in propositional logic and use the inference rules find the glasses.

RK → GK

GK → SB

∼ SB

RL ∨ RK

RL → GC

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Inferencing ExampleLet RK = I was reading the newspaper in the kitchen. GK = My glasses are on the kitchen table. SB = I saw my glasses at breakfast.RL = I was reading the newspaper in the living room. GC = My glasses are on the coffee table.

RK → GKGK → SB

∴ RK → SB by transitivity

RK → SB∼ SB

∴ ∼RK by modus tollens

RL ∨ RK ∼ RK

∴ RL by elimination

RL → GCRL

∴ GC by modus ponenes

Thus the glasses are on the coffee table.

RK → GKGK → SB∼ SBRL ∨ RK RL → GC

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