#3 formal methods – propositional logic

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Prepared by: Sharif Omar Salem – [email protected]

Formal Methods :Propositional Logic

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Prepositional logic is a formal system of representing knowledge

Prepositional logic has:Syntax – what the allowable expressions are. Structure of the

sentence.Semantics – what the expressions mean. MeaningProof theory – how conclusions are drawn from a set of

statements. Reasoning.

Introduction

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Symbols represent factsE.g. “Penguins need a cold environment” is a factThat could be represented by the symbol P

Each fact is called an atomic formulas or atoms

Atomic propositions can be combined using logical connectives –

Order of precedence: ￢ ∧ ∨ → ↔

Propositional Logic - Syntax

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These symbols P, Q,………. etc used to represent propositions, are called atomic formulas, or simply atoms.

To express more complex propositions such as the following compound proposition, we use logical connectives such as → (if-then or imply):“if car brake pedal is pressed, then car stops within five

seconds.”This compound proposition is expressed in propositional

logic as: P → Q

Atomic formula (Atoms)

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We can combine propositions and logical connectives to form complicated formulas.

Well-Formed Formulas: Well-formed formulas in propositional logic are defined recursively as follows:1. An atom is a formula.2. If F is a formula, then (¬F) is a formula, where ¬ is the not operator. 3. If F and G are formulas, then (F G), (F G), (F → G),and (F ↔G) are ∧ ∨

formulas. ( is the and operator, is the or operator , ↔ stands for if and ∧ ∨only if or iff.)

4. All formulas are generated using the above rules.

Well-Formed formulas (WFF)

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Logic is made up sentencesP might be a sentenceP Q is also a sentence∧

If we know the truth values of P and Q, we can work out the truth value of the sentence.

If P and Q are both true then P Q is true, otherwise it is false∧Can use truth tables to ascertain the truth of a sentence

Semantics/ Interpretation

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An interpretation of a propositional formula G is an assignment of truth values to the atoms A1,... , An in G in which every Ai is assigned either T or F, but not both.

The Figure shows the truth table for several simple formulas.

Semantics/ Interpretation

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A literal is an atomic formula or the negation of an atomic formula. A clause is a wff express a fact (premises or conclusion).A clause set is a group of clause express an argument.A formula is in conjunctive normal form (CNF) if it is a conjunction

of disjunction of literals.

A formula is in disjunctive normal form (DNF) if it is a disjunction of conjunction of literals.

Some other definitions

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P - represents the fact “Penguins eat fish”Q - represents the fact “Penguins like fish”

P Q – Penguins eat fish ∧ and penguins like fishP Q – Penguins eat fish ∨ or penguins like fish￢ Q – Penguins do not like fishP → Q

Penguins eat fish therefore penguins like fish.If penguins eat fish then penguins like fish.

P ↔ Q Penguins eat fish therefore penguins like fish

and penguins like fish therefore penguins eat fish.

Propositional Logic – example

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If the train arrives late and there is no taxi at the station, then john is late for this meeting. John is not late for his meeting. The train did arrive late. Therefore, there were taxis at the station.

Propositional Logic – Arguments

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Definition of Argument:An argument is a sequence of statements in which the conjunction of

the initial statements (called the premises/hypotheses) is said to imply the final statement (called the conclusion).

An argument can be presented symbolically as

(P1 Λ P2 Λ ... Λ Pn) Qwhere P1, P2, ..., Pn represent the hypotheses and Q represents the conclusion.

Deriving a logical conclusion by combining many propositions and using formal logic: hence, determining the truth of arguments.

This formula representing the whole argument as hypothesis and conclusion is known as NATURAL DEDUCTION


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What is a valid argument? An argument is valid if Q (conclusion) logically follow from P1, P2, ..., Pn

(hypotheses)Informal answer: Whenever the truth of hypotheses leads to the conclusion

A formula is valid iff it is true under all its interpretations. (Called Tautology) A formula is invalid iff it is not valid.

A valid argument is intrinsically true, i.e. (P1 Λ P2 Λ ... Λ Pn) Q is a tautology.

Note: We need to focus on the relationship of the conclusion to the hypotheses and not just any knowledge we might have about the conclusion Q.

Propositional Logic –Valid Arguments

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Example:P1: Neil Armstrong was the first human to step on the moon.P2 : Mars is a red planet

And the conclusionQ: No human has ever been to Mars.This wff P1 Λ P2 Q is not a tautology ( Not True)

Truth of Hypothesis doesn’t lead to the conclusion. Mean the argument is not valid.


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Example:P1: Tokyo is located in Japan.P2 : Japan is not located in Europe.

And the conclusionQ: Tokyo is not located in EuropeThis wff P1 Λ P2 Q is a tautology ( Always True)

Truth of Hypothesis leads to the conclusion. Mean the argument is valid.


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Russia was a superior power, and either France was not

strong or Napoleon made an error. Napoleon did not make

an error, but if the army did not fail, then France was strong.

Hence the army failed and Russia was a superior power.

Converting it to a propositional form using letters A, B, C and D

A: Russia was a superior powerB: France was strong B: France was not strongC: Napoleon made an error C: Napoleon did not make an

errorD: The army failed D: The army did not fail

Translating Verbal Arguments

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A: Russia was a superior powerB: France was strong B: France was not strongC: Napoleon made an error C: Napoleon did not make an errorD: The army failed D: The army did not fail

Combining, the statements using logic(A Λ (B V C)) hypothesisC hypothesis(D B) hypothesis(D Λ A) conclusion

Combining them, the propositional form is (A Λ (B V C)) Λ C Λ (D B) (D Λ A)


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Example:

Real Madrid is a superior power team, and either FC Barcelona is not

strong or Juardiola make an error. Juardiola does not make an error, but

if FC Barcelona wins the game, then FC Barcelona is strong. Hence FC

Barcelona loses the game and Real Madrid is a superior power team.

Converting atomic prepositions to propositional symbolsConverting it to a propositional form using letters W, X, Y and Z

W: Real Madrid is a superior power team.X: FC Barcelona is strong X: FC Barcelona is not strongY: Juardiola make an error Y: Juardiola do not make an errorZ: FC Barcelona loses the game Z: FC Barcelona wins the game


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W: Real Madrid is a superior power team.X: FC Barcelona is strong X: FC Barcelona is not strongY: Juardiola make an error Y: Juardiola do not make an errorZ: FC Barcelona loses the game Z: FC Barcelona wins the game

Convert verbal argument to propositional logic (hypothesis, conclusion and form)(W Λ (X V Y)) hypothesisY hypothesis(Z X) hypothesis (Z Λ W) conclusion

Argument form is (W Λ (X V Y)) Λ Y Λ (Z X) (Z Λ W)


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Example:

If the program is efficient, it executes quickly. Either the program is

efficient, or it has a bug. However, the program does not execute

quickly. Therefore it has a bug.

Converting Key statements to propositional symbolsE: The program is efficient.Q: The program executes quickly Q: The program does not execute quicklyB: The program has a bug


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Convert verbal argument to propositional logic (hypothesis, conclusion and form)E Q hypothesisE B ˅ hypothesisQ’ hypothesis B conclusion

Argument form is(E Q) (E B) Q’ ˄ ˅ ˄ B


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If the room temperature is hot, then the air conditioner is on. If

the room temperature is cold, then the heater is on. If the

room temperature is neither hot nor cold, then the room

temperature is comfortable. Therefore, If neither the air

conditioner nor the heater is on, then the room temperature

is comfortable.

Translate the argument using propositional Logic.

Excersise

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How to prove Validity of an argument?Truth Table proof

Equivalency Laws deduction proof

Resolution Theorem proof

Propositional Logic –Proof Theory

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If the room temperature is hot, then the air conditioner is on. If the room temperature is cold, then the heater is on. If the room temperature is neither hot nor cold, then the room temperature is comfortable. Therefore, If neither the air conditioner nor the heater is on, then the room temperature is comfortable.Converting Key statements to propositional symbols

H = the room temperature is hot C = the room temperature is cold M = the room temperature is comfortable A = the air conditioner is on G = the heater is on.

Convert propositional logic (hypothesis/conclusion/form) Hypothesis1: F1= H A Hypothesis2: F2= C G Hypothesis3: F3= ¬(H ˅ C) M Conclusion: F4= ¬(A ˅ G) M

Truth Table proof

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Argument Formula: F1 F2˄ F3˄ F4 (H A) (˄ C G) [˄ ¬(H ˅ C) M] [¬(A ˅ G) M]

Truth Table proof

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To prove this proposition with the truth-table technique. You have

to exhaustively checks every interpretation of the formula F4 to

determine if it evaluates to T. The truth table shows that every

interpretation of F4 evaluates to T, thus F4 is valid.

Truth Table proof

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Definition of Proof Sequence:It is a sequence of wffs in which each wff is either a hypothesis or

the result of applying one of the formal system’s derivation rules to earlier wffs in the sequence.

Derivation rules for propositional logic areEquivalence Rules.Inference Rules.Deduction Method.

Equivalency Laws-Deduction proof

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Tautological propositiona tautology is a statement that can never be falseall of the lines of the truth table have the result "true"

Contradictory propositiona contradiction is a statement that can never be trueall of the lines of the truth table have the result "false"

Logical equivalence of two propositionstwo statements are logically equivalent if they will be true in exactly the

same cases and false in exactly the same casesall of the lines of one column of the truth table have all of the same

truth values as the corresponding lines from another column of the truth table

it's indicated using or ↔

Special results in the truth table

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These rules state that certain pairs of wffs are equivalent, hence one can be substituted for the other with no change to truth values.

The set of equivalence rules are summarized here:

Equivalence Rules

Expression Equivalent to Abbreviation for ruleR V SR Λ S

S V RS Λ R

Commutative (comm)

(R V S) V Q (R Λ S) Λ Q

R Λ (S Λ Q)R V (S V Q)

Associative (ass)

(R V S) (R Λ S)

R Λ SR V S

De-Morgan’s Laws(De-Morgan)

R S R V S implication (imp)R (R) Double Negation (dn)

PQ (P Q) Λ (Q P) Equivalence (equ)Q P P Q Contraposition- cont

P P Λ P Self-reference - selfP V P P Self-reference - self

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Example for using Equivalence rule in a proof sequence:Simplify (A V B) V C to an argument.

The result must be an argument in the form ofP1 ^ P2^………..Pn Q

Equivalence Rules

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Example for using Equivalence rule in a proof sequence:Simplify (A V B) V C to an argument.

The result must be an argument in the form ofP1 ^ P2^………..Pn Q

Equivalence Rules

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Inference rules allow us to add a wff to the last part of the proof sequence, if one or more wffs that match the first part already exist in the proof sequence. ( Works in one direction , unlike equivalence rules)

Inference Rules

From Can Derive Abbreviation for ruleR, R S S Modus Ponens- mpR S, S R Modus Tollens- mt

R, S R Λ S Conjunction-conR Λ S R, S Simplification- simR R V S Addition- add

P Q, Q R P R Hypothetical syllogism- hs

P V Q, P Q Disjunctive syllogism- ds

(P Λ Q) R P (Q R) Exportation - expP, P Q Inconsistency - inc

P Λ (Q V R) (P Λ Q) V (P Λ R) Distributive - distP V (Q Λ R) (P V Q) Λ (P V R) Distributive - dist

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To prove an argument of the form

P1 Λ P2 Λ ... Λ Pn R Q

Deduction method allows for the use of R as an additional hypothesis and thus prove

P1 Λ P2 Λ ... Λ Pn Λ R Q

Deduction Method

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Example : Prove (A B) Λ (B C) (A C)

Deduction Method

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Prove that (P Q) (Q P) is a valid argument (called Contraposition – con).

Proofs of inference rules

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Prove the argumentA Λ (B C) Λ [(A Λ B) (D V C)] Λ B D

Proofs using Propositional Logic

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Prove the argumentA Λ (B C) Λ [(A Λ B) (D V C)] Λ B D

First, write down all the hypotheses.1. A2. B C3. (A Λ B) (D V C)4. B


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Prove the argumentA Λ (B C) Λ [(A Λ B) (D V C)] Λ B DFirst, write down all the hypotheses.1. A2. B C3. (A Λ B) (D V C)4. BUse the inference and equivalence rules to get at the conclusion D.5. C 2,4, mp6. A Λ B 1,4, con7. D V C 3,6, mp8. C V D 7, comm9. C D 8, imp

and finally10. D 5,9 impThe idea is to keep focused on the result and sometimes it is very easy to go down

a longer path than necessary.


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Russia was a superior power, and either France was not strong or Napoleon made an error. Napoleon did not make an error, but if the army did not fail, then France was strong. Hence the army failed and Russia was a superior power.

Q: Prove the upper argument ? From previous slides we translate this argument to the following

argument formula

(A Λ (B V C)) Λ C Λ (D B) (D Λ A)

Now we have to proof this propositional formula using proof sequence.

Proving Verbal Arguments

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Prove (A Λ (B V C)) Λ C Λ (D B) (D Λ A)

Proof sequence1. A Λ (B V C) hyp2. C hyp3. D B hyp4. A 1, sim5. B V C 1, sim6. C V B 5, comm7. B 2, 6, ds8. B (D) 3, cont9. (D) 7, 8, mp10. D 9, dn11. D Λ A 4, 10 , con

Verbal Argument Proof

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Example: Real Madrid is a superior power team, and either FC Barcelona is not strong

or Juardiola make an error. Juardiola does not make an error, but if FC Barcelona wins the game, then FC Barcelona is strong. Hence FC Barcelona loses the game and Real Madrid is a superior power team.


argument formula

(W Λ (X V Y)) Λ Y Λ (Z X) (Z Λ W)



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Prove the propositional form using the prove sequence rules (equivalence, inference and deduction)


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Example: If the program is efficient, it executes quickly. Either the program is

efficient, or it has a bug. However, the program does not execute quickly. Therefore it has a bug.


argument formula

(E Q) (E B) Q’ ˄ ˅ ˄ B



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(E Q) (E B) Q’ ˄ ˅ ˄ BProve the propositional form using the prove sequence rules (equivalence, inference and deduction)


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For Proving Verbal Arguments, you need to pass three stepsStep 1: Converting atomic prepositions to propositional

symbols

Step 2: Convert verbal argument to propositional logic (hypothesis, conclusion and form)

Step 3: Prove the propositional form using the prove sequence rules (equivalence, inference and deduction)

Recap

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End of Lecture

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Next Lecture:Predicate Logic

#3 formal methods – propositional logic

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